diff --git a/lapack-netlib/SRC/cgedmd.c b/lapack-netlib/SRC/cgedmd.c index 447b23014..570395c7b 100644 --- a/lapack-netlib/SRC/cgedmd.c +++ b/lapack-netlib/SRC/cgedmd.c @@ -509,3 +509,1162 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; +static integer c__1 = 1; +static integer c__0 = 0; + +/* Subroutine */ int cgedmd_(char *jobs, char *jobz, char *jobr, char *jobf, + integer *whtsvd, integer *m, integer *n, complex *x, integer *ldx, + complex *y, integer *ldy, integer *nrnk, real *tol, integer *k, + complex *eigs, complex *z__, integer *ldz, real *res, complex *b, + integer *ldb, complex *w, integer *ldw, complex *s, integer *lds, + complex *zwork, integer *lzwork, real *rwork, integer *lrwork, + integer *iwork, integer *liwork, integer *info) +{ + /* System generated locals */ + integer x_dim1, x_offset, y_dim1, y_offset, z_dim1, z_offset, b_dim1, + b_offset, w_dim1, w_offset, s_dim1, s_offset, i__1, i__2, i__3, + i__4, i__5; + real r__1, r__2; + complex q__1, q__2; + + /* Local variables */ + complex zone; + real zero, ssum; + integer info1, info2; + real xscl1, xscl2; + integer i__, j; + real scale; + extern /* Subroutine */ int cgemm_(char *, char *, integer *, integer *, + integer *, complex *, complex *, integer *, complex *, integer *, + complex *, complex *, integer *), cgeev_(char *, + char *, integer *, complex *, integer *, complex *, complex *, + integer *, complex *, integer *, complex *, integer *, real *, + integer *); + extern logical lsame_(char *, char *); + logical badxy; + real small; + char jobzl[1]; + extern /* Subroutine */ int caxpy_(integer *, complex *, complex *, + integer *, complex *, integer *); + logical wntex; + complex zzero; + extern real scnrm2_(integer *, complex *, integer *); + extern /* Subroutine */ int cgesdd_(char *, integer *, integer *, complex + *, integer *, real *, complex *, integer *, complex *, integer *, + complex *, integer *, real *, integer *, integer *), + clascl_(char *, integer *, integer *, real *, real *, integer *, + integer *, complex *, integer *, integer *); + extern integer icamax_(integer *, complex *, integer *); + extern real slamch_(char *); + extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer + *), cgesvd_(char *, char *, integer *, integer *, complex *, + integer *, real *, complex *, integer *, complex *, integer *, + complex *, integer *, real *, integer *), clacpy_( + char *, integer *, integer *, complex *, integer *, complex *, + integer *), xerbla_(char *, integer *); + char t_or_n__[1]; + extern /* Subroutine */ int cgejsv_(char *, char *, char *, char *, char * + , char *, integer *, integer *, complex *, integer *, real *, + complex *, integer *, complex *, integer *, complex *, integer *, + real *, integer *, integer *, integer *), classq_(integer *, complex *, integer *, + real *, real *); + logical sccolx, sccoly; + extern logical sisnan_(real *); + integer lwrsdd, mwrsdd, iminwr; + logical wntref, wntvec; + real rootsc; + integer lwrkev, mlwork, mwrkev, numrnk, olwork, lwrsvd, mwrsvd, mlrwrk; + logical lquery, wntres; + char jsvopt[1]; + integer lwrsvj, mwrsvj; + real rdummy[2]; + extern /* Subroutine */ int mecago_(); + integer lwrsvq, mwrsvq; + real ofl, one; + extern /* Subroutine */ int cgesvdq_(char *, char *, char *, char *, char + *, integer *, integer *, complex *, integer *, real *, complex *, + integer *, complex *, integer *, integer *, integer *, integer *, + complex *, integer *, real *, integer *, integer *); + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real32 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ............................................................ */ +/* Purpose */ +/* ======= */ +/* CGEDMD computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, CGEDMD computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, CGEDMD returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* ...................................................................... */ +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ...................................................................... */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product X(:,1:K)*W, where X */ +/* contains a POD basis (leading left singular vectors */ +/* of the data matrix X) and W contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of K, X, W, Z. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will be */ +/* computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: CGESVD (the QR SVD algorithm) */ +/* 2 :: CGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: CGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M>= 0 */ +/* The state space dimension (the row dimension of X, Y). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshot pairs */ +/* (the number of columns of X and Y). */ +/* ..... */ +/* X (input/output) COMPLEX(KIND=WP) M-by-N array */ +/* > On entry, X contains the data snapshot matrix X. It is */ +/* assumed that the column norms of X are in the range of */ +/* the normalized floating point numbers. */ +/* < On exit, the leading K columns of X contain a POD basis, */ +/* i.e. the leading K left singular vectors of the input */ +/* data matrix X, U(:,1:K). All N columns of X contain all */ +/* left singular vectors of the input matrix X. */ +/* See the descriptions of K, Z and W. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= M */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (input/workspace/output) COMPLEX(KIND=WP) M-by-N array */ +/* > On entry, Y contains the data snapshot matrix Y */ +/* < On exit, */ +/* If JOBR == 'R', the leading K columns of Y contain */ +/* the residual vectors for the computed Ritz pairs. */ +/* See the description of RES. */ +/* If JOBR == 'N', Y contains the original input data, */ +/* scaled according to the value of JOBS. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= M */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the descriptions of TOL and K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the POD basis for the data snapshot */ +/* matrix X and the number of the computed Ritz pairs. */ +/* The value of K is determined according to the rule set */ +/* by the parameters NRNK and TOL. */ +/* See the descriptions of NRNK and TOL. */ +/* ..... */ +/* EIGS (output) COMPLEX(KIND=WP) N-by-1 array */ +/* The leading K (K<=N) entries of EIGS contain */ +/* the computed eigenvalues (Ritz values). */ +/* See the descriptions of K, and Z. */ +/* ..... */ +/* Z (workspace/output) COMPLEX(KIND=WP) M-by-N array */ +/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */ +/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */ +/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */ +/* the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) */ +/* is an eigenvector corresponding to EIGS(i). The columns */ +/* of W(1:k,1:K) are the computed eigenvectors of the */ +/* K-by-K Rayleigh quotient. */ +/* See the descriptions of EIGS, X and W. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) N-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs, */ +/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */ +/* See the description of EIGS and Z. */ +/* ..... */ +/* B (output) COMPLEX(KIND=WP) M-by-N array. */ +/* IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:M,1:K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* If JOBF =='N', then B is not referenced. */ +/* See the descriptions of X, W, K. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= M */ +/* The leading dimension of the array B. */ +/* ..... */ +/* W (workspace/output) COMPLEX(KIND=WP) N-by-N array */ +/* On exit, W(1:K,1:K) contains the K computed */ +/* eigenvectors of the matrix Rayleigh quotient. */ +/* The Ritz vectors (returned in Z) are the */ +/* product of X (containing a POD basis for the input */ +/* matrix X) and W. See the descriptions of K, S, X and Z. */ +/* W is also used as a workspace to temporarily store the */ +/* right singular vectors of X. */ +/* ..... */ +/* LDW (input) INTEGER, LDW >= N */ +/* The leading dimension of the array W. */ +/* ..... */ +/* S (workspace/output) COMPLEX(KIND=WP) N-by-N array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by CGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N */ +/* The leading dimension of the array S. */ +/* ..... */ +/* ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array */ +/* ZWORK is used as complex workspace in the complex SVD, as */ +/* specified by WHTSVD (1,2, 3 or 4) and for CGEEV for computing */ +/* the eigenvalues of a Rayleigh quotient. */ +/* If the call to CGEDMD is only workspace query, then */ +/* ZWORK(1) contains the minimal complex workspace length and */ +/* ZWORK(2) is the optimal complex workspace length. */ +/* Hence, the length of work is at least 2. */ +/* See the description of LZWORK. */ +/* ..... */ +/* LZWORK (input) INTEGER */ +/* The minimal length of the workspace vector ZWORK. */ +/* LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_CGEEV), */ +/* where LZWORK_CGEEV = MAX( 1, 2*N ) and the minimal */ +/* LZWORK_SVD is calculated as follows */ +/* If WHTSVD == 1 :: CGESVD :: */ +/* LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) */ +/* If WHTSVD == 2 :: CGESDD :: */ +/* LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) */ +/* If WHTSVD == 3 :: CGESVDQ :: */ +/* LZWORK_SVD = obtainable by a query */ +/* If WHTSVD == 4 :: CGEJSV :: */ +/* LZWORK_SVD = obtainable by a query */ +/* If on entry LZWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths and returns them in */ +/* LZWORK(1) and LZWORK(2), respectively. */ +/* ..... */ +/* RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array */ +/* On exit, RWORK(1:N) contains the singular values of */ +/* X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). */ +/* If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain */ +/* scaling factor RWORK(N+2)/RWORK(N+1) used to scale X */ +/* and Y to avoid overflow in the SVD of X. */ +/* This may be of interest if the scaling option is off */ +/* and as many as possible smallest eigenvalues are */ +/* desired to the highest feasible accuracy. */ +/* If the call to CGEDMD is only workspace query, then */ +/* RWORK(1) contains the minimal workspace length. */ +/* See the description of LRWORK. */ +/* ..... */ +/* LRWORK (input) INTEGER */ +/* The minimal length of the workspace vector RWORK. */ +/* LRWORK is calculated as follows: */ +/* LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_CGEEV), where */ +/* LRWORK_CGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace */ +/* for the SVD subroutine determined by the input parameter */ +/* WHTSVD. */ +/* If WHTSVD == 1 :: CGESVD :: */ +/* LRWORK_SVD = 5*MIN(M,N) */ +/* If WHTSVD == 2 :: CGESDD :: */ +/* LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), */ +/* 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) */ +/* If WHTSVD == 3 :: CGESVDQ :: */ +/* LRWORK_SVD = obtainable by a query */ +/* If WHTSVD == 4 :: CGEJSV :: */ +/* LRWORK_SVD = obtainable by a query */ +/* If on entry LRWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* real workspace length and returns it in RWORK(1). */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for ZWORK, RWORK and */ +/* IWORK. See the descriptions of ZWORK, RWORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ + +/* Local arrays */ +/* ~~~~~~~~~~~~ */ +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* ............................................................ */ + /* Parameter adjustments */ + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --eigs; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + w_dim1 = *ldw; + w_offset = 1 + w_dim1 * 1; + w -= w_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --zwork; + --rwork; + --iwork; + + /* Function Body */ + zero = 0.f; + one = 1.f; + zzero.r = 0.f, zzero.i = 0.f; + zone.r = 1.f, zone.i = 0.f; + +/* Test the input arguments */ + + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + *info = 0; + lquery = *lzwork == -1 || *liwork == -1 || *lrwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || lsame_(jobz, "N") || lsame_( + jobz, "F"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && ! wntvec) { + *info = -3; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -4; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -5; + } else if (*m < 0) { + *info = -6; + } else if (*n < 0 || *n > *m) { + *info = -7; + } else if (*ldx < *m) { + *info = -9; + } else if (*ldy < *m) { + *info = -11; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -12; + } else if (*tol < zero || *tol >= one) { + *info = -13; + } else if (*ldz < *m) { + *info = -17; + } else if ((wntref || wntex) && *ldb < *m) { + *info = -20; + } else if (*ldw < *n) { + *info = -22; + } else if (*lds < *n) { + *info = -24; + } + + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0) { +/* Quick return. All output except K is void. */ +/* INFO=1 signals the void input. */ +/* In case of a workspace query, the default */ +/* minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + rwork[1] = 1.f; + zwork[1].r = 2.f, zwork[1].i = 0.f; + zwork[2].r = 2.f, zwork[2].i = 0.f; + } else { + *k = 0; + } + *info = 1; + return 0; + } + iminwr = 1; + mlrwrk = f2cmax(1,*n); + mlwork = 2; + olwork = 2; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of CGESVD: */ +/* MWRSVD = MAX(1,2*MIN(M,N)+MAX(M,N)) */ +/* Computing MAX */ + i__1 = 1, i__2 = (f2cmin(*m,*n) << 1) + f2cmax(*m,*n); + mwrsvd = f2cmax(i__1,i__2); + mlwork = f2cmax(mlwork,mwrsvd); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = *n + f2cmin(*m,*n) * 5; + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + cgesvd_("O", "S", m, n, &x[x_offset], ldx, &rwork[1], &b[ + b_offset], ldb, &w[w_offset], ldw, &zwork[1], &c_n1, + rdummy, &info1); + lwrsvd = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrsvd); + } + } else if (*whtsvd == 2) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of CGESDD: */ +/* MWRSDD = 2*f2cmin(M,N)*f2cmin(M,N)+2*f2cmin(M,N)+f2cmax(M,N). */ +/* RWORK length: 5*MIN(M,N)*MIN(M,N)+7*MIN(M,N) */ +/* In LAPACK 3.10.1 RWORK is defined differently. */ +/* Below we take f2cmax over the two versions. */ +/* IMINWR = 8*MIN(M,N) */ + mwrsdd = (f2cmin(*m,*n) << 1) * f2cmin(*m,*n) + (f2cmin(*m,*n) << 1) + f2cmax( + *m,*n); + mlwork = f2cmax(mlwork,mwrsdd); + iminwr = f2cmin(*m,*n) << 3; +/* Computing MAX */ +/* Computing MAX */ + i__3 = f2cmin(*m,*n) * 5 * f2cmin(*m,*n) + f2cmin(*m,*n) * 7, i__4 = f2cmin(* + m,*n) * 5 * f2cmin(*m,*n) + f2cmin(*m,*n) * 5, i__3 = f2cmax(i__3, + i__4), i__4 = (f2cmax(*m,*n) << 1) * f2cmin(*m,*n) + (f2cmin(*m,*n) + << 1) * f2cmin(*m,*n) + f2cmin(*m,*n); + i__1 = mlrwrk, i__2 = *n + f2cmax(i__3,i__4); + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + cgesdd_("O", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], + ldb, &w[w_offset], ldw, &zwork[1], &c_n1, rdummy, & + iwork[1], &info1); +/* Computing MAX */ + i__1 = mwrsdd, i__2 = (integer) zwork[1].r; + lwrsdd = f2cmax(i__1,i__2); + olwork = f2cmax(olwork,lwrsdd); + } + } else if (*whtsvd == 3) { + cgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &rwork[ + 1], &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, & + iwork[1], &c_n1, &zwork[1], &c_n1, rdummy, &c_n1, &info1); + iminwr = iwork[1]; + mwrsvq = (integer) zwork[2].r; + mlwork = f2cmax(mlwork,mwrsvq); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = *n + (integer) rdummy[0]; + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + lwrsvq = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrsvq); + } + } else if (*whtsvd == 4) { + *(unsigned char *)jsvopt = 'J'; + cgejsv_("F", "U", jsvopt, "N", "N", "P", m, n, &x[x_offset], ldx, + &rwork[1], &z__[z_offset], ldz, &w[w_offset], ldw, &zwork[ + 1], &c_n1, rdummy, &c_n1, &iwork[1], &info1); + iminwr = iwork[1]; + mwrsvj = (integer) zwork[2].r; + mlwork = f2cmax(mlwork,mwrsvj); +/* Computing MAX */ +/* Computing MAX */ + i__3 = 7, i__4 = (integer) rdummy[0]; + i__1 = mlrwrk, i__2 = *n + f2cmax(i__3,i__4); + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + lwrsvj = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrsvj); + } +/* END SELECT */ + } + if (wntvec || wntex || lsame_(jobz, "F")) { + *(unsigned char *)jobzl = 'V'; + } else { + *(unsigned char *)jobzl = 'N'; + } +/* Workspace calculation to the CGEEV call */ +/* Computing MAX */ + i__1 = 1, i__2 = *n << 1; + mwrkev = f2cmax(i__1,i__2); + mlwork = f2cmax(mlwork,mwrkev); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = *n + (*n << 1); + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + cgeev_("N", jobzl, n, &s[s_offset], lds, &eigs[1], &w[w_offset], + ldw, &w[w_offset], ldw, &zwork[1], &c_n1, &rwork[1], & + info1); +/* LAPACK CALL */ + lwrkev = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrkev); + olwork = f2cmax(2,olwork); + } + + if (*liwork < iminwr && ! lquery) { + *info = -30; + } + if (*lrwork < mlrwrk && ! lquery) { + *info = -28; + } + if (*lzwork < mlwork && ! lquery) { + *info = -26; + } + } + + if (*info != 0) { + i__1 = -(*info); + xerbla_("CGEDMD", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + rwork[1] = (real) mlrwrk; + zwork[1].r = (real) mlwork, zwork[1].i = 0.f; + zwork[2].r = (real) olwork, zwork[2].i = 0.f; + return 0; + } +/* ............................................................ */ + + ofl = slamch_("O") * slamch_("P"); + small = slamch_("S"); + badxy = FALSE_; + +/* <1> Optional scaling of the snapshots (columns of X, Y) */ +/* ========================================================== */ + if (sccolx) { +/* The columns of X will be normalized. */ +/* To prevent overflows, the column norms of X are */ +/* carefully computed using CLASSQ. */ + *k = 0; + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* WORK(i) = SCNRM2( M, X(1,i), 1 ) */ + scale = zero; + classq_(m, &x[i__ * x_dim1 + 1], &c__1, &scale, &ssum); + if (sisnan_(&scale) || sisnan_(&ssum)) { + *k = 0; + *info = -8; + i__2 = -(*info); + xerbla_("CGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of X(:,i) overflows. First, X(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||X(:,i)||_2. */ +/* Next, the norm of X(:,i) is stored without */ +/* overflow as WORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of X(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + r__1 = one / rootsc; + clascl_("G", &c__0, &c__0, &scale, &r__1, m, &c__1, &x[ + i__ * x_dim1 + 1], ldx, &info2); + rwork[i__] = -scale * (rootsc / (real) (*m)); + } else { +/* X(:,i) will be scaled to unit 2-norm */ + rwork[i__] = scale * rootsc; + clascl_("G", &c__0, &c__0, &rwork[i__], &one, m, &c__1, & + x[i__ * x_dim1 + 1], ldx, &info2); +/* X(1:M,i) = (ONE/RWORK(i)) * X(1:M,i) ! INTRINSIC */ +/* LAPAC */ + } + } else { + rwork[i__] = zero; + ++(*k); + } + } + if (*k == *n) { +/* All columns of X are zero. Return error code -8. */ +/* (the 8th input variable had an illegal value) */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("CGEDMD", &i__1); + return 0; + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of Y. */ + if (rwork[i__] > zero) { + r__1 = one / rwork[i__]; + csscal_(m, &r__1, &y[i__ * y_dim1 + 1], &c__1); +/* Y(1:M,i) = (ONE/RWORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (rwork[i__] < zero) { + r__1 = -rwork[i__]; + r__2 = one / (real) (*m); + clascl_("G", &c__0, &c__0, &r__1, &r__2, m, &c__1, &y[i__ * + y_dim1 + 1], ldy, &info2); +/* LAPACK C */ + } else if (c_abs(&y[icamax_(m, &y[i__ * y_dim1 + 1], &c__1) + i__ + * y_dim1]) != zero) { +/* X(:,i) is zero vector. For consistency, */ +/* Y(:,i) should also be zero. If Y(:,i) is not */ +/* zero, then the data might be inconsistent or */ +/* corrupted. If JOBS == 'C', Y(:,i) is set to */ +/* zero and a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + if (lsame_(jobs, "C")) { + csscal_(m, &zero, &y[i__ * y_dim1 + 1], &c__1); + } +/* BLAS CALL */ + } + } + } + + if (sccoly) { +/* The columns of Y will be normalized. */ +/* To prevent overflows, the column norms of Y are */ +/* carefully computed using CLASSQ. */ + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* RWORK(i) = SCNRM2( M, Y(1,i), 1 ) */ + scale = zero; + classq_(m, &y[i__ * y_dim1 + 1], &c__1, &scale, &ssum); + if (sisnan_(&scale) || sisnan_(&ssum)) { + *k = 0; + *info = -10; + i__2 = -(*info); + xerbla_("CGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of Y(:,i) overflows. First, Y(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||Y(:,i)||_2. */ +/* Next, the norm of Y(:,i) is stored without */ +/* overflow as RWORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of Y(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + r__1 = one / rootsc; + clascl_("G", &c__0, &c__0, &scale, &r__1, m, &c__1, &y[ + i__ * y_dim1 + 1], ldy, &info2); + rwork[i__] = -scale * (rootsc / (real) (*m)); + } else { +/* Y(:,i) will be scaled to unit 2-norm */ + rwork[i__] = scale * rootsc; + clascl_("G", &c__0, &c__0, &rwork[i__], &one, m, &c__1, & + y[i__ * y_dim1 + 1], ldy, &info2); +/* Y(1:M,i) = (ONE/RWORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* LAPA */ + } + } else { + rwork[i__] = zero; + } + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of X. */ + if (rwork[i__] > zero) { + r__1 = one / rwork[i__]; + csscal_(m, &r__1, &x[i__ * x_dim1 + 1], &c__1); +/* X(1:M,i) = (ONE/RWORK(i)) * X(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (rwork[i__] < zero) { + r__1 = -rwork[i__]; + r__2 = one / (real) (*m); + clascl_("G", &c__0, &c__0, &r__1, &r__2, m, &c__1, &x[i__ * + x_dim1 + 1], ldx, &info2); +/* LAPACK */ + } else if (c_abs(&x[icamax_(m, &x[i__ * x_dim1 + 1], &c__1) + i__ + * x_dim1]) != zero) { +/* Y(:,i) is zero vector. If X(:,i) is not */ +/* zero, then a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + } + } + } + +/* <2> SVD of the data snapshot matrix X. */ +/* ===================================== */ +/* The left singular vectors are stored in the array X. */ +/* The right singular vectors are in the array W. */ +/* The array W will later on contain the eigenvectors */ +/* of a Rayleigh quotient. */ + numrnk = *n; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { + cgesvd_("O", "S", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], + ldb, &w[w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], & + info1); +/* LA */ + *(unsigned char *)t_or_n__ = 'C'; + } else if (*whtsvd == 2) { + cgesdd_("O", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], ldb, & + w[w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &iwork[1] + , &info1); +/* LAP */ + *(unsigned char *)t_or_n__ = 'C'; + } else if (*whtsvd == 3) { + i__1 = *lrwork - *n; + cgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &rwork[1], + &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, &iwork[1], + liwork, &zwork[1], lzwork, &rwork[*n + 1], &i__1, &info1); +/* LAPACK CA */ + clacpy_("A", m, &numrnk, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK C */ + *(unsigned char *)t_or_n__ = 'C'; + } else if (*whtsvd == 4) { + i__1 = *lrwork - *n; + cgejsv_("F", "U", jsvopt, "N", "N", "P", m, n, &x[x_offset], ldx, & + rwork[1], &z__[z_offset], ldz, &w[w_offset], ldw, &zwork[1], + lzwork, &rwork[*n + 1], &i__1, &iwork[1], &info1); + clacpy_("A", m, n, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK CALL */ + *(unsigned char *)t_or_n__ = 'N'; + xscl1 = rwork[*n + 1]; + xscl2 = rwork[*n + 2]; + if (xscl1 != xscl2) { +/* This is an exceptional situation. If the */ +/* data matrices are not scaled and the */ +/* largest singular value of X overflows. */ +/* In that case CGEJSV can return the SVD */ +/* in scaled form. The scaling factor can be used */ +/* to rescale the data (X and Y). */ + clascl_("G", &c__0, &c__0, &xscl1, &xscl2, m, n, &y[y_offset], + ldy, &info2); + } +/* END SELECT */ + } + + if (info1 > 0) { +/* The SVD selected subroutine did not converge. */ +/* Return with an error code. */ + *info = 2; + return 0; + } + + if (rwork[1] == zero) { +/* The largest computed singular value of (scaled) */ +/* X is zero. Return error code -8 */ +/* (the 8th input variable had an illegal value). */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("CGEDMD", &i__1); + return 0; + } + +/* <3> Determine the numerical rank of the data */ +/* snapshots matrix X. This depends on the */ +/* parameters NRNK and TOL. */ +/* SELECT CASE ( NRNK ) */ + if (*nrnk == -1) { + *k = 1; + i__1 = numrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (rwork[i__] <= rwork[1] * *tol || rwork[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else if (*nrnk == -2) { + *k = 1; + i__1 = numrnk - 1; + for (i__ = 1; i__ <= i__1; ++i__) { + if (rwork[i__ + 1] <= rwork[i__] * *tol || rwork[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else { + *k = 1; + i__1 = *nrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (rwork[i__] <= small) { + myexit_(); + } + ++(*k); + } +/* END SELECT */ + } +/* Now, U = X(1:M,1:K) is the SVD/POD basis for the */ +/* snapshot data in the input matrix X. */ +/* <4> Compute the Rayleigh quotient S = U^H * A * U. */ +/* Depending on the requested outputs, the computation */ +/* is organized to compute additional auxiliary */ +/* matrices (for the residuals and refinements). */ + +/* In all formulas below, we need V_k*Sigma_k^(-1) */ +/* where either V_k is in W(1:N,1:K), or V_k^H is in */ +/* W(1:K,1:N). Here Sigma_k=diag(WORK(1:K)). */ + if (lsame_(t_or_n__, "N")) { + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + r__1 = one / rwork[i__]; + csscal_(n, &r__1, &w[i__ * w_dim1 + 1], &c__1); +/* W(1:N,i) = (ONE/RWORK(i)) * W(1:N,i) ! INTRINSIC */ +/* BLAS CALL */ + } + } else { +/* This non-unit stride access is due to the fact */ +/* that CGESVD, CGESVDQ and CGESDD return the */ +/* adjoint matrix of the right singular vectors. */ +/* DO i = 1, K */ +/* CALL DSCAL( N, ONE/RWORK(i), W(i,1), LDW ) ! BLAS CALL */ +/* ! W(i,1:N) = (ONE/RWORK(i)) * W(i,1:N) ! INTRINSIC */ +/* END DO */ + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + rwork[*n + i__] = one / rwork[i__]; + } + i__1 = *n; + for (j = 1; j <= i__1; ++j) { + i__2 = *k; + for (i__ = 1; i__ <= i__2; ++i__) { + i__3 = i__ + j * w_dim1; + i__4 = *n + i__; + q__2.r = rwork[i__4], q__2.i = zero; + i__5 = i__ + j * w_dim1; + q__1.r = q__2.r * w[i__5].r - q__2.i * w[i__5].i, q__1.i = + q__2.r * w[i__5].i + q__2.i * w[i__5].r; + w[i__3].r = q__1.r, w[i__3].i = q__1.i; + } + } + } + + if (wntref) { + +/* Need A*U(:,1:K)=Y*V_k*inv(diag(WORK(1:K))) */ +/* for computing the refined Ritz vectors */ +/* (optionally, outside CGEDMD). */ + cgemm_("N", t_or_n__, m, k, n, &zone, &y[y_offset], ldy, &w[w_offset], + ldw, &zzero, &z__[z_offset], ldz); +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRI */ +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),W(1:N,1:K)) ! INTRI */ + +/* At this point Z contains */ +/* A * U(:,1:K) = Y * V_k * Sigma_k^(-1), and */ +/* this is needed for computing the residuals. */ +/* This matrix is returned in the array B and */ +/* it can be used to compute refined Ritz vectors. */ +/* BLAS */ + clacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); +/* B(1:M,1:K) = Z(1:M,1:K) ! INTRINSIC */ +/* BLAS CALL */ + cgemm_("C", "N", k, k, m, &zone, &x[x_offset], ldx, &z__[z_offset], + ldz, &zzero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(TANSPOSE(X(1:M,1:K)),Z(1:M,1:K)) ! INTRI */ +/* At this point S = U^H * A * U is the Rayleigh quotient. */ +/* BLAS */ + } else { +/* A * U(:,1:K) is not explicitly needed and the */ +/* computation is organized differently. The Rayleigh */ +/* quotient is computed more efficiently. */ + cgemm_("C", "N", k, n, m, &zone, &x[x_offset], ldx, &y[y_offset], ldy, + &zzero, &z__[z_offset], ldz); +/* Z(1:K,1:N) = MATMUL( TRANSPOSE(X(1:M,1:K)), Y(1:M,1:N) ) ! IN */ + +/* B */ + cgemm_("N", t_or_n__, k, k, n, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRIN */ +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),(W(1:N,1:K))) ! INTRIN */ +/* At this point S = U^H * A * U is the Rayleigh quotient. */ +/* If the residuals are requested, save scaled V_k into Z. */ +/* Recall that V_k or V_k^H is stored in W. */ +/* BLAS */ + if (wntres || wntex) { + if (lsame_(t_or_n__, "N")) { + clacpy_("A", n, k, &w[w_offset], ldw, &z__[z_offset], ldz); + } else { + clacpy_("A", k, n, &w[w_offset], ldw, &z__[z_offset], ldz); + } + } + } + +/* <5> Compute the Ritz values and (if requested) the */ +/* right eigenvectors of the Rayleigh quotient. */ + + cgeev_("N", jobzl, k, &s[s_offset], lds, &eigs[1], &w[w_offset], ldw, &w[ + w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &info1); + +/* W(1:K,1:K) contains the eigenvectors of the Rayleigh */ +/* quotient. See the description of Z. */ +/* Also, see the description of CGEEV. */ +/* LAPACK CA */ + if (info1 > 0) { +/* CGEEV failed to compute the eigenvalues and */ +/* eigenvectors of the Rayleigh quotient. */ + *info = 3; + return 0; + } + +/* <6> Compute the eigenvectors (if requested) and, */ +/* the residuals (if requested). */ + + if (wntvec || wntex) { + if (wntres) { + if (wntref) { +/* Here, if the refinement is requested, we have */ +/* A*U(:,1:K) already computed and stored in Z. */ +/* For the residuals, need Y = A * U(:,1;K) * W. */ + cgemm_("N", "N", m, k, k, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &y[y_offset], ldy); +/* Y(1:M,1:K) = Z(1:M,1:K) * W(1:K,1:K) ! INTRINSIC */ +/* This frees Z; Y contains A * U(:,1:K) * W. */ +/* BLAS CALL */ + } else { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) (or its adjoint) is stored in Z */ + cgemm_(t_or_n__, "N", n, k, k, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + cgemm_("N", "N", m, k, n, &zone, &y[y_offset], ldy, &s[ + s_offset], lds, &zzero, &z__[z_offset], ldz); +/* Save a copy of Z into Y and free Z for holding */ +/* the Ritz vectors. */ + clacpy_("A", m, k, &z__[z_offset], ldz, &y[y_offset], ldy); + if (wntex) { + clacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); + } + } + } else if (wntex) { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) is stored in Z */ + cgemm_(t_or_n__, "N", n, k, k, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + cgemm_("N", "N", m, k, n, &zone, &y[y_offset], ldy, &s[s_offset], + lds, &zzero, &b[b_offset], ldb); +/* The above call replaces the following two calls */ +/* that were used in the developing-testing phase. */ +/* CALL CGEMM( 'N', 'N', M, K, N, ZONE, Y, LDY, S, & */ +/* LDS, ZZERO, Z, LDZ) */ +/* Save a copy of Z into Y and free Z for holding */ +/* the Ritz vectors. */ +/* CALL CLACPY( 'A', M, K, Z, LDZ, B, LDB ) */ + } + +/* Compute the Ritz vectors */ + if (wntvec) { + cgemm_("N", "N", m, k, k, &zone, &x[x_offset], ldx, &w[w_offset], + ldw, &zzero, &z__[z_offset], ldz); + } +/* Z(1:M,1:K) = MATMUL(X(1:M,1:K), W(1:K,1:K)) ! INTRIN */ + +/* BLAS CALL */ + if (wntres) { + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + i__2 = i__; + q__1.r = -eigs[i__2].r, q__1.i = -eigs[i__2].i; + caxpy_(m, &q__1, &z__[i__ * z_dim1 + 1], &c__1, &y[i__ * + y_dim1 + 1], &c__1); +/* Y(1:M,i) = Y(1:M,i) - EIGS(i) * Z(1:M,i) ! */ + + res[i__] = scnrm2_(m, &y[i__ * y_dim1 + 1], &c__1); + + } + } + } + + if (*whtsvd == 4) { + rwork[*n + 1] = xscl1; + rwork[*n + 2] = xscl2; + } + +/* Successful exit. */ + if (! badxy) { + *info = 0; + } else { +/* A warning on possible data inconsistency. */ +/* This should be a rare event. */ + *info = 4; + } +/* ............................................................ */ + return 0; +/* ...... */ +} /* cgedmd_ */ + diff --git a/lapack-netlib/SRC/cgedmdq.c b/lapack-netlib/SRC/cgedmdq.c index 447b23014..6e3a1faca 100644 --- a/lapack-netlib/SRC/cgedmdq.c +++ b/lapack-netlib/SRC/cgedmdq.c @@ -509,3 +509,781 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; + +/* Subroutine */ int cgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq, + char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n, + complex *f, integer *ldf, complex *x, integer *ldx, complex *y, + integer *ldy, integer *nrnk, real *tol, integer *k, complex *eigs, + complex *z__, integer *ldz, real *res, complex *b, integer *ldb, + complex *v, integer *ldv, complex *s, integer *lds, complex *zwork, + integer *lzwork, real *work, integer *lwork, integer *iwork, integer * + liwork, integer *info) +{ + /* System generated locals */ + integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1, + z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset, + i__1, i__2; + + /* Local variables */ + real zero; + integer info1; + extern logical lsame_(char *, char *); + char jobvl[1]; + integer minmn; + logical wantq; + integer mlwqr, olwqr; + logical wntex; + complex zzero; + extern /* Subroutine */ int cgedmd_(char *, char *, char *, char *, + integer *, integer *, integer *, complex *, integer *, complex *, + integer *, integer *, real *, integer *, complex *, complex *, + integer *, real *, complex *, integer *, complex *, integer *, + complex *, integer *, complex *, integer *, real *, integer *, + integer *, integer *, integer *), + cgeqrf_(integer *, integer *, complex *, integer *, complex *, + complex *, integer *, integer *), clacpy_(char *, integer *, + integer *, complex *, integer *, complex *, integer *), + claset_(char *, integer *, integer *, complex *, complex *, + complex *, integer *), xerbla_(char *, integer *); + integer mlwdmd, olwdmd; + logical sccolx, sccoly; + extern /* Subroutine */ int cungqr_(integer *, integer *, integer *, + complex *, integer *, complex *, complex *, integer *, integer *); + integer iminwr; + logical wntvec, wntvcf; + integer mlwgqr; + logical wntref; + integer mlwork, olwgqr, olwork, mlrwrk, mlwmqr, olwmqr; + logical lquery, wntres, wnttrf, wntvcq; + extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *, + integer *, complex *, integer *, complex *, complex *, integer *, + complex *, integer *, integer *); + real one; + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real32 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ..... */ +/* Purpose */ +/* ======= */ +/* CGEDMDQ computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices, using a QR factorization */ +/* based compression of the data. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, CGEDMDQ computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, CGEDMDQ returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office. */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ...................................................................... */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. The data snapshots are the columns */ +/* of F. The leading N-1 columns of F are denoted X and the */ +/* trailing N-1 columns are denoted Y. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Z*V, where Z */ +/* is orthonormal and V contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of F, V, Z. */ +/* 'Q' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Q*Z, where Z */ +/* contains the eigenvectors of the compression of the */ +/* underlying discretised operator onto the span of */ +/* the data snapshots. See the descriptions of F, V, Z. */ +/* Q is from the inital QR facorization. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will */ +/* be computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBQ (input) CHARACTER*1 */ +/* Specifies whether to explicitly compute and return the */ +/* unitary matrix from the QR factorization. */ +/* 'Q' :: The matrix Q of the QR factorization of the data */ +/* snapshot matrix is computed and stored in the */ +/* array F. See the description of F. */ +/* 'N' :: The matrix Q is not explicitly computed. */ +/* ..... */ +/* JOBT (input) CHARACTER*1 */ +/* Specifies whether to return the upper triangular factor */ +/* from the QR factorization. */ +/* 'R' :: The matrix R of the QR factorization of the data */ +/* snapshot matrix F is returned in the array Y. */ +/* See the description of Y and Further details. */ +/* 'N' :: The matrix R is not returned. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* To be useful on exit, this option needs JOBQ='Q'. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: CGESVD (the QR SVD algorithm) */ +/* 2 :: CGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: CGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M >= 0 */ +/* The state space dimension (the number of rows of F). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshots from a single trajectory, */ +/* taken at equidistant discrete times. This is the */ +/* number of columns of F. */ +/* ..... */ +/* F (input/output) COMPLEX(KIND=WP) M-by-N array */ +/* > On entry, */ +/* the columns of F are the sequence of data snapshots */ +/* from a single trajectory, taken at equidistant discrete */ +/* times. It is assumed that the column norms of F are */ +/* in the range of the normalized floating point numbers. */ +/* < On exit, */ +/* If JOBQ == 'Q', the array F contains the orthogonal */ +/* matrix/factor of the QR factorization of the initial */ +/* data snapshots matrix F. See the description of JOBQ. */ +/* If JOBQ == 'N', the entries in F strictly below the main */ +/* diagonal contain, column-wise, the information on the */ +/* Householder vectors, as returned by CGEQRF. The */ +/* remaining information to restore the orthogonal matrix */ +/* of the initial QR factorization is stored in ZWORK(1:MIN(M,N)). */ +/* See the description of ZWORK. */ +/* ..... */ +/* LDF (input) INTEGER, LDF >= M */ +/* The leading dimension of the array F. */ +/* ..... */ +/* X (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array */ +/* X is used as workspace to hold representations of the */ +/* leading N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, the leading K columns of X contain the leading */ +/* K left singular vectors of the above described content */ +/* of X. To lift them to the space of the left singular */ +/* vectors U(:,1:K) of the input data, pre-multiply with the */ +/* Q factor from the initial QR factorization. */ +/* See the descriptions of F, K, V and Z. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= N */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N) array */ +/* Y is used as workspace to hold representations of the */ +/* trailing N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, */ +/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */ +/* triangular factor from the QR factorization of the data */ +/* snapshot matrix F. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= N */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N-1 :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the description of K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the SVD/POD basis for the leading N-1 */ +/* data snapshots (columns of F) and the number of the */ +/* computed Ritz pairs. The value of K is determined */ +/* according to the rule set by the parameters NRNK and */ +/* TOL. See the descriptions of NRNK and TOL. */ +/* ..... */ +/* EIGS (output) COMPLEX(KIND=WP) (N-1)-by-1 array */ +/* The leading K (K<=N-1) entries of EIGS contain */ +/* the computed eigenvalues (Ritz values). */ +/* See the descriptions of K, and Z. */ +/* ..... */ +/* Z (workspace/output) COMPLEX(KIND=WP) M-by-(N-1) array */ +/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */ +/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */ +/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */ +/* Z*V, where Z contains orthonormal matrix (the product of */ +/* Q from the initial QR factorization and the SVD/POD_basis */ +/* returned by CGEDMD in X) and the second factor (the */ +/* eigenvectors of the Rayleigh quotient) is in the array V, */ +/* as returned by CGEDMD. That is, X(:,1:K)*V(:,i) */ +/* is an eigenvector corresponding to EIGS(i). The columns */ +/* of V(1:K,1:K) are the computed eigenvectors of the */ +/* K-by-K Rayleigh quotient. */ +/* See the descriptions of EIGS, X and V. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs, */ +/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */ +/* See the description of EIGS and Z. */ +/* ..... */ +/* B (output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array. */ +/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:N,1;K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* In both cases, the content of B can be lifted to the */ +/* original dimension of the input data by pre-multiplying */ +/* with the Q factor from the initial QR factorization. */ +/* Here A denotes a compression of the underlying operator. */ +/* See the descriptions of F and X. */ +/* If JOBF =='N', then B is not referenced. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= MIN(M,N) */ +/* The leading dimension of the array B. */ +/* ..... */ +/* V (workspace/output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */ +/* On exit, V(1:K,1:K) V contains the K eigenvectors of */ +/* the Rayleigh quotient. The Ritz vectors */ +/* (returned in Z) are the product of Q from the initial QR */ +/* factorization (see the description of F) X (see the */ +/* description of X) and V. */ +/* ..... */ +/* LDV (input) INTEGER, LDV >= N-1 */ +/* The leading dimension of the array V. */ +/* ..... */ +/* S (output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by CGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N-1 */ +/* The leading dimension of the array S. */ +/* ..... */ +/* ZWORK (workspace/output) COMPLEX(KIND=WP) LWORK-by-1 array */ +/* On exit, */ +/* ZWORK(1:MIN(M,N)) contains the scalar factors of the */ +/* elementary reflectors as returned by CGEQRF of the */ +/* M-by-N input matrix F. */ +/* If the call to CGEDMDQ is only workspace query, then */ +/* ZWORK(1) contains the minimal complex workspace length and */ +/* ZWORK(2) is the optimal complex workspace length. */ +/* Hence, the length of work is at least 2. */ +/* See the description of LZWORK. */ +/* ..... */ +/* LZWORK (input) INTEGER */ +/* The minimal length of the workspace vector ZWORK. */ +/* LZWORK is calculated as follows: */ +/* Let MLWQR = N (minimal workspace for CGEQRF[M,N]) */ +/* MLWDMD = minimal workspace for CGEDMD (see the */ +/* description of LWORK in CGEDMD) */ +/* MLWMQR = N (minimal workspace for */ +/* ZUNMQR['L','N',M,N,N]) */ +/* MLWGQR = N (minimal workspace for ZUNGQR[M,N,N]) */ +/* MINMN = MIN(M,N) */ +/* Then */ +/* LZWORK = MAX(2, MIN(M,N)+MLWQR, MINMN+MLWDMD) */ +/* is further updated as follows: */ +/* if JOBZ == 'V' or JOBZ == 'F' THEN */ +/* LZWORK = MAX( LZWORK, MINMN+MLWMQR ) */ +/* if JOBQ == 'Q' THEN */ +/* LZWORK = MAX( ZLWORK, MINMN+MLWGQR) */ + +/* ..... */ +/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */ +/* On exit, */ +/* WORK(1:N-1) contains the singular values of */ +/* the input submatrix F(1:M,1:N-1). */ +/* If the call to CGEDMDQ is only workspace query, then */ +/* WORK(1) contains the minimal workspace length and */ +/* WORK(2) is the optimal workspace length. hence, the */ +/* length of work is at least 2. */ +/* See the description of LWORK. */ +/* ..... */ +/* LWORK (input) INTEGER */ +/* The minimal length of the workspace vector WORK. */ +/* LWORK is the same as in CGEDMD, because in CGEDMDQ */ +/* only CGEDMD requires real workspace for snapshots */ +/* of dimensions MIN(M,N)-by-(N-1). */ +/* If on entry LWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* Let M1=MIN(M,N), N1=N-1. Then */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ +/* COMPLEX(KIND=WP), PARAMETER :: ZONE = ( 1.0_WP, 0.0_WP ) */ + +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ + +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ + +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* External subroutines */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* .......................................................... */ + /* Parameter adjustments */ + f_dim1 = *ldf; + f_offset = 1 + f_dim1 * 1; + f -= f_offset; + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --eigs; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + v_dim1 = *ldv; + v_offset = 1 + v_dim1 * 1; + v -= v_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --zwork; + --work; + --iwork; + + /* Function Body */ + one = 1.f; + zero = 0.f; + zzero.r = 0.f, zzero.i = 0.f; + +/* Test the input arguments */ + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntvcf = lsame_(jobz, "F"); + wntvcq = lsame_(jobz, "Q"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + wantq = lsame_(jobq, "Q"); + wnttrf = lsame_(jobt, "R"); + minmn = f2cmin(*m,*n); + *info = 0; + lquery = *lwork == -1 || *liwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && lsame_(jobz, "N")) { + *info = -3; + } else if (! (wantq || lsame_(jobq, "N"))) { + *info = -4; + } else if (! (wnttrf || lsame_(jobt, "N"))) { + *info = -5; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -6; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -7; + } else if (*m < 0) { + *info = -8; + } else if (*n < 0 || *n > *m + 1) { + *info = -9; + } else if (*ldf < *m) { + *info = -11; + } else if (*ldx < minmn) { + *info = -13; + } else if (*ldy < minmn) { + *info = -15; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -16; + } else if (*tol < zero || *tol >= one) { + *info = -17; + } else if (*ldz < *m) { + *info = -21; + } else if ((wntref || wntex) && *ldb < minmn) { + *info = -24; + } else if (*ldv < *n - 1) { + *info = -26; + } else if (*lds < *n - 1) { + *info = -28; + } + + if (wntvec || wntvcf || wntvcq) { + *(unsigned char *)jobvl = 'V'; + } else { + *(unsigned char *)jobvl = 'N'; + } + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0 || *n == 1) { +/* All output except K is void. INFO=1 signals */ +/* the void input. In case of a workspace query, */ +/* the minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + work[1] = 2.f; + work[2] = 2.f; + } else { + *k = 0; + } + *info = 1; + return 0; + } + mlrwrk = 2; + mlwork = 2; + olwork = 2; + iminwr = 1; + mlwqr = f2cmax(1,*n); +/* Minimal workspace length for CGEQRF. */ +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + cgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[1], &c_n1, & + info1); + olwqr = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwqr; + olwork = f2cmax(i__1,i__2); + } + i__1 = *n - 1; + cgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], + ldx, &y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset] + , ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[ + s_offset], lds, &zwork[1], lzwork, &work[1], &c_n1, &iwork[1], + liwork, &info1); + mlwdmd = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwdmd; + mlwork = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = (integer) work[1]; + mlrwrk = f2cmax(i__1,i__2); + iminwr = f2cmax(iminwr,iwork[1]); + if (lquery) { + olwdmd = (integer) zwork[2].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwdmd; + olwork = f2cmax(i__1,i__2); + } + if (wntvec || wntvcf) { + mlwmqr = f2cmax(1,*n); +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwmqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + cunmqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &zwork[1], + &z__[z_offset], ldz, &zwork[1], &c_n1, &info1); + olwmqr = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwmqr; + olwork = f2cmax(i__1,i__2); + } + } + if (wantq) { + mlwgqr = f2cmax(1,*n); +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwgqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + cungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], & + zwork[1], &c_n1, &info1); + olwgqr = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwgqr; + olwork = f2cmax(i__1,i__2); + } + } + if (*liwork < iminwr && ! lquery) { + *info = -34; + } + if (*lwork < mlrwrk && ! lquery) { + *info = -32; + } + if (*lzwork < mlwork && ! lquery) { + *info = -30; + } + } + if (*info != 0) { + i__1 = -(*info); + xerbla_("CGEDMDQ", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + zwork[1].r = (real) mlwork, zwork[1].i = 0.f; + zwork[2].r = (real) olwork, zwork[2].i = 0.f; + work[1] = (real) mlrwrk; + work[2] = (real) mlrwrk; + return 0; + } +/* ..... */ +/* Initial QR factorization that is used to represent the */ +/* snapshots as elements of lower dimensional subspace. */ +/* For large scale computation with M >>N , at this place */ +/* one can use an out of core QRF. */ + + i__1 = *lzwork - minmn; + cgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[minmn + 1], &i__1, & + info1); + +/* Define X and Y as the snapshots representations in the */ +/* orthogonal basis computed in the QR factorization. */ +/* X corresponds to the leading N-1 and Y to the trailing */ +/* N-1 snapshots. */ + i__1 = *n - 1; + claset_("L", &minmn, &i__1, &zzero, &zzero, &x[x_offset], ldx); + i__1 = *n - 1; + clacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx); + i__1 = *n - 1; + clacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy); + if (*m >= 3) { + i__1 = minmn - 2; + i__2 = *n - 2; + claset_("L", &i__1, &i__2, &zzero, &zzero, &y[y_dim1 + 3], ldy); + } + +/* Compute the DMD of the projected snapshot pairs (X,Y) */ + i__1 = *n - 1; + i__2 = *lzwork - minmn; + cgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx, + &y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset], ldz, & + res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[s_offset], lds, & + zwork[minmn + 1], &i__2, &work[1], lwork, &iwork[1], liwork, & + info1); + if (info1 == 2 || info1 == 3) { +/* Return with error code. See CGEDMD for details. */ + *info = info1; + return 0; + } else { + *info = info1; + } + +/* The Ritz vectors (Koopman modes) can be explicitly */ +/* formed or returned in factored form. */ + if (wntvec) { +/* Compute the eigenvectors explicitly. */ + if (*m > minmn) { + i__1 = *m - minmn; + claset_("A", &i__1, k, &zzero, &zzero, &z__[minmn + 1 + z_dim1], + ldz); + } + i__1 = *lzwork - minmn; + cunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[ + z_offset], ldz, &zwork[minmn + 1], &i__1, &info1); + } else if (wntvcf) { +/* Return the Ritz vectors (eigenvectors) in factored */ +/* form Z*V, where Z contains orthonormal matrix (the */ +/* product of Q from the initial QR factorization and */ +/* the SVD/POD_basis returned by CGEDMD in X) and the */ +/* second factor (the eigenvectors of the Rayleigh */ +/* quotient) is in the array V, as returned by CGEDMD. */ + clacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz); + if (*m > *n) { + i__1 = *m - *n; + claset_("A", &i__1, k, &zzero, &zzero, &z__[*n + 1 + z_dim1], ldz); + } + i__1 = *lzwork - minmn; + cunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[ + z_offset], ldz, &zwork[minmn + 1], &i__1, &info1); + } + +/* Some optional output variables: */ + +/* The upper triangular factor R in the initial QR */ +/* factorization is optionally returned in the array Y. */ +/* This is useful if this call to CGEDMDQ is to be */ +/* followed by a streaming DMD that is implemented in a */ +/* QR compressed form. */ + if (wnttrf) { +/* Return the upper triangular R in Y */ + claset_("A", &minmn, n, &zzero, &zzero, &y[y_offset], ldy); + clacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy); + } + +/* The orthonormal/unitary factor Q in the initial QR */ +/* factorization is optionally returned in the array F. */ +/* Same as with the triangular factor above, this is */ +/* useful in a streaming DMD. */ + if (wantq) { +/* Q overwrites F */ + i__1 = *lzwork - minmn; + cungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], &zwork[minmn + + 1], &i__1, &info1); + } + + return 0; + +} /* cgedmdq_ */ + diff --git a/lapack-netlib/SRC/dgedmd.c b/lapack-netlib/SRC/dgedmd.c index 447b23014..66b4d5da6 100644 --- a/lapack-netlib/SRC/dgedmd.c +++ b/lapack-netlib/SRC/dgedmd.c @@ -509,3 +509,1245 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; +static integer c__1 = 1; +static integer c__0 = 0; +static integer c__2 = 2; + +/* Subroutine */ int dgedmd_(char *jobs, char *jobz, char *jobr, char *jobf, + integer *whtsvd, integer *m, integer *n, doublereal *x, integer *ldx, + doublereal *y, integer *ldy, integer *nrnk, doublereal *tol, integer * + k, doublereal *reig, doublereal *imeig, doublereal *z__, integer *ldz, + doublereal *res, doublereal *b, integer *ldb, doublereal *w, integer + *ldw, doublereal *s, integer *lds, doublereal *work, integer *lwork, + integer *iwork, integer *liwork, integer *info) +{ + /* System generated locals */ + integer x_dim1, x_offset, y_dim1, y_offset, z_dim1, z_offset, b_dim1, + b_offset, w_dim1, w_offset, s_dim1, s_offset, i__1, i__2; + doublereal d__1, d__2; + + /* Local variables */ + doublereal zero, ssum; + integer info1, info2; + extern doublereal dnrm2_(integer *, doublereal *, integer *); + doublereal xscl1, xscl2; + integer i__, j; + extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, + integer *); + doublereal scale; + extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *, + integer *, doublereal *, doublereal *, integer *, doublereal *, + integer *, doublereal *, doublereal *, integer *), + dgeev_(char *, char *, integer *, doublereal *, integer *, + doublereal *, doublereal *, doublereal *, integer *, doublereal *, + integer *, doublereal *, integer *, integer *); + extern logical lsame_(char *, char *); + logical badxy; + doublereal small; + char jobzl[1]; + extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *, + integer *, doublereal *, integer *); + logical wntex; + doublereal ab[4] /* was [2][2] */; + extern doublereal dlamch_(char *), dlange_(char *, integer *, + integer *, doublereal *, integer *, doublereal *); + extern /* Subroutine */ int dgesdd_(char *, integer *, integer *, + doublereal *, integer *, doublereal *, doublereal *, integer *, + doublereal *, integer *, doublereal *, integer *, integer *, + integer *), dlascl_(char *, integer *, integer *, + doublereal *, doublereal *, integer *, integer *, doublereal *, + integer *, integer *); + extern integer idamax_(integer *, doublereal *, integer *); + extern logical disnan_(doublereal *); + extern /* Subroutine */ int dgesvd_(char *, char *, integer *, integer *, + doublereal *, integer *, doublereal *, doublereal *, integer *, + doublereal *, integer *, doublereal *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal + *, integer *, doublereal *, integer *), xerbla_(char *, + integer *); + char t_or_n__[1]; + extern /* Subroutine */ int dgejsv_(char *, char *, char *, char *, char * + , char *, integer *, integer *, doublereal *, integer *, + doublereal *, doublereal *, integer *, doublereal *, integer *, + doublereal *, integer *, integer *, integer *), dlassq_(integer *, doublereal *, + integer *, doublereal *, doublereal *); + logical sccolx, sccoly; + integer lwrsdd, mwrsdd, iminwr; + logical wntref, wntvec; + doublereal rootsc; + integer lwrkev, mlwork, mwrkev, numrnk, olwork; + doublereal rdummy[2]; + integer lwrsvd, mwrsvd; + logical lquery, wntres; + char jsvopt[1]; + extern /* Subroutine */ int mecago_(); + integer mwrsvj, lwrsvq, mwrsvq; + doublereal rdummy2[2], ofl, one; + extern /* Subroutine */ int dgesvdq_(char *, char *, char *, char *, char + *, integer *, integer *, doublereal *, integer *, doublereal *, + doublereal *, integer *, doublereal *, integer *, integer *, + integer *, integer *, doublereal *, integer *, doublereal *, + integer *, integer *); + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real64 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ............................................................ */ +/* Purpose */ +/* ======= */ +/* DGEDMD computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, DGEDMD computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, DGEDMD returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* ...................................................................... */ +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ............................................................ */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product X(:,1:K)*W, where X */ +/* contains a POD basis (leading left singular vectors */ +/* of the data matrix X) and W contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of K, X, W, Z. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will be */ +/* computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: DGESVD (the QR SVD algorithm) */ +/* 2 :: DGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: DGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: DGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M>= 0 */ +/* The state space dimension (the row dimension of X, Y). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshot pairs */ +/* (the number of columns of X and Y). */ +/* ..... */ +/* X (input/output) REAL(KIND=WP) M-by-N array */ +/* > On entry, X contains the data snapshot matrix X. It is */ +/* assumed that the column norms of X are in the range of */ +/* the normalized floating point numbers. */ +/* < On exit, the leading K columns of X contain a POD basis, */ +/* i.e. the leading K left singular vectors of the input */ +/* data matrix X, U(:,1:K). All N columns of X contain all */ +/* left singular vectors of the input matrix X. */ +/* See the descriptions of K, Z and W. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= M */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (input/workspace/output) REAL(KIND=WP) M-by-N array */ +/* > On entry, Y contains the data snapshot matrix Y */ +/* < On exit, */ +/* If JOBR == 'R', the leading K columns of Y contain */ +/* the residual vectors for the computed Ritz pairs. */ +/* See the description of RES. */ +/* If JOBR == 'N', Y contains the original input data, */ +/* scaled according to the value of JOBS. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= M */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1). */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ + +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the descriptions of TOL and K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the POD basis for the data snapshot */ +/* matrix X and the number of the computed Ritz pairs. */ +/* The value of K is determined according to the rule set */ +/* by the parameters NRNK and TOL. */ +/* See the descriptions of NRNK and TOL. */ +/* ..... */ +/* REIG (output) REAL(KIND=WP) N-by-1 array */ +/* The leading K (K<=N) entries of REIG contain */ +/* the real parts of the computed eigenvalues */ +/* REIG(1:K) + sqrt(-1)*IMEIG(1:K). */ +/* See the descriptions of K, IMEIG, and Z. */ +/* ..... */ +/* IMEIG (output) REAL(KIND=WP) N-by-1 array */ +/* The leading K (K<=N) entries of IMEIG contain */ +/* the imaginary parts of the computed eigenvalues */ +/* REIG(1:K) + sqrt(-1)*IMEIG(1:K). */ +/* The eigenvalues are determined as follows: */ +/* If IMEIG(i) == 0, then the corresponding eigenvalue is */ +/* real, LAMBDA(i) = REIG(i). */ +/* If IMEIG(i)>0, then the corresponding complex */ +/* conjugate pair of eigenvalues reads */ +/* LAMBDA(i) = REIG(i) + sqrt(-1)*IMAG(i) */ +/* LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i) */ +/* That is, complex conjugate pairs have consecutive */ +/* indices (i,i+1), with the positive imaginary part */ +/* listed first. */ +/* See the descriptions of K, REIG, and Z. */ +/* ..... */ +/* Z (workspace/output) REAL(KIND=WP) M-by-N array */ +/* If JOBZ =='V' then */ +/* Z contains real Ritz vectors as follows: */ +/* If IMEIG(i)=0, then Z(:,i) is an eigenvector of */ +/* the i-th Ritz value; ||Z(:,i)||_2=1. */ +/* If IMEIG(i) > 0 (and IMEIG(i+1) < 0) then */ +/* [Z(:,i) Z(:,i+1)] span an invariant subspace and */ +/* the Ritz values extracted from this subspace are */ +/* REIG(i) + sqrt(-1)*IMEIG(i) and */ +/* REIG(i) - sqrt(-1)*IMEIG(i). */ +/* The corresponding eigenvectors are */ +/* Z(:,i) + sqrt(-1)*Z(:,i+1) and */ +/* Z(:,i) - sqrt(-1)*Z(:,i+1), respectively. */ +/* || Z(:,i:i+1)||_F = 1. */ +/* If JOBZ == 'F', then the above descriptions hold for */ +/* the columns of X(:,1:K)*W(1:K,1:K), where the columns */ +/* of W(1:k,1:K) are the computed eigenvectors of the */ +/* K-by-K Rayleigh quotient. The columns of W(1:K,1:K) */ +/* are similarly structured: If IMEIG(i) == 0 then */ +/* X(:,1:K)*W(:,i) is an eigenvector, and if IMEIG(i)>0 */ +/* then X(:,1:K)*W(:,i)+sqrt(-1)*X(:,1:K)*W(:,i+1) and */ +/* X(:,1:K)*W(:,i)-sqrt(-1)*X(:,1:K)*W(:,i+1) */ +/* are the eigenvectors of LAMBDA(i), LAMBDA(i+1). */ +/* See the descriptions of REIG, IMEIG, X and W. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) N-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs. */ +/* If LAMBDA(i) is real, then */ +/* RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2. */ +/* If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair */ +/* then */ +/* RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F */ +/* where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ] */ +/* [-imag(LAMBDA(i)) real(LAMBDA(i)) ]. */ +/* It holds that */ +/* RES(i) = || A*ZC(:,i) - LAMBDA(i) *ZC(:,i) ||_2 */ +/* RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2 */ +/* where ZC(:,i) = Z(:,i) + sqrt(-1)*Z(:,i+1) */ +/* ZC(:,i+1) = Z(:,i) - sqrt(-1)*Z(:,i+1) */ +/* See the description of REIG, IMEIG and Z. */ +/* ..... */ +/* B (output) REAL(KIND=WP) M-by-N array. */ +/* IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:M,1;K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* If JOBF =='N', then B is not referenced. */ +/* See the descriptions of X, W, K. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= M */ +/* The leading dimension of the array B. */ +/* ..... */ +/* W (workspace/output) REAL(KIND=WP) N-by-N array */ +/* On exit, W(1:K,1:K) contains the K computed */ +/* eigenvectors of the matrix Rayleigh quotient (real and */ +/* imaginary parts for each complex conjugate pair of the */ +/* eigenvalues). The Ritz vectors (returned in Z) are the */ +/* product of X (containing a POD basis for the input */ +/* matrix X) and W. See the descriptions of K, S, X and Z. */ +/* W is also used as a workspace to temporarily store the */ +/* right singular vectors of X. */ +/* ..... */ +/* LDW (input) INTEGER, LDW >= N */ +/* The leading dimension of the array W. */ +/* ..... */ +/* S (workspace/output) REAL(KIND=WP) N-by-N array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by DGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N */ +/* The leading dimension of the array S. */ +/* ..... */ +/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */ +/* On exit, WORK(1:N) contains the singular values of */ +/* X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). */ +/* If WHTSVD==4, then WORK(N+1) and WORK(N+2) contain */ +/* scaling factor WORK(N+2)/WORK(N+1) used to scale X */ +/* and Y to avoid overflow in the SVD of X. */ +/* This may be of interest if the scaling option is off */ +/* and as many as possible smallest eigenvalues are */ +/* desired to the highest feasible accuracy. */ +/* If the call to DGEDMD is only workspace query, then */ +/* WORK(1) contains the minimal workspace length and */ +/* WORK(2) is the optimal workspace length. Hence, the */ +/* leng of work is at least 2. */ +/* See the description of LWORK. */ +/* ..... */ +/* LWORK (input) INTEGER */ +/* The minimal length of the workspace vector WORK. */ +/* LWORK is calculated as follows: */ +/* If WHTSVD == 1 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)). */ +/* If JOBZ == 'N' then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)). */ +/* Here LWORK_SVD = MAX(1,3*N+M,5*N) is the minimal */ +/* workspace length of DGESVD. */ +/* If WHTSVD == 2 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)) */ +/* If JOBZ == 'N', then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)) */ +/* Here LWORK_SVD = MAX(M, 5*N*N+4*N)+3*N*N is the */ +/* minimal workspace length of DGESDD. */ +/* If WHTSVD == 3 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N)) */ +/* If JOBZ == 'N', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N)) */ +/* Here LWORK_SVD = N+M+MAX(3*N+1, */ +/* MAX(1,3*N+M,5*N),MAX(1,N)) */ +/* is the minimal workspace length of DGESVDQ. */ +/* If WHTSVD == 4 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N)) */ +/* If JOBZ == 'N', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N)) */ +/* Here LWORK_SVD = MAX(7,2*M+N,6*N+2*N*N) is the */ +/* minimal workspace length of DGEJSV. */ +/* The above expressions are not simplified in order to */ +/* make the usage of WORK more transparent, and for */ +/* easier checking. In any case, LWORK >= 2. */ +/* If on entry LWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ +/* Local arrays */ +/* ~~~~~~~~~~~~ */ +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* ............................................................ */ + /* Parameter adjustments */ + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --reig; + --imeig; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + w_dim1 = *ldw; + w_offset = 1 + w_dim1 * 1; + w -= w_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --work; + --iwork; + + /* Function Body */ + one = 1.f; + zero = 0.f; + +/* Test the input arguments */ + + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + *info = 0; + lquery = *lwork == -1 || *liwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || lsame_(jobz, "N") || lsame_( + jobz, "F"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && ! wntvec) { + *info = -3; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -4; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -5; + } else if (*m < 0) { + *info = -6; + } else if (*n < 0 || *n > *m) { + *info = -7; + } else if (*ldx < *m) { + *info = -9; + } else if (*ldy < *m) { + *info = -11; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -12; + } else if (*tol < zero || *tol >= one) { + *info = -13; + } else if (*ldz < *m) { + *info = -18; + } else if ((wntref || wntex) && *ldb < *m) { + *info = -21; + } else if (*ldw < *n) { + *info = -23; + } else if (*lds < *n) { + *info = -25; + } + + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0) { +/* Quick return. All output except K is void. */ +/* INFO=1 signals the void input. */ +/* In case of a workspace query, the default */ +/* minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + work[1] = 2.; + work[2] = 2.; + } else { + *k = 0; + } + *info = 1; + return 0; + } + mlwork = f2cmax(2,*n); + olwork = f2cmax(2,*n); + iminwr = 1; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of DGESVD: */ +/* MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) */ +/* Computing MAX */ + i__1 = 1, i__2 = f2cmin(*m,*n) * 3 + f2cmax(*m,*n), i__1 = f2cmax(i__1, + i__2), i__2 = f2cmin(*m,*n) * 5; + mwrsvd = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsvd; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + dgesvd_("O", "S", m, n, &x[x_offset], ldx, &work[1], &b[ + b_offset], ldb, &w[w_offset], ldw, rdummy, &c_n1, & + info1); +/* Computing MAX */ + i__1 = mwrsvd, i__2 = (integer) rdummy[0]; + lwrsvd = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrsvd; + olwork = f2cmax(i__1,i__2); + } + } else if (*whtsvd == 2) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of DGESDD: */ +/* MWRSDD = 3*MIN(M,N)*MIN(M,N) + */ +/* MAX( MAX(M,N),5*MIN(M,N)*MIN(M,N)+4*MIN(M,N) ) */ +/* IMINWR = 8*MIN(M,N) */ +/* Computing MAX */ + i__1 = f2cmax(*m,*n), i__2 = f2cmin(*m,*n) * 5 * f2cmin(*m,*n) + (f2cmin(*m,* + n) << 2); + mwrsdd = f2cmin(*m,*n) * 3 * f2cmin(*m,*n) + f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsdd; + mlwork = f2cmax(i__1,i__2); + iminwr = f2cmin(*m,*n) << 3; + if (lquery) { + dgesdd_("O", m, n, &x[x_offset], ldx, &work[1], &b[b_offset], + ldb, &w[w_offset], ldw, rdummy, &c_n1, &iwork[1], & + info1); +/* Computing MAX */ + i__1 = mwrsdd, i__2 = (integer) rdummy[0]; + lwrsdd = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrsdd; + olwork = f2cmax(i__1,i__2); + } + } else if (*whtsvd == 3) { +/* LWQP3 = 3*N+1 */ +/* LWORQ = MAX(N, 1) */ +/* MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) */ +/* MWRSVQ = N + MAX( LWQP3, MWRSVD, LWORQ ) + MAX(M,2) */ +/* MLWORK = N + MWRSVQ */ +/* IMINWR = M+N-1 */ + dgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &work[ + 1], &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, & + iwork[1], liwork, rdummy, &c_n1, rdummy2, &c_n1, &info1); + iminwr = iwork[1]; + mwrsvq = (integer) rdummy[1]; +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsvq + (integer) rdummy2[0]; + mlwork = f2cmax(i__1,i__2); + if (lquery) { +/* Computing MAX */ + i__1 = mwrsvq, i__2 = (integer) rdummy[0]; + lwrsvq = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrsvq + (integer) rdummy2[0]; + olwork = f2cmax(i__1,i__2); + } + } else if (*whtsvd == 4) { + *(unsigned char *)jsvopt = 'J'; +/* MWRSVJ = MAX( 7, 2*M+N, 6*N+2*N*N ) ! for JSVOPT='V' */ +/* Computing MAX */ + i__1 = 7, i__2 = (*m << 1) + *n, i__1 = f2cmax(i__1,i__2), i__2 = (* + n << 2) + *n * *n, i__1 = f2cmax(i__1,i__2), i__2 = (*n << 1) + + *n * *n + 6; + mwrsvj = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsvj; + mlwork = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = 3, i__2 = *m + *n * 3; + iminwr = f2cmax(i__1,i__2); + if (lquery) { +/* Computing MAX */ + i__1 = olwork, i__2 = *n + mwrsvj; + olwork = f2cmax(i__1,i__2); + } +/* END SELECT */ + } + if (wntvec || wntex || lsame_(jobz, "F")) { + *(unsigned char *)jobzl = 'V'; + } else { + *(unsigned char *)jobzl = 'N'; + } +/* Workspace calculation to the DGEEV call */ + if (lsame_(jobzl, "V")) { +/* Computing MAX */ + i__1 = 1, i__2 = *n << 2; + mwrkev = f2cmax(i__1,i__2); + } else { +/* Computing MAX */ + i__1 = 1, i__2 = *n * 3; + mwrkev = f2cmax(i__1,i__2); + } +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrkev; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + dgeev_("N", jobzl, n, &s[s_offset], lds, &reig[1], &imeig[1], &w[ + w_offset], ldw, &w[w_offset], ldw, rdummy, &c_n1, &info1); +/* Computing MAX */ + i__1 = mwrkev, i__2 = (integer) rdummy[0]; + lwrkev = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrkev; + olwork = f2cmax(i__1,i__2); + } + + if (*liwork < iminwr && ! lquery) { + *info = -29; + } + if (*lwork < mlwork && ! lquery) { + *info = -27; + } + } + + if (*info != 0) { + i__1 = -(*info); + xerbla_("DGEDMD", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + work[1] = (doublereal) mlwork; + work[2] = (doublereal) olwork; + return 0; + } +/* ............................................................ */ + + ofl = dlamch_("O"); + small = dlamch_("S"); + badxy = FALSE_; + +/* <1> Optional scaling of the snapshots (columns of X, Y) */ +/* ========================================================== */ + if (sccolx) { +/* The columns of X will be normalized. */ +/* To prevent overflows, the column norms of X are */ +/* carefully computed using DLASSQ. */ + *k = 0; + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* WORK(i) = DNRM2( M, X(1,i), 1 ) */ + scale = zero; + dlassq_(m, &x[i__ * x_dim1 + 1], &c__1, &scale, &ssum); + if (disnan_(&scale) || disnan_(&ssum)) { + *k = 0; + *info = -8; + i__2 = -(*info); + xerbla_("DGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of X(:,i) overflows. First, X(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||X(:,i)||_2. */ +/* Next, the norm of X(:,i) is stored without */ +/* overflow as WORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of X(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + d__1 = one / rootsc; + dlascl_("G", &c__0, &c__0, &scale, &d__1, m, &c__1, &x[ + i__ * x_dim1 + 1], m, &info2); + work[i__] = -scale * (rootsc / (doublereal) (*m)); + } else { +/* X(:,i) will be scaled to unit 2-norm */ + work[i__] = scale * rootsc; + dlascl_("G", &c__0, &c__0, &work[i__], &one, m, &c__1, &x[ + i__ * x_dim1 + 1], m, &info2); +/* X(1:M,i) = (ONE/WORK(i)) * X(1:M,i) ! INTRINSIC */ +/* LAPACK */ + } + } else { + work[i__] = zero; + ++(*k); + } + } + if (*k == *n) { +/* All columns of X are zero. Return error code -8. */ +/* (the 8th input variable had an illegal value) */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("DGEDMD", &i__1); + return 0; + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of Y. */ + if (work[i__] > zero) { + d__1 = one / work[i__]; + dscal_(m, &d__1, &y[i__ * y_dim1 + 1], &c__1); +/* Y(1:M,i) = (ONE/WORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (work[i__] < zero) { + d__1 = -work[i__]; + d__2 = one / (doublereal) (*m); + dlascl_("G", &c__0, &c__0, &d__1, &d__2, m, &c__1, &y[i__ * + y_dim1 + 1], m, &info2); +/* LAPACK CAL */ + } else if (y[idamax_(m, &y[i__ * y_dim1 + 1], &c__1) + i__ * + y_dim1] != zero) { +/* X(:,i) is zero vector. For consistency, */ +/* Y(:,i) should also be zero. If Y(:,i) is not */ +/* zero, then the data might be inconsistent or */ +/* corrupted. If JOBS == 'C', Y(:,i) is set to */ +/* zero and a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + if (lsame_(jobs, "C")) { + dscal_(m, &zero, &y[i__ * y_dim1 + 1], &c__1); + } +/* BLAS CALL */ + } + } + } + + if (sccoly) { +/* The columns of Y will be normalized. */ +/* To prevent overflows, the column norms of Y are */ +/* carefully computed using DLASSQ. */ + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* WORK(i) = DNRM2( M, Y(1,i), 1 ) */ + scale = zero; + dlassq_(m, &y[i__ * y_dim1 + 1], &c__1, &scale, &ssum); + if (disnan_(&scale) || disnan_(&ssum)) { + *k = 0; + *info = -10; + i__2 = -(*info); + xerbla_("DGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of Y(:,i) overflows. First, Y(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||Y(:,i)||_2. */ +/* Next, the norm of Y(:,i) is stored without */ +/* overflow as WORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of Y(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + d__1 = one / rootsc; + dlascl_("G", &c__0, &c__0, &scale, &d__1, m, &c__1, &y[ + i__ * y_dim1 + 1], m, &info2); + work[i__] = -scale * (rootsc / (doublereal) (*m)); + } else { +/* X(:,i) will be scaled to unit 2-norm */ + work[i__] = scale * rootsc; + dlascl_("G", &c__0, &c__0, &work[i__], &one, m, &c__1, &y[ + i__ * y_dim1 + 1], m, &info2); +/* Y(1:M,i) = (ONE/WORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* LAPACK */ + } + } else { + work[i__] = zero; + } + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of X. */ + if (work[i__] > zero) { + d__1 = one / work[i__]; + dscal_(m, &d__1, &x[i__ * x_dim1 + 1], &c__1); +/* X(1:M,i) = (ONE/WORK(i)) * X(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (work[i__] < zero) { + d__1 = -work[i__]; + d__2 = one / (doublereal) (*m); + dlascl_("G", &c__0, &c__0, &d__1, &d__2, m, &c__1, &x[i__ * + x_dim1 + 1], m, &info2); +/* LAPACK CAL */ + } else if (x[idamax_(m, &x[i__ * x_dim1 + 1], &c__1) + i__ * + x_dim1] != zero) { +/* Y(:,i) is zero vector. If X(:,i) is not */ +/* zero, then a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + } + } + } + +/* <2> SVD of the data snapshot matrix X. */ +/* ===================================== */ +/* The left singular vectors are stored in the array X. */ +/* The right singular vectors are in the array W. */ +/* The array W will later on contain the eigenvectors */ +/* of a Rayleigh quotient. */ + numrnk = *n; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { + i__1 = *lwork - *n; + dgesvd_("O", "S", m, n, &x[x_offset], ldx, &work[1], &b[b_offset], + ldb, &w[w_offset], ldw, &work[*n + 1], &i__1, &info1); +/* LAPACK CAL */ + *(unsigned char *)t_or_n__ = 'T'; + } else if (*whtsvd == 2) { + i__1 = *lwork - *n; + dgesdd_("O", m, n, &x[x_offset], ldx, &work[1], &b[b_offset], ldb, &w[ + w_offset], ldw, &work[*n + 1], &i__1, &iwork[1], &info1); +/* LAPACK CAL */ + *(unsigned char *)t_or_n__ = 'T'; + } else if (*whtsvd == 3) { + i__1 = *lwork - *n - f2cmax(2,*m); + i__2 = f2cmax(2,*m); + dgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &work[1], & + z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, &iwork[1], + liwork, &work[*n + f2cmax(2,*m) + 1], &i__1, &work[*n + 1], & + i__2, &info1); +/* L */ + dlacpy_("A", m, &numrnk, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK C */ + *(unsigned char *)t_or_n__ = 'T'; + } else if (*whtsvd == 4) { + i__1 = *lwork - *n; + dgejsv_("F", "U", jsvopt, "N", "N", "P", m, n, &x[x_offset], ldx, & + work[1], &z__[z_offset], ldz, &w[w_offset], ldw, &work[*n + 1] + , &i__1, &iwork[1], &info1); +/* LAPACK CALL */ + dlacpy_("A", m, n, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK CALL */ + *(unsigned char *)t_or_n__ = 'N'; + xscl1 = work[*n + 1]; + xscl2 = work[*n + 2]; + if (xscl1 != xscl2) { +/* This is an exceptional situation. If the */ +/* data matrices are not scaled and the */ +/* largest singular value of X overflows. */ +/* In that case DGEJSV can return the SVD */ +/* in scaled form. The scaling factor can be used */ +/* to rescale the data (X and Y). */ + dlascl_("G", &c__0, &c__0, &xscl1, &xscl2, m, n, &y[y_offset], + ldy, &info2); + } +/* END SELECT */ + } + + if (info1 > 0) { +/* The SVD selected subroutine did not converge. */ +/* Return with an error code. */ + *info = 2; + return 0; + } + + if (work[1] == zero) { +/* The largest computed singular value of (scaled) */ +/* X is zero. Return error code -8 */ +/* (the 8th input variable had an illegal value). */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("DGEDMD", &i__1); + return 0; + } + +/* <3> Determine the numerical rank of the data */ +/* snapshots matrix X. This depends on the */ +/* parameters NRNK and TOL. */ +/* SELECT CASE ( NRNK ) */ + if (*nrnk == -1) { + *k = 1; + i__1 = numrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (work[i__] <= work[1] * *tol || work[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else if (*nrnk == -2) { + *k = 1; + i__1 = numrnk - 1; + for (i__ = 1; i__ <= i__1; ++i__) { + if (work[i__ + 1] <= work[i__] * *tol || work[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else { + *k = 1; + i__1 = *nrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (work[i__] <= small) { + myexit_(); + } + ++(*k); + } +/* END SELECT */ + } +/* Now, U = X(1:M,1:K) is the SVD/POD basis for the */ +/* snapshot data in the input matrix X. */ +/* <4> Compute the Rayleigh quotient S = U^T * A * U. */ +/* Depending on the requested outputs, the computation */ +/* is organized to compute additional auxiliary */ +/* matrices (for the residuals and refinements). */ + +/* In all formulas below, we need V_k*Sigma_k^(-1) */ +/* where either V_k is in W(1:N,1:K), or V_k^T is in */ +/* W(1:K,1:N). Here Sigma_k=diag(WORK(1:K)). */ + if (lsame_(t_or_n__, "N")) { + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + d__1 = one / work[i__]; + dscal_(n, &d__1, &w[i__ * w_dim1 + 1], &c__1); +/* W(1:N,i) = (ONE/WORK(i)) * W(1:N,i) ! INTRINSIC */ +/* BLAS CALL */ + } + } else { +/* This non-unit stride access is due to the fact */ +/* that DGESVD, DGESVDQ and DGESDD return the */ +/* transposed matrix of the right singular vectors. */ +/* DO i = 1, K */ +/* CALL DSCAL( N, ONE/WORK(i), W(i,1), LDW ) ! BLAS CALL */ +/* ! W(i,1:N) = (ONE/WORK(i)) * W(i,1:N) ! INTRINSIC */ +/* END DO */ + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + work[*n + i__] = one / work[i__]; + } + i__1 = *n; + for (j = 1; j <= i__1; ++j) { + i__2 = *k; + for (i__ = 1; i__ <= i__2; ++i__) { + w[i__ + j * w_dim1] = work[*n + i__] * w[i__ + j * w_dim1]; + } + } + } + + if (wntref) { + +/* Need A*U(:,1:K)=Y*V_k*inv(diag(WORK(1:K))) */ +/* for computing the refined Ritz vectors */ +/* (optionally, outside DGEDMD). */ + dgemm_("N", t_or_n__, m, k, n, &one, &y[y_offset], ldy, &w[w_offset], + ldw, &zero, &z__[z_offset], ldz); +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRI */ +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),W(1:N,1:K)) ! INTRI */ + +/* At this point Z contains */ +/* A * U(:,1:K) = Y * V_k * Sigma_k^(-1), and */ +/* this is needed for computing the residuals. */ +/* This matrix is returned in the array B and */ +/* it can be used to compute refined Ritz vectors. */ +/* BLAS */ + dlacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); +/* B(1:M,1:K) = Z(1:M,1:K) ! INTRINSIC */ +/* BLAS CALL */ + dgemm_("T", "N", k, k, m, &one, &x[x_offset], ldx, &z__[z_offset], + ldz, &zero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(TANSPOSE(X(1:M,1:K)),Z(1:M,1:K)) ! INTRI */ +/* At this point S = U^T * A * U is the Rayleigh quotient. */ +/* BLAS */ + } else { +/* A * U(:,1:K) is not explicitly needed and the */ +/* computation is organized differently. The Rayleigh */ +/* quotient is computed more efficiently. */ + dgemm_("T", "N", k, n, m, &one, &x[x_offset], ldx, &y[y_offset], ldy, + &zero, &z__[z_offset], ldz); +/* Z(1:K,1:N) = MATMUL( TRANSPOSE(X(1:M,1:K)), Y(1:M,1:N) ) ! IN */ +/* In the two DGEMM calls here, can use K for LDZ. */ +/* B */ + dgemm_("N", t_or_n__, k, k, n, &one, &z__[z_offset], ldz, &w[w_offset] + , ldw, &zero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRIN */ +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),(W(1:N,1:K))) ! INTRIN */ +/* At this point S = U^T * A * U is the Rayleigh quotient. */ +/* If the residuals are requested, save scaled V_k into Z. */ +/* Recall that V_k or V_k^T is stored in W. */ +/* BLAS */ + if (wntres || wntex) { + if (lsame_(t_or_n__, "N")) { + dlacpy_("A", n, k, &w[w_offset], ldw, &z__[z_offset], ldz); + } else { + dlacpy_("A", k, n, &w[w_offset], ldw, &z__[z_offset], ldz); + } + } + } + +/* <5> Compute the Ritz values and (if requested) the */ +/* right eigenvectors of the Rayleigh quotient. */ + + i__1 = *lwork - *n; + dgeev_("N", jobzl, k, &s[s_offset], lds, &reig[1], &imeig[1], &w[w_offset] + , ldw, &w[w_offset], ldw, &work[*n + 1], &i__1, &info1); + +/* W(1:K,1:K) contains the eigenvectors of the Rayleigh */ +/* quotient. Even in the case of complex spectrum, all */ +/* computation is done in real arithmetic. REIG and */ +/* IMEIG are the real and the imaginary parts of the */ +/* eigenvalues, so that the spectrum is given as */ +/* REIG(:) + sqrt(-1)*IMEIG(:). Complex conjugate pairs */ +/* are listed at consecutive positions. For such a */ +/* complex conjugate pair of the eigenvalues, the */ +/* corresponding eigenvectors are also a complex */ +/* conjugate pair with the real and imaginary parts */ +/* stored column-wise in W at the corresponding */ +/* consecutive column indices. See the description of Z. */ +/* Also, see the description of DGEEV. */ +/* LAPACK C */ + if (info1 > 0) { +/* DGEEV failed to compute the eigenvalues and */ +/* eigenvectors of the Rayleigh quotient. */ + *info = 3; + return 0; + } + +/* <6> Compute the eigenvectors (if requested) and, */ +/* the residuals (if requested). */ + + if (wntvec || wntex) { + if (wntres) { + if (wntref) { +/* Here, if the refinement is requested, we have */ +/* A*U(:,1:K) already computed and stored in Z. */ +/* For the residuals, need Y = A * U(:,1;K) * W. */ + dgemm_("N", "N", m, k, k, &one, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zero, &y[y_offset], ldy); +/* Y(1:M,1:K) = Z(1:M,1:K) * W(1:K,1:K) ! INTRINSIC */ +/* This frees Z; Y contains A * U(:,1:K) * W. */ +/* BLAS CALL */ + } else { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) is stored in Z */ + dgemm_(t_or_n__, "N", n, k, k, &one, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + dgemm_("N", "N", m, k, n, &one, &y[y_offset], ldy, &s[ + s_offset], lds, &zero, &z__[z_offset], ldz); +/* Save a copy of Z into Y and free Z for holding */ +/* the Ritz vectors. */ + dlacpy_("A", m, k, &z__[z_offset], ldz, &y[y_offset], ldy); + if (wntex) { + dlacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); + } + } + } else if (wntex) { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) is stored in Z */ + dgemm_(t_or_n__, "N", n, k, k, &one, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + dgemm_("N", "N", m, k, n, &one, &y[y_offset], ldy, &s[s_offset], + lds, &zero, &b[b_offset], ldb); +/* The above call replaces the following two calls */ +/* that were used in the developing-testing phase. */ +/* CALL DGEMM( 'N', 'N', M, K, N, ONE, Y, LDY, S, & */ +/* LDS, ZERO, Z, LDZ) */ +/* Save a copy of Z into B and free Z for holding */ +/* the Ritz vectors. */ +/* CALL DLACPY( 'A', M, K, Z, LDZ, B, LDB ) */ + } + +/* Compute the real form of the Ritz vectors */ + if (wntvec) { + dgemm_("N", "N", m, k, k, &one, &x[x_offset], ldx, &w[w_offset], + ldw, &zero, &z__[z_offset], ldz); + } +/* Z(1:M,1:K) = MATMUL(X(1:M,1:K), W(1:K,1:K)) ! INTRINSIC */ + +/* BLAS CALL */ + if (wntres) { + i__ = 1; + while(i__ <= *k) { + if (imeig[i__] == zero) { +/* have a real eigenvalue with real eigenvector */ + d__1 = -reig[i__]; + daxpy_(m, &d__1, &z__[i__ * z_dim1 + 1], &c__1, &y[i__ * + y_dim1 + 1], &c__1); +/* Y(1:M,i) = Y(1:M,i) - REIG(i) * Z(1:M,i) ! */ + + res[i__] = dnrm2_(m, &y[i__ * y_dim1 + 1], &c__1); + + ++i__; + } else { +/* Have a complex conjugate pair */ +/* REIG(i) +- sqrt(-1)*IMEIG(i). */ +/* Since all computation is done in real */ +/* arithmetic, the formula for the residual */ +/* is recast for real representation of the */ +/* complex conjugate eigenpair. See the */ +/* description of RES. */ + ab[0] = reig[i__]; + ab[1] = -imeig[i__]; + ab[2] = imeig[i__]; + ab[3] = reig[i__]; + d__1 = -one; + dgemm_("N", "N", m, &c__2, &c__2, &d__1, &z__[i__ * + z_dim1 + 1], ldz, ab, &c__2, &one, &y[i__ * + y_dim1 + 1], ldy); +/* Y(1:M,i:i+1) = Y(1:M,i:i+1) - Z(1:M,i:i+1) * AB ! INT */ +/* BL */ + res[i__] = dlange_("F", m, &c__2, &y[i__ * y_dim1 + 1], + ldy, &work[*n + 1]); +/* LA */ + res[i__ + 1] = res[i__]; + i__ += 2; + } + } + } + } + + if (*whtsvd == 4) { + work[*n + 1] = xscl1; + work[*n + 2] = xscl2; + } + +/* Successful exit. */ + if (! badxy) { + *info = 0; + } else { +/* A warning on possible data inconsistency. */ +/* This should be a rare event. */ + *info = 4; + } +/* ............................................................ */ + return 0; +/* ...... */ +} /* dgedmd_ */ + diff --git a/lapack-netlib/SRC/dgedmdq.c b/lapack-netlib/SRC/dgedmdq.c index 447b23014..a743a3156 100644 --- a/lapack-netlib/SRC/dgedmdq.c +++ b/lapack-netlib/SRC/dgedmdq.c @@ -509,3 +509,792 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; + +/* Subroutine */ int dgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq, + char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n, + doublereal *f, integer *ldf, doublereal *x, integer *ldx, doublereal * + y, integer *ldy, integer *nrnk, doublereal *tol, integer *k, + doublereal *reig, doublereal *imeig, doublereal *z__, integer *ldz, + doublereal *res, doublereal *b, integer *ldb, doublereal *v, integer * + ldv, doublereal *s, integer *lds, doublereal *work, integer *lwork, + integer *iwork, integer *liwork, integer *info) +{ + /* System generated locals */ + integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1, + z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset, + i__1, i__2; + + /* Local variables */ + doublereal zero; + integer info1; + extern logical lsame_(char *, char *); + char jobvl[1]; + integer minmn; + logical wantq; + integer mlwqr, olwqr; + logical wntex; + extern /* Subroutine */ int dgedmd_(char *, char *, char *, char *, + integer *, integer *, integer *, doublereal *, integer *, + doublereal *, integer *, integer *, doublereal *, integer *, + doublereal *, doublereal *, doublereal *, integer *, doublereal *, + doublereal *, integer *, doublereal *, integer *, doublereal *, + integer *, doublereal *, integer *, integer *, integer *, integer + *), dgeqrf_(integer *, integer *, + doublereal *, integer *, doublereal *, doublereal *, integer *, + integer *), dlacpy_(char *, integer *, integer *, doublereal *, + integer *, doublereal *, integer *), dlaset_(char *, + integer *, integer *, doublereal *, doublereal *, doublereal *, + integer *), xerbla_(char *, integer *); + integer mlwdmd, olwdmd; + logical sccolx, sccoly; + extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *, + doublereal *, integer *, doublereal *, doublereal *, integer *, + integer *), dormqr_(char *, char *, integer *, integer *, integer + *, doublereal *, integer *, doublereal *, doublereal *, integer *, + doublereal *, integer *, integer *); + integer iminwr; + logical wntvec, wntvcf; + integer mlwgqr; + logical wntref; + integer mlwork, olwgqr, olwork; + doublereal rdummy[2]; + integer mlwmqr, olwmqr; + logical lquery, wntres, wnttrf, wntvcq; + doublereal one; + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real64 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ..... */ +/* Purpose */ +/* ======= */ +/* DGEDMDQ computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices, using a QR factorization */ +/* based compression of the data. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, DGEDMDQ computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, DGEDMDQ returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office. */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ...................................................................... */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. The data snapshots are the columns */ +/* of F. The leading N-1 columns of F are denoted X and the */ +/* trailing N-1 columns are denoted Y. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Z*V, where Z */ +/* is orthonormal and V contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of F, V, Z. */ +/* 'Q' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Q*Z, where Z */ +/* contains the eigenvectors of the compression of the */ +/* underlying discretized operator onto the span of */ +/* the data snapshots. See the descriptions of F, V, Z. */ +/* Q is from the initial QR factorization. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will */ +/* be computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBQ (input) CHARACTER*1 */ +/* Specifies whether to explicitly compute and return the */ +/* orthogonal matrix from the QR factorization. */ +/* 'Q' :: The matrix Q of the QR factorization of the data */ +/* snapshot matrix is computed and stored in the */ +/* array F. See the description of F. */ +/* 'N' :: The matrix Q is not explicitly computed. */ +/* ..... */ +/* JOBT (input) CHARACTER*1 */ +/* Specifies whether to return the upper triangular factor */ +/* from the QR factorization. */ +/* 'R' :: The matrix R of the QR factorization of the data */ +/* snapshot matrix F is returned in the array Y. */ +/* See the description of Y and Further details. */ +/* 'N' :: The matrix R is not returned. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* To be useful on exit, this option needs JOBQ='Q'. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: DGESVD (the QR SVD algorithm) */ +/* 2 :: DGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: DGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: DGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M >= 0 */ +/* The state space dimension (the number of rows of F). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshots from a single trajectory, */ +/* taken at equidistant discrete times. This is the */ +/* number of columns of F. */ +/* ..... */ +/* F (input/output) REAL(KIND=WP) M-by-N array */ +/* > On entry, */ +/* the columns of F are the sequence of data snapshots */ +/* from a single trajectory, taken at equidistant discrete */ +/* times. It is assumed that the column norms of F are */ +/* in the range of the normalized floating point numbers. */ +/* < On exit, */ +/* If JOBQ == 'Q', the array F contains the orthogonal */ +/* matrix/factor of the QR factorization of the initial */ +/* data snapshots matrix F. See the description of JOBQ. */ +/* If JOBQ == 'N', the entries in F strictly below the main */ +/* diagonal contain, column-wise, the information on the */ +/* Householder vectors, as returned by DGEQRF. The */ +/* remaining information to restore the orthogonal matrix */ +/* of the initial QR factorization is stored in WORK(1:N). */ +/* See the description of WORK. */ +/* ..... */ +/* LDF (input) INTEGER, LDF >= M */ +/* The leading dimension of the array F. */ +/* ..... */ +/* X (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array */ +/* X is used as workspace to hold representations of the */ +/* leading N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, the leading K columns of X contain the leading */ +/* K left singular vectors of the above described content */ +/* of X. To lift them to the space of the left singular */ +/* vectors U(:,1:K)of the input data, pre-multiply with the */ +/* Q factor from the initial QR factorization. */ +/* See the descriptions of F, K, V and Z. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= N */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array */ +/* Y is used as workspace to hold representations of the */ +/* trailing N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, */ +/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */ +/* triangular factor from the QR factorization of the data */ +/* snapshot matrix F. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= N */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N-1 :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the description of K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the SVD/POD basis for the leading N-1 */ +/* data snapshots (columns of F) and the number of the */ +/* computed Ritz pairs. The value of K is determined */ +/* according to the rule set by the parameters NRNK and */ +/* TOL. See the descriptions of NRNK and TOL. */ +/* ..... */ +/* REIG (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* The leading K (K<=N) entries of REIG contain */ +/* the real parts of the computed eigenvalues */ +/* REIG(1:K) + sqrt(-1)*IMEIG(1:K). */ +/* See the descriptions of K, IMEIG, Z. */ +/* ..... */ +/* IMEIG (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* The leading K (K0, then the corresponding complex */ +/* conjugate pair of eigenvalues reads */ +/* LAMBDA(i) = REIG(i) + sqrt(-1)*IMAG(i) */ +/* LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i) */ +/* That is, complex conjugate pairs have consequtive */ +/* indices (i,i+1), with the positive imaginary part */ +/* listed first. */ +/* See the descriptions of K, REIG, Z. */ +/* ..... */ +/* Z (workspace/output) REAL(KIND=WP) M-by-(N-1) array */ +/* If JOBZ =='V' then */ +/* Z contains real Ritz vectors as follows: */ +/* If IMEIG(i)=0, then Z(:,i) is an eigenvector of */ +/* the i-th Ritz value. */ +/* If IMEIG(i) > 0 (and IMEIG(i+1) < 0) then */ +/* [Z(:,i) Z(:,i+1)] span an invariant subspace and */ +/* the Ritz values extracted from this subspace are */ +/* REIG(i) + sqrt(-1)*IMEIG(i) and */ +/* REIG(i) - sqrt(-1)*IMEIG(i). */ +/* The corresponding eigenvectors are */ +/* Z(:,i) + sqrt(-1)*Z(:,i+1) and */ +/* Z(:,i) - sqrt(-1)*Z(:,i+1), respectively. */ +/* If JOBZ == 'F', then the above descriptions hold for */ +/* the columns of Z*V, where the columns of V are the */ +/* eigenvectors of the K-by-K Rayleigh quotient, and Z is */ +/* orthonormal. The columns of V are similarly structured: */ +/* If IMEIG(i) == 0 then Z*V(:,i) is an eigenvector, and if */ +/* IMEIG(i) > 0 then Z*V(:,i)+sqrt(-1)*Z*V(:,i+1) and */ +/* Z*V(:,i)-sqrt(-1)*Z*V(:,i+1) */ +/* are the eigenvectors of LAMBDA(i), LAMBDA(i+1). */ +/* See the descriptions of REIG, IMEIG, X and V. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs. */ +/* If LAMBDA(i) is real, then */ +/* RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2. */ +/* If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair */ +/* then */ +/* RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F */ +/* where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ] */ +/* [-imag(LAMBDA(i)) real(LAMBDA(i)) ]. */ +/* It holds that */ +/* RES(i) = || A*ZC(:,i) - LAMBDA(i) *ZC(:,i) ||_2 */ +/* RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2 */ +/* where ZC(:,i) = Z(:,i) + sqrt(-1)*Z(:,i+1) */ +/* ZC(:,i+1) = Z(:,i) - sqrt(-1)*Z(:,i+1) */ +/* See the description of Z. */ +/* ..... */ +/* B (output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array. */ +/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:N,1;K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* In both cases, the content of B can be lifted to the */ +/* original dimension of the input data by pre-multiplying */ +/* with the Q factor from the initial QR factorization. */ +/* Here A denotes a compression of the underlying operator. */ +/* See the descriptions of F and X. */ +/* If JOBF =='N', then B is not referenced. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= MIN(M,N) */ +/* The leading dimension of the array B. */ +/* ..... */ +/* V (workspace/output) REAL(KIND=WP) (N-1)-by-(N-1) array */ +/* On exit, V(1:K,1:K) contains the K eigenvectors of */ +/* the Rayleigh quotient. The eigenvectors of a complex */ +/* conjugate pair of eigenvalues are returned in real form */ +/* as explained in the description of Z. The Ritz vectors */ +/* (returned in Z) are the product of X and V; see */ +/* the descriptions of X and Z. */ +/* ..... */ +/* LDV (input) INTEGER, LDV >= N-1 */ +/* The leading dimension of the array V. */ +/* ..... */ +/* S (output) REAL(KIND=WP) (N-1)-by-(N-1) array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by DGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N-1 */ +/* The leading dimension of the array S. */ +/* ..... */ +/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */ +/* On exit, */ +/* WORK(1:MIN(M,N)) contains the scalar factors of the */ +/* elementary reflectors as returned by DGEQRF of the */ +/* M-by-N input matrix F. */ +/* WORK(MIN(M,N)+1:MIN(M,N)+N-1) contains the singular values of */ +/* the input submatrix F(1:M,1:N-1). */ +/* If the call to DGEDMDQ is only workspace query, then */ +/* WORK(1) contains the minimal workspace length and */ +/* WORK(2) is the optimal workspace length. Hence, the */ +/* length of work is at least 2. */ +/* See the description of LWORK. */ +/* ..... */ +/* LWORK (input) INTEGER */ +/* The minimal length of the workspace vector WORK. */ +/* LWORK is calculated as follows: */ +/* Let MLWQR = N (minimal workspace for DGEQRF[M,N]) */ +/* MLWDMD = minimal workspace for DGEDMD (see the */ +/* description of LWORK in DGEDMD) for */ +/* snapshots of dimensions MIN(M,N)-by-(N-1) */ +/* MLWMQR = N (minimal workspace for */ +/* DORMQR['L','N',M,N,N]) */ +/* MLWGQR = N (minimal workspace for DORGQR[M,N,N]) */ +/* Then */ +/* LWORK = MAX(N+MLWQR, N+MLWDMD) */ +/* is updated as follows: */ +/* if JOBZ == 'V' or JOBZ == 'F' THEN */ +/* LWORK = MAX( LWORK, MIN(M,N)+N-1+MLWMQR ) */ +/* if JOBQ == 'Q' THEN */ +/* LWORK = MAX( LWORK, MIN(M,N)+N-1+MLWGQR) */ +/* If on entry LWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* Let M1=MIN(M,N), N1=N-1. Then */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ + +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ + +/* Local array */ +/* ~~~~~~~~~~~ */ + +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ + +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* External subroutines */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* .......................................................... */ + /* Parameter adjustments */ + f_dim1 = *ldf; + f_offset = 1 + f_dim1 * 1; + f -= f_offset; + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --reig; + --imeig; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + v_dim1 = *ldv; + v_offset = 1 + v_dim1 * 1; + v -= v_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --work; + --iwork; + + /* Function Body */ + zero = 0.f; + one = 1.f; + +/* Test the input arguments */ + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntvcf = lsame_(jobz, "F"); + wntvcq = lsame_(jobz, "Q"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + wantq = lsame_(jobq, "Q"); + wnttrf = lsame_(jobt, "R"); + minmn = f2cmin(*m,*n); + *info = 0; + lquery = *lwork == -1 || *liwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && lsame_(jobz, "N")) { + *info = -3; + } else if (! (wantq || lsame_(jobq, "N"))) { + *info = -4; + } else if (! (wnttrf || lsame_(jobt, "N"))) { + *info = -5; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -6; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -7; + } else if (*m < 0) { + *info = -8; + } else if (*n < 0 || *n > *m + 1) { + *info = -9; + } else if (*ldf < *m) { + *info = -11; + } else if (*ldx < minmn) { + *info = -13; + } else if (*ldy < minmn) { + *info = -15; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -16; + } else if (*tol < zero || *tol >= one) { + *info = -17; + } else if (*ldz < *m) { + *info = -22; + } else if ((wntref || wntex) && *ldb < minmn) { + *info = -25; + } else if (*ldv < *n - 1) { + *info = -27; + } else if (*lds < *n - 1) { + *info = -29; + } + + if (wntvec || wntvcf || wntvcq) { + *(unsigned char *)jobvl = 'V'; + } else { + *(unsigned char *)jobvl = 'N'; + } + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0 || *n == 1) { +/* All output except K is void. INFO=1 signals */ +/* the void input. In case of a workspace query, */ +/* the minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + work[1] = 2.; + work[2] = 2.; + } else { + *k = 0; + } + *info = 1; + return 0; + } + mlwqr = f2cmax(1,*n); +/* Minimal workspace length for DGEQRF. */ + mlwork = minmn + mlwqr; + if (lquery) { + dgeqrf_(m, n, &f[f_offset], ldf, &work[1], rdummy, &c_n1, &info1); + olwqr = (integer) rdummy[0]; + olwork = f2cmin(*m,*n) + olwqr; + } + i__1 = *n - 1; + dgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], + ldx, &y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], & + z__[z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset], + ldv, &s[s_offset], lds, &work[1], &c_n1, &iwork[1], liwork, & + info1); + mlwdmd = (integer) work[1]; +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwdmd; + mlwork = f2cmax(i__1,i__2); + iminwr = iwork[1]; + if (lquery) { + olwdmd = (integer) work[2]; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwdmd; + olwork = f2cmax(i__1,i__2); + } + if (wntvec || wntvcf) { + mlwmqr = f2cmax(1,*n); +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + *n - 1 + mlwmqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + dormqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &work[1], & + z__[z_offset], ldz, &work[1], &c_n1, &info1); + olwmqr = (integer) work[1]; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + *n - 1 + olwmqr; + olwork = f2cmax(i__1,i__2); + } + } + if (wantq) { + mlwgqr = *n; +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + *n - 1 + mlwgqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + dorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[ + 1], &c_n1, &info1); + olwgqr = (integer) work[1]; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + *n - 1 + olwgqr; + olwork = f2cmax(i__1,i__2); + } + } + iminwr = f2cmax(1,iminwr); + mlwork = f2cmax(2,mlwork); + if (*lwork < mlwork && ! lquery) { + *info = -31; + } + if (*liwork < iminwr && ! lquery) { + *info = -33; + } + } + if (*info != 0) { + i__1 = -(*info); + xerbla_("DGEDMDQ", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + work[1] = (doublereal) mlwork; + work[2] = (doublereal) olwork; + return 0; + } +/* ..... */ +/* Initial QR factorization that is used to represent the */ +/* snapshots as elements of lower dimensional subspace. */ +/* For large scale computation with M >>N , at this place */ +/* one can use an out of core QRF. */ + + i__1 = *lwork - minmn; + dgeqrf_(m, n, &f[f_offset], ldf, &work[1], &work[minmn + 1], &i__1, & + info1); + +/* Define X and Y as the snapshots representations in the */ +/* orthogonal basis computed in the QR factorization. */ +/* X corresponds to the leading N-1 and Y to the trailing */ +/* N-1 snapshots. */ + i__1 = *n - 1; + dlaset_("L", &minmn, &i__1, &zero, &zero, &x[x_offset], ldx); + i__1 = *n - 1; + dlacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx); + i__1 = *n - 1; + dlacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy); + if (*m >= 3) { + i__1 = minmn - 2; + i__2 = *n - 2; + dlaset_("L", &i__1, &i__2, &zero, &zero, &y[y_dim1 + 3], ldy); + } + +/* Compute the DMD of the projected snapshot pairs (X,Y) */ + i__1 = *n - 1; + i__2 = *lwork - minmn; + dgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx, + &y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], &z__[ + z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[ + s_offset], lds, &work[minmn + 1], &i__2, &iwork[1], liwork, & + info1); + if (info1 == 2 || info1 == 3) { +/* Return with error code. See DGEDMD for details. */ + *info = info1; + return 0; + } else { + *info = info1; + } + +/* The Ritz vectors (Koopman modes) can be explicitly */ +/* formed or returned in factored form. */ + if (wntvec) { +/* Compute the eigenvectors explicitly. */ + if (*m > minmn) { + i__1 = *m - minmn; + dlaset_("A", &i__1, k, &zero, &zero, &z__[minmn + 1 + z_dim1], + ldz); + } + i__1 = *lwork - (minmn + *n - 1); + dormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[ + z_offset], ldz, &work[minmn + *n], &i__1, &info1); + } else if (wntvcf) { +/* Return the Ritz vectors (eigenvectors) in factored */ +/* form Z*V, where Z contains orthonormal matrix (the */ +/* product of Q from the initial QR factorization and */ +/* the SVD/POD_basis returned by DGEDMD in X) and the */ +/* second factor (the eigenvectors of the Rayleigh */ +/* quotient) is in the array V, as returned by DGEDMD. */ + dlacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz); + if (*m > *n) { + i__1 = *m - *n; + dlaset_("A", &i__1, k, &zero, &zero, &z__[*n + 1 + z_dim1], ldz); + } + i__1 = *lwork - (minmn + *n - 1); + dormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[ + z_offset], ldz, &work[minmn + *n], &i__1, &info1); + } + +/* Some optional output variables: */ + +/* The upper triangular factor R in the initial QR */ +/* factorization is optionally returned in the array Y. */ +/* This is useful if this call to DGEDMDQ is to be */ +/* followed by a streaming DMD that is implemented in a */ +/* QR compressed form. */ + if (wnttrf) { +/* Return the upper triangular R in Y */ + dlaset_("A", &minmn, n, &zero, &zero, &y[y_offset], ldy); + dlacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy); + } + +/* The orthonormal/orthogonal factor Q in the initial QR */ +/* factorization is optionally returned in the array F. */ +/* Same as with the triangular factor above, this is */ +/* useful in a streaming DMD. */ + if (wantq) { +/* Q overwrites F */ + i__1 = *lwork - (minmn + *n - 1); + dorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[minmn + + *n], &i__1, &info1); + } + + return 0; + +} /* dgedmdq_ */ + diff --git a/lapack-netlib/SRC/sgedmd.c b/lapack-netlib/SRC/sgedmd.c index 447b23014..c8f3a5964 100644 --- a/lapack-netlib/SRC/sgedmd.c +++ b/lapack-netlib/SRC/sgedmd.c @@ -509,3 +509,1238 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; +static integer c__1 = 1; +static integer c__0 = 0; +static integer c__2 = 2; + +/* Subroutine */ int sgedmd_(char *jobs, char *jobz, char *jobr, char *jobf, + integer *whtsvd, integer *m, integer *n, real *x, integer *ldx, real * + y, integer *ldy, integer *nrnk, real *tol, integer *k, real *reig, + real *imeig, real *z__, integer *ldz, real *res, real *b, integer * + ldb, real *w, integer *ldw, real *s, integer *lds, real *work, + integer *lwork, integer *iwork, integer *liwork, integer *info) +{ + /* System generated locals */ + integer x_dim1, x_offset, y_dim1, y_offset, z_dim1, z_offset, b_dim1, + b_offset, w_dim1, w_offset, s_dim1, s_offset, i__1, i__2; + real r__1, r__2; + + /* Local variables */ + real zero, ssum; + integer info1, info2; + real xscl1, xscl2; + extern real snrm2_(integer *, real *, integer *); + integer i__, j; + real scale; + extern logical lsame_(char *, char *); + extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); + logical badxy; + real small; + extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *, + integer *, real *, real *, integer *, real *, integer *, real *, + real *, integer *), sgeev_(char *, char *, + integer *, real *, integer *, real *, real *, real *, integer *, + real *, integer *, real *, integer *, integer *); + char jobzl[1]; + extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, + real *, integer *); + logical wntex; + real ab[4] /* was [2][2] */; + extern real slamch_(char *), slange_(char *, integer *, integer *, + real *, integer *, real *); + extern /* Subroutine */ int sgesdd_(char *, integer *, integer *, real *, + integer *, real *, real *, integer *, real *, integer *, real *, + integer *, integer *, integer *), xerbla_(char *, integer + *); + char t_or_n__[1]; + extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, + real *, integer *, integer *, real *, integer *, integer *); + extern integer isamax_(integer *, real *, integer *); + logical sccolx, sccoly; + extern logical sisnan_(real *); + extern /* Subroutine */ int sgesvd_(char *, char *, integer *, integer *, + real *, integer *, real *, real *, integer *, real *, integer *, + real *, integer *, integer *); + integer lwrsdd, mwrsdd; + extern /* Subroutine */ int sgejsv_(char *, char *, char *, char *, char * + , char *, integer *, integer *, real *, integer *, real *, real *, + integer *, real *, integer *, real *, integer *, integer *, + integer *), + slacpy_(char *, integer *, integer *, real *, integer *, real *, + integer *); + integer iminwr; + logical wntref, wntvec; + real rootsc; + integer lwrkev, mlwork, mwrkev, numrnk, olwork; + real rdummy[2]; + integer lwrsvd, mwrsvd; + logical lquery, wntres; + char jsvopt[1]; + extern /* Subroutine */ int slassq_(integer *, real *, integer *, real *, + real *), mecago_(); + integer mwrsvj, lwrsvq, mwrsvq; + real rdummy2[2], ofl, one; + extern /* Subroutine */ int sgesvdq_(char *, char *, char *, char *, char + *, integer *, integer *, real *, integer *, real *, real *, + integer *, real *, integer *, integer *, integer *, integer *, + real *, integer *, real *, integer *, integer *); + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real32 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ............................................................ */ +/* Purpose */ +/* ======= */ +/* SGEDMD computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, SGEDMD computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, SGEDMD returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* ...................................................................... */ +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ...................................................................... */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product X(:,1:K)*W, where X */ +/* contains a POD basis (leading left singular vectors */ +/* of the data matrix X) and W contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of K, X, W, Z. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will be */ +/* computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: SGESVD (the QR SVD algorithm) */ +/* 2 :: SGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: SGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: SGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M>= 0 */ +/* The state space dimension (the row dimension of X, Y). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshot pairs */ +/* (the number of columns of X and Y). */ +/* ..... */ +/* X (input/output) REAL(KIND=WP) M-by-N array */ +/* > On entry, X contains the data snapshot matrix X. It is */ +/* assumed that the column norms of X are in the range of */ +/* the normalized floating point numbers. */ +/* < On exit, the leading K columns of X contain a POD basis, */ +/* i.e. the leading K left singular vectors of the input */ +/* data matrix X, U(:,1:K). All N columns of X contain all */ +/* left singular vectors of the input matrix X. */ +/* See the descriptions of K, Z and W. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= M */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (input/workspace/output) REAL(KIND=WP) M-by-N array */ +/* > On entry, Y contains the data snapshot matrix Y */ +/* < On exit, */ +/* If JOBR == 'R', the leading K columns of Y contain */ +/* the residual vectors for the computed Ritz pairs. */ +/* See the description of RES. */ +/* If JOBR == 'N', Y contains the original input data, */ +/* scaled according to the value of JOBS. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= M */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the descriptions of TOL and K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the POD basis for the data snapshot */ +/* matrix X and the number of the computed Ritz pairs. */ +/* The value of K is determined according to the rule set */ +/* by the parameters NRNK and TOL. */ +/* See the descriptions of NRNK and TOL. */ +/* ..... */ +/* REIG (output) REAL(KIND=WP) N-by-1 array */ +/* The leading K (K<=N) entries of REIG contain */ +/* the real parts of the computed eigenvalues */ +/* REIG(1:K) + sqrt(-1)*IMEIG(1:K). */ +/* See the descriptions of K, IMEIG, and Z. */ +/* ..... */ +/* IMEIG (output) REAL(KIND=WP) N-by-1 array */ +/* The leading K (K<=N) entries of IMEIG contain */ +/* the imaginary parts of the computed eigenvalues */ +/* REIG(1:K) + sqrt(-1)*IMEIG(1:K). */ +/* The eigenvalues are determined as follows: */ +/* If IMEIG(i) == 0, then the corresponding eigenvalue is */ +/* real, LAMBDA(i) = REIG(i). */ +/* If IMEIG(i)>0, then the corresponding complex */ +/* conjugate pair of eigenvalues reads */ +/* LAMBDA(i) = REIG(i) + sqrt(-1)*IMAG(i) */ +/* LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i) */ +/* That is, complex conjugate pairs have consecutive */ +/* indices (i,i+1), with the positive imaginary part */ +/* listed first. */ +/* See the descriptions of K, REIG, and Z. */ +/* ..... */ +/* Z (workspace/output) REAL(KIND=WP) M-by-N array */ +/* If JOBZ =='V' then */ +/* Z contains real Ritz vectors as follows: */ +/* If IMEIG(i)=0, then Z(:,i) is an eigenvector of */ +/* the i-th Ritz value; ||Z(:,i)||_2=1. */ +/* If IMEIG(i) > 0 (and IMEIG(i+1) < 0) then */ +/* [Z(:,i) Z(:,i+1)] span an invariant subspace and */ +/* the Ritz values extracted from this subspace are */ +/* REIG(i) + sqrt(-1)*IMEIG(i) and */ +/* REIG(i) - sqrt(-1)*IMEIG(i). */ +/* The corresponding eigenvectors are */ +/* Z(:,i) + sqrt(-1)*Z(:,i+1) and */ +/* Z(:,i) - sqrt(-1)*Z(:,i+1), respectively. */ +/* || Z(:,i:i+1)||_F = 1. */ +/* If JOBZ == 'F', then the above descriptions hold for */ +/* the columns of X(:,1:K)*W(1:K,1:K), where the columns */ +/* of W(1:k,1:K) are the computed eigenvectors of the */ +/* K-by-K Rayleigh quotient. The columns of W(1:K,1:K) */ +/* are similarly structured: If IMEIG(i) == 0 then */ +/* X(:,1:K)*W(:,i) is an eigenvector, and if IMEIG(i)>0 */ +/* then X(:,1:K)*W(:,i)+sqrt(-1)*X(:,1:K)*W(:,i+1) and */ +/* X(:,1:K)*W(:,i)-sqrt(-1)*X(:,1:K)*W(:,i+1) */ +/* are the eigenvectors of LAMBDA(i), LAMBDA(i+1). */ +/* See the descriptions of REIG, IMEIG, X and W. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) N-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs. */ +/* If LAMBDA(i) is real, then */ +/* RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2. */ +/* If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair */ +/* then */ +/* RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F */ +/* where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ] */ +/* [-imag(LAMBDA(i)) real(LAMBDA(i)) ]. */ +/* It holds that */ +/* RES(i) = || A*ZC(:,i) - LAMBDA(i) *ZC(:,i) ||_2 */ +/* RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2 */ +/* where ZC(:,i) = Z(:,i) + sqrt(-1)*Z(:,i+1) */ +/* ZC(:,i+1) = Z(:,i) - sqrt(-1)*Z(:,i+1) */ +/* See the description of REIG, IMEIG and Z. */ +/* ..... */ +/* B (output) REAL(KIND=WP) M-by-N array. */ +/* IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:M,1;K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* If JOBF =='N', then B is not referenced. */ +/* See the descriptions of X, W, K. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= M */ +/* The leading dimension of the array B. */ +/* ..... */ +/* W (workspace/output) REAL(KIND=WP) N-by-N array */ +/* On exit, W(1:K,1:K) contains the K computed */ +/* eigenvectors of the matrix Rayleigh quotient (real and */ +/* imaginary parts for each complex conjugate pair of the */ +/* eigenvalues). The Ritz vectors (returned in Z) are the */ +/* product of X (containing a POD basis for the input */ +/* matrix X) and W. See the descriptions of K, S, X and Z. */ +/* W is also used as a workspace to temporarily store the */ +/* left singular vectors of X. */ +/* ..... */ +/* LDW (input) INTEGER, LDW >= N */ +/* The leading dimension of the array W. */ +/* ..... */ +/* S (workspace/output) REAL(KIND=WP) N-by-N array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by SGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N */ +/* The leading dimension of the array S. */ +/* ..... */ +/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */ +/* On exit, WORK(1:N) contains the singular values of */ +/* X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). */ +/* If WHTSVD==4, then WORK(N+1) and WORK(N+2) contain */ +/* scaling factor WORK(N+2)/WORK(N+1) used to scale X */ +/* and Y to avoid overflow in the SVD of X. */ +/* This may be of interest if the scaling option is off */ +/* and as many as possible smallest eigenvalues are */ +/* desired to the highest feasible accuracy. */ +/* If the call to SGEDMD is only workspace query, then */ +/* WORK(1) contains the minimal workspace length and */ +/* WORK(2) is the optimal workspace length. Hence, the */ +/* length of work is at least 2. */ +/* See the description of LWORK. */ +/* ..... */ +/* LWORK (input) INTEGER */ +/* The minimal length of the workspace vector WORK. */ +/* LWORK is calculated as follows: */ +/* If WHTSVD == 1 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)). */ +/* If JOBZ == 'N' then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)). */ +/* Here LWORK_SVD = MAX(1,3*N+M,5*N) is the minimal */ +/* workspace length of SGESVD. */ +/* If WHTSVD == 2 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,4*N)) */ +/* If JOBZ == 'N', then */ +/* LWORK >= MAX(2, N + LWORK_SVD, N+MAX(1,3*N)) */ +/* Here LWORK_SVD = MAX(M, 5*N*N+4*N)+3*N*N is the */ +/* minimal workspace length of SGESDD. */ +/* If WHTSVD == 3 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N)) */ +/* If JOBZ == 'N', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N)) */ +/* Here LWORK_SVD = N+M+MAX(3*N+1, */ +/* MAX(1,3*N+M,5*N),MAX(1,N)) */ +/* is the minimal workspace length of SGESVDQ. */ +/* If WHTSVD == 4 :: */ +/* If JOBZ == 'V', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,4*N)) */ +/* If JOBZ == 'N', then */ +/* LWORK >= MAX(2, N+LWORK_SVD,N+MAX(1,3*N)) */ +/* Here LWORK_SVD = MAX(7,2*M+N,6*N+2*N*N) is the */ +/* minimal workspace length of SGEJSV. */ +/* The above expressions are not simplified in order to */ +/* make the usage of WORK more transparent, and for */ +/* easier checking. In any case, LWORK >= 2. */ +/* If on entry LWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ +/* Local arrays */ +/* ~~~~~~~~~~~~ */ +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* ............................................................ */ + /* Parameter adjustments */ + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --reig; + --imeig; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + w_dim1 = *ldw; + w_offset = 1 + w_dim1 * 1; + w -= w_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --work; + --iwork; + + /* Function Body */ + one = 1.f; + zero = 0.f; + +/* Test the input arguments */ + + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + *info = 0; + lquery = *lwork == -1 || *liwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || lsame_(jobz, "N") || lsame_( + jobz, "F"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && ! wntvec) { + *info = -3; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -4; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -5; + } else if (*m < 0) { + *info = -6; + } else if (*n < 0 || *n > *m) { + *info = -7; + } else if (*ldx < *m) { + *info = -9; + } else if (*ldy < *m) { + *info = -11; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -12; + } else if (*tol < zero || *tol >= one) { + *info = -13; + } else if (*ldz < *m) { + *info = -18; + } else if ((wntref || wntex) && *ldb < *m) { + *info = -21; + } else if (*ldw < *n) { + *info = -23; + } else if (*lds < *n) { + *info = -25; + } + + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0) { +/* Quick return. All output except K is void. */ +/* INFO=1 signals the void input. */ +/* In case of a workspace query, the default */ +/* minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + work[1] = 2.f; + work[2] = 2.f; + } else { + *k = 0; + } + *info = 1; + return 0; + } + mlwork = f2cmax(2,*n); + olwork = f2cmax(2,*n); + iminwr = 1; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of SGESVD: */ +/* MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) */ +/* Computing MAX */ + i__1 = 1, i__2 = f2cmin(*m,*n) * 3 + f2cmax(*m,*n), i__1 = f2cmax(i__1, + i__2), i__2 = f2cmin(*m,*n) * 5; + mwrsvd = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsvd; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + sgesvd_("O", "S", m, n, &x[x_offset], ldx, &work[1], &b[ + b_offset], ldb, &w[w_offset], ldw, rdummy, &c_n1, & + info1); +/* Computing MAX */ + i__1 = mwrsvd, i__2 = (integer) rdummy[0]; + lwrsvd = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrsvd; + olwork = f2cmax(i__1,i__2); + } + } else if (*whtsvd == 2) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of SGESDD: */ +/* MWRSDD = 3*MIN(M,N)*MIN(M,N) + */ +/* MAX( MAX(M,N),5*MIN(M,N)*MIN(M,N)+4*MIN(M,N) ) */ +/* IMINWR = 8*MIN(M,N) */ +/* Computing MAX */ + i__1 = f2cmax(*m,*n), i__2 = f2cmin(*m,*n) * 5 * f2cmin(*m,*n) + (f2cmin(*m,* + n) << 2); + mwrsdd = f2cmin(*m,*n) * 3 * f2cmin(*m,*n) + f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsdd; + mlwork = f2cmax(i__1,i__2); + iminwr = f2cmin(*m,*n) << 3; + if (lquery) { + sgesdd_("O", m, n, &x[x_offset], ldx, &work[1], &b[b_offset], + ldb, &w[w_offset], ldw, rdummy, &c_n1, &iwork[1], & + info1); +/* Computing MAX */ + i__1 = mwrsdd, i__2 = (integer) rdummy[0]; + lwrsdd = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrsdd; + olwork = f2cmax(i__1,i__2); + } + } else if (*whtsvd == 3) { +/* LWQP3 = 3*N+1 */ +/* LWORQ = MAX(N, 1) */ +/* MWRSVD = MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) */ +/* MWRSVQ = N + MAX( LWQP3, MWRSVD, LWORQ )+ MAX(M,2) */ +/* MLWORK = N + MWRSVQ */ +/* IMINWR = M+N-1 */ + sgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &work[ + 1], &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, & + iwork[1], &c_n1, rdummy, &c_n1, rdummy2, &c_n1, &info1); + iminwr = iwork[1]; + mwrsvq = (integer) rdummy[1]; +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsvq + (integer) rdummy2[0]; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + lwrsvq = (integer) rdummy[0]; +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrsvq + (integer) rdummy2[0]; + olwork = f2cmax(i__1,i__2); + } + } else if (*whtsvd == 4) { + *(unsigned char *)jsvopt = 'J'; +/* MWRSVJ = MAX( 7, 2*M+N, 6*N+2*N*N )! for JSVOPT='V' */ +/* Computing MAX */ + i__1 = 7, i__2 = (*m << 1) + *n, i__1 = f2cmax(i__1,i__2), i__2 = (* + n << 2) + *n * *n, i__1 = f2cmax(i__1,i__2), i__2 = (*n << 1) + + *n * *n + 6; + mwrsvj = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrsvj; + mlwork = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = 3, i__2 = *m + *n * 3; + iminwr = f2cmax(i__1,i__2); + if (lquery) { +/* Computing MAX */ + i__1 = olwork, i__2 = *n + mwrsvj; + olwork = f2cmax(i__1,i__2); + } + } +/* END SELECT */ + if (wntvec || wntex || lsame_(jobz, "F")) { + *(unsigned char *)jobzl = 'V'; + } else { + *(unsigned char *)jobzl = 'N'; + } +/* Workspace calculation to the SGEEV call */ + if (lsame_(jobzl, "V")) { +/* Computing MAX */ + i__1 = 1, i__2 = *n << 2; + mwrkev = f2cmax(i__1,i__2); + } else { +/* Computing MAX */ + i__1 = 1, i__2 = *n * 3; + mwrkev = f2cmax(i__1,i__2); + } +/* Computing MAX */ + i__1 = mlwork, i__2 = *n + mwrkev; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + sgeev_("N", jobzl, n, &s[s_offset], lds, &reig[1], &imeig[1], &w[ + w_offset], ldw, &w[w_offset], ldw, rdummy, &c_n1, &info1); +/* Computing MAX */ + i__1 = mwrkev, i__2 = (integer) rdummy[0]; + lwrkev = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = olwork, i__2 = *n + lwrkev; + olwork = f2cmax(i__1,i__2); + } + + if (*liwork < iminwr && ! lquery) { + *info = -29; + } + if (*lwork < mlwork && ! lquery) { + *info = -27; + } + } + + if (*info != 0) { + i__1 = -(*info); + xerbla_("SGEDMD", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + work[1] = (real) mlwork; + work[2] = (real) olwork; + return 0; + } +/* ............................................................ */ + + ofl = slamch_("O"); + small = slamch_("S"); + badxy = FALSE_; + +/* <1> Optional scaling of the snapshots (columns of X, Y) */ +/* ========================================================== */ + if (sccolx) { +/* The columns of X will be normalized. */ +/* To prevent overflows, the column norms of X are */ +/* carefully computed using SLASSQ. */ + *k = 0; + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* WORK(i) = DNRM2( M, X(1,i), 1 ) */ + scale = zero; + slassq_(m, &x[i__ * x_dim1 + 1], &c__1, &scale, &ssum); + if (sisnan_(&scale) || sisnan_(&ssum)) { + *k = 0; + *info = -8; + i__2 = -(*info); + xerbla_("SGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of X(:,i) overflows. First, X(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||X(:,i)||_2. */ +/* Next, the norm of X(:,i) is stored without */ +/* overflow as WORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of X(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + r__1 = one / rootsc; + slascl_("G", &c__0, &c__0, &scale, &r__1, m, &c__1, &x[ + i__ * x_dim1 + 1], m, &info2); + work[i__] = -scale * (rootsc / (real) (*m)); + } else { +/* X(:,i) will be scaled to unit 2-norm */ + work[i__] = scale * rootsc; + slascl_("G", &c__0, &c__0, &work[i__], &one, m, &c__1, &x[ + i__ * x_dim1 + 1], m, &info2); +/* X(1:M,i) = (ONE/WORK(i)) * X(1:M,i) ! INTRINSIC */ +/* LAPACK */ + } + } else { + work[i__] = zero; + ++(*k); + } + } + if (*k == *n) { +/* All columns of X are zero. Return error code -8. */ +/* (the 8th input variable had an illegal value) */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("SGEDMD", &i__1); + return 0; + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of Y. */ + if (work[i__] > zero) { + r__1 = one / work[i__]; + sscal_(m, &r__1, &y[i__ * y_dim1 + 1], &c__1); +/* Y(1:M,i) = (ONE/WORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (work[i__] < zero) { + r__1 = -work[i__]; + r__2 = one / (real) (*m); + slascl_("G", &c__0, &c__0, &r__1, &r__2, m, &c__1, &y[i__ * + y_dim1 + 1], m, &info2); +/* LAPACK CA */ + } else if (y[isamax_(m, &y[i__ * y_dim1 + 1], &c__1) + i__ * + y_dim1] != zero) { +/* X(:,i) is zero vector. For consistency, */ +/* Y(:,i) should also be zero. If Y(:,i) is not */ +/* zero, then the data might be inconsistent or */ +/* corrupted. If JOBS == 'C', Y(:,i) is set to */ +/* zero and a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + if (lsame_(jobs, "C")) { + sscal_(m, &zero, &y[i__ * y_dim1 + 1], &c__1); + } +/* BLAS CALL */ + } + } + } + + if (sccoly) { +/* The columns of Y will be normalized. */ +/* To prevent overflows, the column norms of Y are */ +/* carefully computed using SLASSQ. */ + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* WORK(i) = DNRM2( M, Y(1,i), 1 ) */ + scale = zero; + slassq_(m, &y[i__ * y_dim1 + 1], &c__1, &scale, &ssum); + if (sisnan_(&scale) || sisnan_(&ssum)) { + *k = 0; + *info = -10; + i__2 = -(*info); + xerbla_("SGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of Y(:,i) overflows. First, Y(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||Y(:,i)||_2. */ +/* Next, the norm of Y(:,i) is stored without */ +/* overflow as WORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of Y(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + r__1 = one / rootsc; + slascl_("G", &c__0, &c__0, &scale, &r__1, m, &c__1, &y[ + i__ * y_dim1 + 1], m, &info2); + work[i__] = -scale * (rootsc / (real) (*m)); + } else { +/* X(:,i) will be scaled to unit 2-norm */ + work[i__] = scale * rootsc; + slascl_("G", &c__0, &c__0, &work[i__], &one, m, &c__1, &y[ + i__ * y_dim1 + 1], m, &info2); +/* Y(1:M,i) = (ONE/WORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* LAPACK */ + } + } else { + work[i__] = zero; + } + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of X. */ + if (work[i__] > zero) { + r__1 = one / work[i__]; + sscal_(m, &r__1, &x[i__ * x_dim1 + 1], &c__1); +/* X(1:M,i) = (ONE/WORK(i)) * X(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (work[i__] < zero) { + r__1 = -work[i__]; + r__2 = one / (real) (*m); + slascl_("G", &c__0, &c__0, &r__1, &r__2, m, &c__1, &x[i__ * + x_dim1 + 1], m, &info2); +/* LAPACK CA */ + } else if (x[isamax_(m, &x[i__ * x_dim1 + 1], &c__1) + i__ * + x_dim1] != zero) { +/* Y(:,i) is zero vector. If X(:,i) is not */ +/* zero, then a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + } + } + } + +/* <2> SVD of the data snapshot matrix X. */ +/* ===================================== */ +/* The left singular vectors are stored in the array X. */ +/* The right singular vectors are in the array W. */ +/* The array W will later on contain the eigenvectors */ +/* of a Rayleigh quotient. */ + numrnk = *n; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { + i__1 = *lwork - *n; + sgesvd_("O", "S", m, n, &x[x_offset], ldx, &work[1], &b[b_offset], + ldb, &w[w_offset], ldw, &work[*n + 1], &i__1, &info1); +/* LAPACK CAL */ + *(unsigned char *)t_or_n__ = 'T'; + } else if (*whtsvd == 2) { + i__1 = *lwork - *n; + sgesdd_("O", m, n, &x[x_offset], ldx, &work[1], &b[b_offset], ldb, &w[ + w_offset], ldw, &work[*n + 1], &i__1, &iwork[1], &info1); +/* LAPACK CAL */ + *(unsigned char *)t_or_n__ = 'T'; + } else if (*whtsvd == 3) { + i__1 = *lwork - *n - f2cmax(2,*m); + i__2 = f2cmax(2,*m); + sgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &work[1], & + z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, &iwork[1], + liwork, &work[*n + f2cmax(2,*m) + 1], &i__1, &work[*n + 1], & + i__2, &info1); + + slacpy_("A", m, &numrnk, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK C */ + *(unsigned char *)t_or_n__ = 'T'; + } else if (*whtsvd == 4) { + i__1 = *lwork - *n; + sgejsv_("F", "U", jsvopt, "N", "N", "P", m, n, &x[x_offset], ldx, & + work[1], &z__[z_offset], ldz, &w[w_offset], ldw, &work[*n + 1] + , &i__1, &iwork[1], &info1); +/* LAPACK CALL */ + slacpy_("A", m, n, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK CALL */ + *(unsigned char *)t_or_n__ = 'N'; + xscl1 = work[*n + 1]; + xscl2 = work[*n + 2]; + if (xscl1 != xscl2) { +/* This is an exceptional situation. If the */ +/* data matrices are not scaled and the */ +/* largest singular value of X overflows. */ +/* In that case SGEJSV can return the SVD */ +/* in scaled form. The scaling factor can be used */ +/* to rescale the data (X and Y). */ + slascl_("G", &c__0, &c__0, &xscl1, &xscl2, m, n, &y[y_offset], + ldy, &info2); + } +/* END SELECT */ + } + + if (info1 > 0) { +/* The SVD selected subroutine did not converge. */ +/* Return with an error code. */ + *info = 2; + return 0; + } + + if (work[1] == zero) { +/* The largest computed singular value of (scaled) */ +/* X is zero. Return error code -8 */ +/* (the 8th input variable had an illegal value). */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("SGEDMD", &i__1); + return 0; + } + +/* <3> Determine the numerical rank of the data */ +/* snapshots matrix X. This depends on the */ +/* parameters NRNK and TOL. */ +/* SELECT CASE ( NRNK ) */ + if (*nrnk == -1) { + *k = 1; + i__1 = numrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (work[i__] <= work[1] * *tol || work[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else if (*nrnk == -2) { + *k = 1; + i__1 = numrnk - 1; + for (i__ = 1; i__ <= i__1; ++i__) { + if (work[i__ + 1] <= work[i__] * *tol || work[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else { + *k = 1; + i__1 = *nrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (work[i__] <= small) { + myexit_(); + } + ++(*k); + } +/* END SELECT */ + } +/* Now, U = X(1:M,1:K) is the SVD/POD basis for the */ +/* snapshot data in the input matrix X. */ +/* <4> Compute the Rayleigh quotient S = U^T * A * U. */ +/* Depending on the requested outputs, the computation */ +/* is organized to compute additional auxiliary */ +/* matrices (for the residuals and refinements). */ + +/* In all formulas below, we need V_k*Sigma_k^(-1) */ +/* where either V_k is in W(1:N,1:K), or V_k^T is in */ +/* W(1:K,1:N). Here Sigma_k=diag(WORK(1:K)). */ + if (lsame_(t_or_n__, "N")) { + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + r__1 = one / work[i__]; + sscal_(n, &r__1, &w[i__ * w_dim1 + 1], &c__1); +/* W(1:N,i) = (ONE/WORK(i)) * W(1:N,i) ! INTRINSIC */ +/* BLAS CALL */ + } + } else { +/* This non-unit stride access is due to the fact */ +/* that SGESVD, SGESVDQ and SGESDD return the */ +/* transposed matrix of the right singular vectors. */ +/* DO i = 1, K */ +/* CALL SSCAL( N, ONE/WORK(i), W(i,1), LDW ) ! BLAS CALL */ +/* ! W(i,1:N) = (ONE/WORK(i)) * W(i,1:N) ! INTRINSIC */ +/* END DO */ + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + work[*n + i__] = one / work[i__]; + } + i__1 = *n; + for (j = 1; j <= i__1; ++j) { + i__2 = *k; + for (i__ = 1; i__ <= i__2; ++i__) { + w[i__ + j * w_dim1] = work[*n + i__] * w[i__ + j * w_dim1]; + } + } + } + + if (wntref) { + +/* Need A*U(:,1:K)=Y*V_k*inv(diag(WORK(1:K))) */ +/* for computing the refined Ritz vectors */ +/* (optionally, outside SGEDMD). */ + sgemm_("N", t_or_n__, m, k, n, &one, &y[y_offset], ldy, &w[w_offset], + ldw, &zero, &z__[z_offset], ldz); +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRI */ +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),W(1:N,1:K)) ! INTRI */ + +/* At this point Z contains */ +/* A * U(:,1:K) = Y * V_k * Sigma_k^(-1), and */ +/* this is needed for computing the residuals. */ +/* This matrix is returned in the array B and */ +/* it can be used to compute refined Ritz vectors. */ +/* BLAS */ + slacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); +/* B(1:M,1:K) = Z(1:M,1:K) ! INTRINSIC */ +/* BLAS CALL */ + sgemm_("T", "N", k, k, m, &one, &x[x_offset], ldx, &z__[z_offset], + ldz, &zero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(TANSPOSE(X(1:M,1:K)),Z(1:M,1:K)) ! INTRI */ +/* At this point S = U^T * A * U is the Rayleigh quotient. */ +/* BLAS */ + } else { +/* A * U(:,1:K) is not explicitly needed and the */ +/* computation is organized differently. The Rayleigh */ +/* quotient is computed more efficiently. */ + sgemm_("T", "N", k, n, m, &one, &x[x_offset], ldx, &y[y_offset], ldy, + &zero, &z__[z_offset], ldz); +/* Z(1:K,1:N) = MATMUL( TRANSPOSE(X(1:M,1:K)), Y(1:M,1:N) ) ! IN */ +/* In the two SGEMM calls here, can use K for LDZ */ +/* B */ + sgemm_("N", t_or_n__, k, k, n, &one, &z__[z_offset], ldz, &w[w_offset] + , ldw, &zero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRIN */ +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),(W(1:N,1:K))) ! INTRIN */ +/* At this point S = U^T * A * U is the Rayleigh quotient. */ +/* If the residuals are requested, save scaled V_k into Z. */ +/* Recall that V_k or V_k^T is stored in W. */ +/* BLAS */ + if (wntres || wntex) { + if (lsame_(t_or_n__, "N")) { + slacpy_("A", n, k, &w[w_offset], ldw, &z__[z_offset], ldz); + } else { + slacpy_("A", k, n, &w[w_offset], ldw, &z__[z_offset], ldz); + } + } + } + +/* <5> Compute the Ritz values and (if requested) the */ +/* right eigenvectors of the Rayleigh quotient. */ + + i__1 = *lwork - *n; + sgeev_("N", jobzl, k, &s[s_offset], lds, &reig[1], &imeig[1], &w[w_offset] + , ldw, &w[w_offset], ldw, &work[*n + 1], &i__1, &info1); + +/* W(1:K,1:K) contains the eigenvectors of the Rayleigh */ +/* quotient. Even in the case of complex spectrum, all */ +/* computation is done in real arithmetic. REIG and */ +/* IMEIG are the real and the imaginary parts of the */ +/* eigenvalues, so that the spectrum is given as */ +/* REIG(:) + sqrt(-1)*IMEIG(:). Complex conjugate pairs */ +/* are listed at consecutive positions. For such a */ +/* complex conjugate pair of the eigenvalues, the */ +/* corresponding eigenvectors are also a complex */ +/* conjugate pair with the real and imaginary parts */ +/* stored column-wise in W at the corresponding */ +/* consecutive column indices. See the description of Z. */ +/* Also, see the description of SGEEV. */ +/* LAPACK C */ + if (info1 > 0) { +/* SGEEV failed to compute the eigenvalues and */ +/* eigenvectors of the Rayleigh quotient. */ + *info = 3; + return 0; + } + +/* <6> Compute the eigenvectors (if requested) and, */ +/* the residuals (if requested). */ + + if (wntvec || wntex) { + if (wntres) { + if (wntref) { +/* Here, if the refinement is requested, we have */ +/* A*U(:,1:K) already computed and stored in Z. */ +/* For the residuals, need Y = A * U(:,1;K) * W. */ + sgemm_("N", "N", m, k, k, &one, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zero, &y[y_offset], ldy); +/* Y(1:M,1:K) = Z(1:M,1:K) * W(1:K,1:K) ! INTRINSIC */ +/* This frees Z; Y contains A * U(:,1:K) * W. */ +/* BLAS CALL */ + } else { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) is stored in Z */ + sgemm_(t_or_n__, "N", n, k, k, &one, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + sgemm_("N", "N", m, k, n, &one, &y[y_offset], ldy, &s[ + s_offset], lds, &zero, &z__[z_offset], ldz); +/* Save a copy of Z into Y and free Z for holding */ +/* the Ritz vectors. */ + slacpy_("A", m, k, &z__[z_offset], ldz, &y[y_offset], ldy); + if (wntex) { + slacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); + } + } + } else if (wntex) { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) is stored in Z */ + sgemm_(t_or_n__, "N", n, k, k, &one, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + sgemm_("N", "N", m, k, n, &one, &y[y_offset], ldy, &s[s_offset], + lds, &zero, &b[b_offset], ldb); +/* The above call replaces the following two calls */ +/* that were used in the developing-testing phase. */ +/* CALL SGEMM( 'N', 'N', M, K, N, ONE, Y, LDY, S, & */ +/* LDS, ZERO, Z, LDZ) */ +/* Save a copy of Z into B and free Z for holding */ +/* the Ritz vectors. */ +/* CALL SLACPY( 'A', M, K, Z, LDZ, B, LDB ) */ + } + +/* Compute the real form of the Ritz vectors */ + if (wntvec) { + sgemm_("N", "N", m, k, k, &one, &x[x_offset], ldx, &w[w_offset], + ldw, &zero, &z__[z_offset], ldz); + } +/* Z(1:M,1:K) = MATMUL(X(1:M,1:K), W(1:K,1:K)) ! INTRINSIC */ + +/* BLAS CALL */ + if (wntres) { + i__ = 1; + while(i__ <= *k) { + if (imeig[i__] == zero) { +/* have a real eigenvalue with real eigenvector */ + r__1 = -reig[i__]; + saxpy_(m, &r__1, &z__[i__ * z_dim1 + 1], &c__1, &y[i__ * + y_dim1 + 1], &c__1); +/* Y(1:M,i) = Y(1:M,i) - REIG(i) * Z(1:M,i) ! */ + + res[i__] = snrm2_(m, &y[i__ * y_dim1 + 1], &c__1); + ++i__; + } else { +/* Have a complex conjugate pair */ +/* REIG(i) +- sqrt(-1)*IMEIG(i). */ +/* Since all computation is done in real */ +/* arithmetic, the formula for the residual */ +/* is recast for real representation of the */ +/* complex conjugate eigenpair. See the */ +/* description of RES. */ + ab[0] = reig[i__]; + ab[1] = -imeig[i__]; + ab[2] = imeig[i__]; + ab[3] = reig[i__]; + r__1 = -one; + sgemm_("N", "N", m, &c__2, &c__2, &r__1, &z__[i__ * + z_dim1 + 1], ldz, ab, &c__2, &one, &y[i__ * + y_dim1 + 1], ldy); +/* Y(1:M,i:i+1) = Y(1:M,i:i+1) - Z(1:M,i:i+1) * AB ! INT */ +/* BL */ + res[i__] = slange_("F", m, &c__2, &y[i__ * y_dim1 + 1], + ldy, &work[*n + 1]); +/* LA */ + res[i__ + 1] = res[i__]; + i__ += 2; + } + } + } + } + + if (*whtsvd == 4) { + work[*n + 1] = xscl1; + work[*n + 2] = xscl2; + } + +/* Successful exit. */ + if (! badxy) { + *info = 0; + } else { +/* A warning on possible data inconsistency. */ +/* This should be a rare event. */ + *info = 4; + } +/* ............................................................ */ + return 0; +/* ...... */ +} /* sgedmd_ */ + diff --git a/lapack-netlib/SRC/sgedmdq.c b/lapack-netlib/SRC/sgedmdq.c index 447b23014..0adf3bda3 100644 --- a/lapack-netlib/SRC/sgedmdq.c +++ b/lapack-netlib/SRC/sgedmdq.c @@ -509,3 +509,788 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; + +/* Subroutine */ int sgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq, + char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n, real + *f, integer *ldf, real *x, integer *ldx, real *y, integer *ldy, + integer *nrnk, real *tol, integer *k, real *reig, real *imeig, real * + z__, integer *ldz, real *res, real *b, integer *ldb, real *v, integer + *ldv, real *s, integer *lds, real *work, integer *lwork, integer * + iwork, integer *liwork, integer *info) +{ + /* System generated locals */ + integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1, + z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset, + i__1, i__2; + + /* Local variables */ + real zero; + integer info1; + extern logical lsame_(char *, char *); + char jobvl[1]; + integer minmn; + logical wantq; + integer mlwqr, olwqr; + logical wntex; + extern /* Subroutine */ int sgedmd_(char *, char *, char *, char *, + integer *, integer *, integer *, real *, integer *, real *, + integer *, integer *, real *, integer *, real *, real *, real *, + integer *, real *, real *, integer *, real *, integer *, real *, + integer *, real *, integer *, integer *, integer *, integer *), xerbla_(char *, integer *); + integer mlwdmd, olwdmd; + extern /* Subroutine */ int sgeqrf_(integer *, integer *, real *, integer + *, real *, real *, integer *, integer *); + logical sccolx, sccoly; + extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, + integer *, real *, integer *), slaset_(char *, integer *, + integer *, real *, real *, real *, integer *); + integer iminwr; + logical wntvec, wntvcf; + integer mlwgqr; + logical wntref; + integer mlwork, olwgqr, olwork; + real rdummy[2]; + integer mlwmqr, olwmqr; + logical lquery, wntres, wnttrf, wntvcq; + extern /* Subroutine */ int sorgqr_(integer *, integer *, integer *, real + *, integer *, real *, real *, integer *, integer *), sormqr_(char + *, char *, integer *, integer *, integer *, real *, integer *, + real *, real *, integer *, real *, integer *, integer *); + real one; + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real32 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ..... */ +/* Purpose */ +/* ======= */ +/* SGEDMDQ computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices, using a QR factorization */ +/* based compression of the data. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, SGEDMDQ computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, SGEDMDQ returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office. */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ...................................................................... */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. The data snapshots are the columns */ +/* of F. The leading N-1 columns of F are denoted X and the */ +/* trailing N-1 columns are denoted Y. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Z*V, where Z */ +/* is orthonormal and V contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of F, V, Z. */ +/* 'Q' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Q*Z, where Z */ +/* contains the eigenvectors of the compression of the */ +/* underlying discretized operator onto the span of */ +/* the data snapshots. See the descriptions of F, V, Z. */ +/* Q is from the initial QR factorization. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will */ +/* be computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBQ (input) CHARACTER*1 */ +/* Specifies whether to explicitly compute and return the */ +/* orthogonal matrix from the QR factorization. */ +/* 'Q' :: The matrix Q of the QR factorization of the data */ +/* snapshot matrix is computed and stored in the */ +/* array F. See the description of F. */ +/* 'N' :: The matrix Q is not explicitly computed. */ +/* ..... */ +/* JOBT (input) CHARACTER*1 */ +/* Specifies whether to return the upper triangular factor */ +/* from the QR factorization. */ +/* 'R' :: The matrix R of the QR factorization of the data */ +/* snapshot matrix F is returned in the array Y. */ +/* See the description of Y and Further details. */ +/* 'N' :: The matrix R is not returned. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* To be useful on exit, this option needs JOBQ='Q'. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: SGESVD (the QR SVD algorithm) */ +/* 2 :: SGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: SGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: SGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M >= 0 */ +/* The state space dimension (the number of rows of F) */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshots from a single trajectory, */ +/* taken at equidistant discrete times. This is the */ +/* number of columns of F. */ +/* ..... */ +/* F (input/output) REAL(KIND=WP) M-by-N array */ +/* > On entry, */ +/* the columns of F are the sequence of data snapshots */ +/* from a single trajectory, taken at equidistant discrete */ +/* times. It is assumed that the column norms of F are */ +/* in the range of the normalized floating point numbers. */ +/* < On exit, */ +/* If JOBQ == 'Q', the array F contains the orthogonal */ +/* matrix/factor of the QR factorization of the initial */ +/* data snapshots matrix F. See the description of JOBQ. */ +/* If JOBQ == 'N', the entries in F strictly below the main */ +/* diagonal contain, column-wise, the information on the */ +/* Householder vectors, as returned by SGEQRF. The */ +/* remaining information to restore the orthogonal matrix */ +/* of the initial QR factorization is stored in WORK(1:N). */ +/* See the description of WORK. */ +/* ..... */ +/* LDF (input) INTEGER, LDF >= M */ +/* The leading dimension of the array F. */ +/* ..... */ +/* X (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array */ +/* X is used as workspace to hold representations of the */ +/* leading N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, the leading K columns of X contain the leading */ +/* K left singular vectors of the above described content */ +/* of X. To lift them to the space of the left singular */ +/* vectors U(:,1:K)of the input data, pre-multiply with the */ +/* Q factor from the initial QR factorization. */ +/* See the descriptions of F, K, V and Z. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= N */ +/* The leading dimension of the array X */ +/* ..... */ +/* Y (workspace/output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array */ +/* Y is used as workspace to hold representations of the */ +/* trailing N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, */ +/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */ +/* triangular factor from the QR factorization of the data */ +/* snapshot matrix F. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= N */ +/* The leading dimension of the array Y */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N-1 :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the description of K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the SVD/POD basis for the leading N-1 */ +/* data snapshots (columns of F) and the number of the */ +/* computed Ritz pairs. The value of K is determined */ +/* according to the rule set by the parameters NRNK and */ +/* TOL. See the descriptions of NRNK and TOL. */ +/* ..... */ +/* REIG (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* The leading K (K<=N) entries of REIG contain */ +/* the real parts of the computed eigenvalues */ +/* REIG(1:K) + sqrt(-1)*IMEIG(1:K). */ +/* See the descriptions of K, IMEIG, Z. */ +/* ..... */ +/* IMEIG (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* The leading K (K0, then the corresponding complex */ +/* conjugate pair of eigenvalues reads */ +/* LAMBDA(i) = REIG(i) + sqrt(-1)*IMAG(i) */ +/* LAMBDA(i+1) = REIG(i) - sqrt(-1)*IMAG(i) */ +/* That is, complex conjugate pairs have consecutive */ +/* indices (i,i+1), with the positive imaginary part */ +/* listed first. */ +/* See the descriptions of K, REIG, Z. */ +/* ..... */ +/* Z (workspace/output) REAL(KIND=WP) M-by-(N-1) array */ +/* If JOBZ =='V' then */ +/* Z contains real Ritz vectors as follows: */ +/* If IMEIG(i)=0, then Z(:,i) is an eigenvector of */ +/* the i-th Ritz value. */ +/* If IMEIG(i) > 0 (and IMEIG(i+1) < 0) then */ +/* [Z(:,i) Z(:,i+1)] span an invariant subspace and */ +/* the Ritz values extracted from this subspace are */ +/* REIG(i) + sqrt(-1)*IMEIG(i) and */ +/* REIG(i) - sqrt(-1)*IMEIG(i). */ +/* The corresponding eigenvectors are */ +/* Z(:,i) + sqrt(-1)*Z(:,i+1) and */ +/* Z(:,i) - sqrt(-1)*Z(:,i+1), respectively. */ +/* If JOBZ == 'F', then the above descriptions hold for */ +/* the columns of Z*V, where the columns of V are the */ +/* eigenvectors of the K-by-K Rayleigh quotient, and Z is */ +/* orthonormal. The columns of V are similarly structured: */ +/* If IMEIG(i) == 0 then Z*V(:,i) is an eigenvector, and if */ +/* IMEIG(i) > 0 then Z*V(:,i)+sqrt(-1)*Z*V(:,i+1) and */ +/* Z*V(:,i)-sqrt(-1)*Z*V(:,i+1) */ +/* are the eigenvectors of LAMBDA(i), LAMBDA(i+1). */ +/* See the descriptions of REIG, IMEIG, X and V. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs. */ +/* If LAMBDA(i) is real, then */ +/* RES(i) = || A * Z(:,i) - LAMBDA(i)*Z(:,i))||_2. */ +/* If [LAMBDA(i), LAMBDA(i+1)] is a complex conjugate pair */ +/* then */ +/* RES(i)=RES(i+1) = || A * Z(:,i:i+1) - Z(:,i:i+1) *B||_F */ +/* where B = [ real(LAMBDA(i)) imag(LAMBDA(i)) ] */ +/* [-imag(LAMBDA(i)) real(LAMBDA(i)) ]. */ +/* It holds that */ +/* RES(i) = || A*ZC(:,i) - LAMBDA(i) *ZC(:,i) ||_2 */ +/* RES(i+1) = || A*ZC(:,i+1) - LAMBDA(i+1)*ZC(:,i+1) ||_2 */ +/* where ZC(:,i) = Z(:,i) + sqrt(-1)*Z(:,i+1) */ +/* ZC(:,i+1) = Z(:,i) - sqrt(-1)*Z(:,i+1) */ +/* See the description of Z. */ +/* ..... */ +/* B (output) REAL(KIND=WP) MIN(M,N)-by-(N-1) array. */ +/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:N,1;K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* In both cases, the content of B can be lifted to the */ +/* original dimension of the input data by pre-multiplying */ +/* with the Q factor from the initial QR factorization. */ +/* Here A denotes a compression of the underlying operator. */ +/* See the descriptions of F and X. */ +/* If JOBF =='N', then B is not referenced. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= MIN(M,N) */ +/* The leading dimension of the array B. */ +/* ..... */ +/* V (workspace/output) REAL(KIND=WP) (N-1)-by-(N-1) array */ +/* On exit, V(1:K,1:K) contains the K eigenvectors of */ +/* the Rayleigh quotient. The eigenvectors of a complex */ +/* conjugate pair of eigenvalues are returned in real form */ +/* as explained in the description of Z. The Ritz vectors */ +/* (returned in Z) are the product of X and V; see */ +/* the descriptions of X and Z. */ +/* ..... */ +/* LDV (input) INTEGER, LDV >= N-1 */ +/* The leading dimension of the array V. */ +/* ..... */ +/* S (output) REAL(KIND=WP) (N-1)-by-(N-1) array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by SGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N-1 */ +/* The leading dimension of the array S. */ +/* ..... */ +/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */ +/* On exit, */ +/* WORK(1:MIN(M,N)) contains the scalar factors of the */ +/* elementary reflectors as returned by SGEQRF of the */ +/* M-by-N input matrix F. */ +/* WORK(MIN(M,N)+1:MIN(M,N)+N-1) contains the singular values of */ +/* the input submatrix F(1:M,1:N-1). */ +/* If the call to SGEDMDQ is only workspace query, then */ +/* WORK(1) contains the minimal workspace length and */ +/* WORK(2) is the optimal workspace length. Hence, the */ +/* length of work is at least 2. */ +/* See the description of LWORK. */ +/* ..... */ +/* LWORK (input) INTEGER */ +/* The minimal length of the workspace vector WORK. */ +/* LWORK is calculated as follows: */ +/* Let MLWQR = N (minimal workspace for SGEQRF[M,N]) */ +/* MLWDMD = minimal workspace for SGEDMD (see the */ +/* description of LWORK in SGEDMD) for */ +/* snapshots of dimensions MIN(M,N)-by-(N-1) */ +/* MLWMQR = N (minimal workspace for */ +/* SORMQR['L','N',M,N,N]) */ +/* MLWGQR = N (minimal workspace for SORGQR[M,N,N]) */ +/* Then */ +/* LWORK = MAX(N+MLWQR, N+MLWDMD) */ +/* is updated as follows: */ +/* if JOBZ == 'V' or JOBZ == 'F' THEN */ +/* LWORK = MAX( LWORK,MIN(M,N)+N-1 +MLWMQR ) */ +/* if JOBQ == 'Q' THEN */ +/* LWORK = MAX( LWORK,MIN(M,N)+N-1+MLWGQR) */ +/* If on entry LWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* Let M1=MIN(M,N), N1=N-1. Then */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1) */ +/* If on entry LIWORK = -1, then a worskpace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ + +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ + +/* Local array */ +/* ~~~~~~~~~~~ */ + +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ + +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* External subroutines */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ + /* Parameter adjustments */ + f_dim1 = *ldf; + f_offset = 1 + f_dim1 * 1; + f -= f_offset; + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --reig; + --imeig; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + v_dim1 = *ldv; + v_offset = 1 + v_dim1 * 1; + v -= v_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --work; + --iwork; + + /* Function Body */ + one = 1.f; + zero = 0.f; +/* .......................................................... */ + +/* Test the input arguments */ + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntvcf = lsame_(jobz, "F"); + wntvcq = lsame_(jobz, "Q"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + wantq = lsame_(jobq, "Q"); + wnttrf = lsame_(jobt, "R"); + minmn = f2cmin(*m,*n); + *info = 0; + lquery = *lwork == -1 || *liwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && lsame_(jobz, "N")) { + *info = -3; + } else if (! (wantq || lsame_(jobq, "N"))) { + *info = -4; + } else if (! (wnttrf || lsame_(jobt, "N"))) { + *info = -5; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -6; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -7; + } else if (*m < 0) { + *info = -8; + } else if (*n < 0 || *n > *m + 1) { + *info = -9; + } else if (*ldf < *m) { + *info = -11; + } else if (*ldx < minmn) { + *info = -13; + } else if (*ldy < minmn) { + *info = -15; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -16; + } else if (*tol < zero || *tol >= one) { + *info = -17; + } else if (*ldz < *m) { + *info = -22; + } else if ((wntref || wntex) && *ldb < minmn) { + *info = -25; + } else if (*ldv < *n - 1) { + *info = -27; + } else if (*lds < *n - 1) { + *info = -29; + } + + if (wntvec || wntvcf) { + *(unsigned char *)jobvl = 'V'; + } else { + *(unsigned char *)jobvl = 'N'; + } + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0 || *n == 1) { +/* All output except K is void. INFO=1 signals */ +/* the void input. In case of a workspace query, */ +/* the minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + work[1] = 2.f; + work[2] = 2.f; + } else { + *k = 0; + } + *info = 1; + return 0; + } + mlwqr = f2cmax(1,*n); +/* Minimal workspace length for SGEQRF. */ + mlwork = f2cmin(*m,*n) + mlwqr; + if (lquery) { + sgeqrf_(m, n, &f[f_offset], ldf, &work[1], rdummy, &c_n1, &info1); + olwqr = (integer) rdummy[0]; + olwork = f2cmin(*m,*n) + olwqr; + } + i__1 = *n - 1; + sgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], + ldx, &y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], & + z__[z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset], + ldv, &s[s_offset], lds, &work[1], &c_n1, &iwork[1], liwork, & + info1); + mlwdmd = (integer) work[1]; +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwdmd; + mlwork = f2cmax(i__1,i__2); + iminwr = iwork[1]; + if (lquery) { + olwdmd = (integer) work[2]; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwdmd; + olwork = f2cmax(i__1,i__2); + } + if (wntvec || wntvcf) { + mlwmqr = f2cmax(1,*n); +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + *n - 1 + mlwmqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + sormqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &work[1], & + z__[z_offset], ldz, &work[1], &c_n1, &info1); + olwmqr = (integer) work[1]; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + *n - 1 + olwmqr; + olwork = f2cmax(i__1,i__2); + } + } + if (wantq) { + mlwgqr = *n; +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + *n - 1 + mlwgqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + sorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[ + 1], &c_n1, &info1); + olwgqr = (integer) work[1]; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + *n - 1 + olwgqr; + olwork = f2cmax(i__1,i__2); + } + } + iminwr = f2cmax(1,iminwr); + mlwork = f2cmax(2,mlwork); + if (*lwork < mlwork && ! lquery) { + *info = -31; + } + if (*liwork < iminwr && ! lquery) { + *info = -33; + } + } + if (*info != 0) { + i__1 = -(*info); + xerbla_("SGEDMDQ", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + work[1] = (real) mlwork; + work[2] = (real) olwork; + return 0; + } +/* ..... */ +/* Initial QR factorization that is used to represent the */ +/* snapshots as elements of lower dimensional subspace. */ +/* For large scale computation with M >>N , at this place */ +/* one can use an out of core QRF. */ + + i__1 = *lwork - minmn; + sgeqrf_(m, n, &f[f_offset], ldf, &work[1], &work[minmn + 1], &i__1, & + info1); + +/* Define X and Y as the snapshots representations in the */ +/* orthogonal basis computed in the QR factorization. */ +/* X corresponds to the leading N-1 and Y to the trailing */ +/* N-1 snapshots. */ + i__1 = *n - 1; + slaset_("L", &minmn, &i__1, &zero, &zero, &x[x_offset], ldx); + i__1 = *n - 1; + slacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx); + i__1 = *n - 1; + slacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy); + if (*m >= 3) { + i__1 = minmn - 2; + i__2 = *n - 2; + slaset_("L", &i__1, &i__2, &zero, &zero, &y[y_dim1 + 3], ldy); + } + +/* Compute the DMD of the projected snapshot pairs (X,Y) */ + i__1 = *n - 1; + i__2 = *lwork - minmn; + sgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx, + &y[y_offset], ldy, nrnk, tol, k, &reig[1], &imeig[1], &z__[ + z_offset], ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[ + s_offset], lds, &work[minmn + 1], &i__2, &iwork[1], liwork, & + info1); + if (info1 == 2 || info1 == 3) { +/* Return with error code. */ + *info = info1; + return 0; + } else { + *info = info1; + } + +/* The Ritz vectors (Koopman modes) can be explicitly */ +/* formed or returned in factored form. */ + if (wntvec) { +/* Compute the eigenvectors explicitly. */ + if (*m > minmn) { + i__1 = *m - minmn; + slaset_("A", &i__1, k, &zero, &zero, &z__[minmn + 1 + z_dim1], + ldz); + } + i__1 = *lwork - (minmn + *n - 1); + sormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[ + z_offset], ldz, &work[minmn + *n], &i__1, &info1); + } else if (wntvcf) { +/* Return the Ritz vectors (eigenvectors) in factored */ +/* form Z*V, where Z contains orthonormal matrix (the */ +/* product of Q from the initial QR factorization and */ +/* the SVD/POD_basis returned by SGEDMD in X) and the */ +/* second factor (the eigenvectors of the Rayleigh */ +/* quotient) is in the array V, as returned by SGEDMD. */ + slacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz); + if (*m > *n) { + i__1 = *m - *n; + slaset_("A", &i__1, k, &zero, &zero, &z__[*n + 1 + z_dim1], ldz); + } + i__1 = *lwork - (minmn + *n - 1); + sormqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &work[1], &z__[ + z_offset], ldz, &work[minmn + *n], &i__1, &info1); + } + +/* Some optional output variables: */ + +/* The upper triangular factor in the initial QR */ +/* factorization is optionally returned in the array Y. */ +/* This is useful if this call to SGEDMDQ is to be */ +/* followed by a streaming DMD that is implemented in a */ +/* QR compressed form. */ + if (wnttrf) { +/* Return the upper triangular R in Y */ + slaset_("A", &minmn, n, &zero, &zero, &y[y_offset], ldy); + slacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy); + } + +/* The orthonormal/orthogonal factor in the initial QR */ +/* factorization is optionally returned in the array F. */ +/* Same as with the triangular factor above, this is */ +/* useful in a streaming DMD. */ + if (wantq) { +/* Q overwrites F */ + i__1 = *lwork - (minmn + *n - 1); + sorgqr_(m, &minmn, &minmn, &f[f_offset], ldf, &work[1], &work[minmn + + *n], &i__1, &info1); + } + + return 0; + +} /* sgedmdq_ */ + diff --git a/lapack-netlib/SRC/zgedmd.c b/lapack-netlib/SRC/zgedmd.c index 447b23014..c1b39ba3e 100644 --- a/lapack-netlib/SRC/zgedmd.c +++ b/lapack-netlib/SRC/zgedmd.c @@ -509,3 +509,1168 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; +static integer c__1 = 1; +static integer c__0 = 0; + +/* Subroutine */ int zgedmd_(char *jobs, char *jobz, char *jobr, char *jobf, + integer *whtsvd, integer *m, integer *n, doublecomplex *x, integer * + ldx, doublecomplex *y, integer *ldy, integer *nrnk, doublereal *tol, + integer *k, doublecomplex *eigs, doublecomplex *z__, integer *ldz, + doublereal *res, doublecomplex *b, integer *ldb, doublecomplex *w, + integer *ldw, doublecomplex *s, integer *lds, doublecomplex *zwork, + integer *lzwork, doublereal *rwork, integer *lrwork, integer *iwork, + integer *liwork, integer *info) +{ + /* System generated locals */ + integer x_dim1, x_offset, y_dim1, y_offset, z_dim1, z_offset, b_dim1, + b_offset, w_dim1, w_offset, s_dim1, s_offset, i__1, i__2, i__3, + i__4, i__5; + doublereal d__1, d__2; + doublecomplex z__1, z__2; + + /* Local variables */ + doublecomplex zone; + doublereal zero, ssum; + integer info1, info2; + doublereal xscl1, xscl2; + integer i__, j; + doublereal scale; + extern logical lsame_(char *, char *); + logical badxy; + doublereal small; + extern /* Subroutine */ int zgemm_(char *, char *, integer *, integer *, + integer *, doublecomplex *, doublecomplex *, integer *, + doublecomplex *, integer *, doublecomplex *, doublecomplex *, + integer *); + char jobzl[1]; + extern /* Subroutine */ int zgeev_(char *, char *, integer *, + doublecomplex *, integer *, doublecomplex *, doublecomplex *, + integer *, doublecomplex *, integer *, doublecomplex *, integer *, + doublereal *, integer *); + logical wntex; + doublecomplex zzero; + extern /* Subroutine */ int zaxpy_(integer *, doublecomplex *, + doublecomplex *, integer *, doublecomplex *, integer *); + extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_( + char *); + extern logical disnan_(doublereal *); + extern /* Subroutine */ int xerbla_(char *, integer *); + char t_or_n__[1]; + extern /* Subroutine */ int zdscal_(integer *, doublereal *, + doublecomplex *, integer *), zgesdd_(char *, integer *, integer *, + doublecomplex *, integer *, doublereal *, doublecomplex *, + integer *, doublecomplex *, integer *, doublecomplex *, integer *, + doublereal *, integer *, integer *), zlascl_(char *, + integer *, integer *, doublereal *, doublereal *, integer *, + integer *, doublecomplex *, integer *, integer *); + extern integer izamax_(integer *, doublecomplex *, integer *); + logical sccolx, sccoly; + integer lwrsdd, mwrsdd; + extern /* Subroutine */ int zgesvd_(char *, char *, integer *, integer *, + doublecomplex *, integer *, doublereal *, doublecomplex *, + integer *, doublecomplex *, integer *, doublecomplex *, integer *, + doublereal *, integer *), zlacpy_(char *, + integer *, integer *, doublecomplex *, integer *, doublecomplex *, + integer *); + integer iminwr; + logical wntref, wntvec; + doublereal rootsc; + integer lwrkev, mlwork, mwrkev, numrnk, olwork, lwrsvd, mwrsvd, mlrwrk; + logical lquery, wntres; + char jsvopt[1]; + integer lwrsvj, mwrsvj; + doublereal rdummy[2]; + extern /* Subroutine */ int zgejsv_(char *, char *, char *, char *, char * + , char *, integer *, integer *, doublecomplex *, integer *, + doublereal *, doublecomplex *, integer *, doublecomplex *, + integer *, doublecomplex *, integer *, doublereal *, integer *, + integer *, integer *), zlassq_(integer *, doublecomplex *, integer *, + doublereal *, doublereal *), mecago_(); + integer lwrsvq, mwrsvq; + doublereal ofl, one; + extern /* Subroutine */ int zgesvdq_(char *, char *, char *, char *, char + *, integer *, integer *, doublecomplex *, integer *, doublereal *, + doublecomplex *, integer *, doublecomplex *, integer *, integer * + , integer *, integer *, doublecomplex *, integer *, doublereal *, + integer *, integer *); + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real64 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ............................................................ */ +/* Purpose */ +/* ======= */ +/* ZGEDMD computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, ZGEDMD computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, ZGEDMD returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* ...................................................................... */ +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ............................................................ */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product X(:,1:K)*W, where X */ +/* contains a POD basis (leading left singular vectors */ +/* of the data matrix X) and W contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of K, X, W, Z. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will be */ +/* computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: ZGESVD (the QR SVD algorithm) */ +/* 2 :: ZGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: ZGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: ZGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M>= 0 */ +/* The state space dimension (the row dimension of X, Y). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshot pairs */ +/* (the number of columns of X and Y). */ +/* ..... */ +/* X (input/output) COMPLEX(KIND=WP) M-by-N array */ +/* > On entry, X contains the data snapshot matrix X. It is */ +/* assumed that the column norms of X are in the range of */ +/* the normalized floating point numbers. */ +/* < On exit, the leading K columns of X contain a POD basis, */ +/* i.e. the leading K left singular vectors of the input */ +/* data matrix X, U(:,1:K). All N columns of X contain all */ +/* left singular vectors of the input matrix X. */ +/* See the descriptions of K, Z and W. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= M */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (input/workspace/output) COMPLEX(KIND=WP) M-by-N array */ +/* > On entry, Y contains the data snapshot matrix Y */ +/* < On exit, */ +/* If JOBR == 'R', the leading K columns of Y contain */ +/* the residual vectors for the computed Ritz pairs. */ +/* See the description of RES. */ +/* If JOBR == 'N', Y contains the original input data, */ +/* scaled according to the value of JOBS. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= M */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the descriptions of TOL and K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the POD basis for the data snapshot */ +/* matrix X and the number of the computed Ritz pairs. */ +/* The value of K is determined according to the rule set */ +/* by the parameters NRNK and TOL. */ +/* See the descriptions of NRNK and TOL. */ +/* ..... */ +/* EIGS (output) COMPLEX(KIND=WP) N-by-1 array */ +/* The leading K (K<=N) entries of EIGS contain */ +/* the computed eigenvalues (Ritz values). */ +/* See the descriptions of K, and Z. */ +/* ..... */ +/* Z (workspace/output) COMPLEX(KIND=WP) M-by-N array */ +/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */ +/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */ +/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */ +/* the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) */ +/* is an eigenvector corresponding to EIGS(i). The columns */ +/* of W(1:k,1:K) are the computed eigenvectors of the */ +/* K-by-K Rayleigh quotient. */ +/* See the descriptions of EIGS, X and W. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) N-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs, */ +/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */ +/* See the description of EIGS and Z. */ +/* ..... */ +/* B (output) COMPLEX(KIND=WP) M-by-N array. */ +/* IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:M,1:K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* If JOBF =='N', then B is not referenced. */ +/* See the descriptions of X, W, K. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= M */ +/* The leading dimension of the array B. */ +/* ..... */ +/* W (workspace/output) COMPLEX(KIND=WP) N-by-N array */ +/* On exit, W(1:K,1:K) contains the K computed */ +/* eigenvectors of the matrix Rayleigh quotient. */ +/* The Ritz vectors (returned in Z) are the */ +/* product of X (containing a POD basis for the input */ +/* matrix X) and W. See the descriptions of K, S, X and Z. */ +/* W is also used as a workspace to temporarily store the */ +/* right singular vectors of X. */ +/* ..... */ +/* LDW (input) INTEGER, LDW >= N */ +/* The leading dimension of the array W. */ +/* ..... */ +/* S (workspace/output) COMPLEX(KIND=WP) N-by-N array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by ZGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N */ +/* The leading dimension of the array S. */ +/* ..... */ +/* ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array */ +/* ZWORK is used as complex workspace in the complex SVD, as */ +/* specified by WHTSVD (1,2, 3 or 4) and for ZGEEV for computing */ +/* the eigenvalues of a Rayleigh quotient. */ +/* If the call to ZGEDMD is only workspace query, then */ +/* ZWORK(1) contains the minimal complex workspace length and */ +/* ZWORK(2) is the optimal complex workspace length. */ +/* Hence, the length of work is at least 2. */ +/* See the description of LZWORK. */ +/* ..... */ +/* LZWORK (input) INTEGER */ +/* The minimal length of the workspace vector ZWORK. */ +/* LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_ZGEEV), */ +/* where LZWORK_ZGEEV = MAX( 1, 2*N ) and the minimal */ +/* LZWORK_SVD is calculated as follows */ +/* If WHTSVD == 1 :: ZGESVD :: */ +/* LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) */ +/* If WHTSVD == 2 :: ZGESDD :: */ +/* LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) */ +/* If WHTSVD == 3 :: ZGESVDQ :: */ +/* LZWORK_SVD = obtainable by a query */ +/* If WHTSVD == 4 :: ZGEJSV :: */ +/* LZWORK_SVD = obtainable by a query */ +/* If on entry LZWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths and returns them in */ +/* LZWORK(1) and LZWORK(2), respectively. */ +/* ..... */ +/* RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array */ +/* On exit, RWORK(1:N) contains the singular values of */ +/* X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). */ +/* If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain */ +/* scaling factor RWORK(N+2)/RWORK(N+1) used to scale X */ +/* and Y to avoid overflow in the SVD of X. */ +/* This may be of interest if the scaling option is off */ +/* and as many as possible smallest eigenvalues are */ +/* desired to the highest feasible accuracy. */ +/* If the call to ZGEDMD is only workspace query, then */ +/* RWORK(1) contains the minimal workspace length. */ +/* See the description of LRWORK. */ +/* ..... */ +/* LRWORK (input) INTEGER */ +/* The minimal length of the workspace vector RWORK. */ +/* LRWORK is calculated as follows: */ +/* LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_ZGEEV), where */ +/* LRWORK_ZGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace */ +/* for the SVD subroutine determined by the input parameter */ +/* WHTSVD. */ +/* If WHTSVD == 1 :: ZGESVD :: */ +/* LRWORK_SVD = 5*MIN(M,N) */ +/* If WHTSVD == 2 :: ZGESDD :: */ +/* LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), */ +/* 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) */ +/* If WHTSVD == 3 :: ZGESVDQ :: */ +/* LRWORK_SVD = obtainable by a query */ +/* If WHTSVD == 4 :: ZGEJSV :: */ +/* LRWORK_SVD = obtainable by a query */ +/* If on entry LRWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* real workspace length and returns it in RWORK(1). */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for ZWORK, RWORK and */ +/* IWORK. See the descriptions of ZWORK, RWORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ + +/* Local arrays */ +/* ~~~~~~~~~~~~ */ +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* ............................................................ */ + /* Parameter adjustments */ + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --eigs; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + w_dim1 = *ldw; + w_offset = 1 + w_dim1 * 1; + w -= w_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --zwork; + --rwork; + --iwork; + + /* Function Body */ + zero = 0.f; + one = 1.f; + zzero.r = 0.f, zzero.i = 0.f; + zone.r = 1.f, zone.i = 0.f; + +/* Test the input arguments */ + + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + *info = 0; + lquery = *lzwork == -1 || *liwork == -1 || *lrwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || lsame_(jobz, "N") || lsame_( + jobz, "F"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && ! wntvec) { + *info = -3; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -4; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -5; + } else if (*m < 0) { + *info = -6; + } else if (*n < 0 || *n > *m) { + *info = -7; + } else if (*ldx < *m) { + *info = -9; + } else if (*ldy < *m) { + *info = -11; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -12; + } else if (*tol < zero || *tol >= one) { + *info = -13; + } else if (*ldz < *m) { + *info = -17; + } else if ((wntref || wntex) && *ldb < *m) { + *info = -20; + } else if (*ldw < *n) { + *info = -22; + } else if (*lds < *n) { + *info = -24; + } + + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0) { +/* Quick return. All output except K is void. */ +/* INFO=1 signals the void input. */ +/* In case of a workspace query, the default */ +/* minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + rwork[1] = 1.; + zwork[1].r = 2., zwork[1].i = 0.; + zwork[2].r = 2., zwork[2].i = 0.; + } else { + *k = 0; + } + *info = 1; + return 0; + } + iminwr = 1; + mlrwrk = f2cmax(1,*n); + mlwork = 2; + olwork = 2; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of ZGESVD: */ +/* MWRSVD = MAX(1,2*MIN(M,N)+MAX(M,N)) */ +/* Computing MAX */ + i__1 = 1, i__2 = (f2cmin(*m,*n) << 1) + f2cmax(*m,*n); + mwrsvd = f2cmax(i__1,i__2); + mlwork = f2cmax(mlwork,mwrsvd); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = *n + f2cmin(*m,*n) * 5; + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + zgesvd_("O", "S", m, n, &x[x_offset], ldx, &rwork[1], &b[ + b_offset], ldb, &w[w_offset], ldw, &zwork[1], &c_n1, + rdummy, &info1); + lwrsvd = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrsvd); + } + } else if (*whtsvd == 2) { +/* The following is specified as the minimal */ +/* length of WORK in the definition of ZGESDD: */ +/* MWRSDD = 2*f2cmin(M,N)*f2cmin(M,N)+2*f2cmin(M,N)+f2cmax(M,N). */ +/* RWORK length: 5*MIN(M,N)*MIN(M,N)+7*MIN(M,N) */ +/* In LAPACK 3.10.1 RWORK is defined differently. */ +/* Below we take f2cmax over the two versions. */ +/* IMINWR = 8*MIN(M,N) */ + mwrsdd = (f2cmin(*m,*n) << 1) * f2cmin(*m,*n) + (f2cmin(*m,*n) << 1) + f2cmax( + *m,*n); + mlwork = f2cmax(mlwork,mwrsdd); + iminwr = f2cmin(*m,*n) << 3; +/* Computing MAX */ +/* Computing MAX */ + i__3 = f2cmin(*m,*n) * 5 * f2cmin(*m,*n) + f2cmin(*m,*n) * 7, i__4 = f2cmin(* + m,*n) * 5 * f2cmin(*m,*n) + f2cmin(*m,*n) * 5, i__3 = f2cmax(i__3, + i__4), i__4 = (f2cmax(*m,*n) << 1) * f2cmin(*m,*n) + (f2cmin(*m,*n) + << 1) * f2cmin(*m,*n) + f2cmin(*m,*n); + i__1 = mlrwrk, i__2 = *n + f2cmax(i__3,i__4); + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + zgesdd_("O", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], + ldb, &w[w_offset], ldw, &zwork[1], &c_n1, rdummy, & + iwork[1], &info1); +/* Computing MAX */ + i__1 = mwrsdd, i__2 = (integer) zwork[1].r; + lwrsdd = f2cmax(i__1,i__2); +/* Possible bug in ZGESDD optimal workspace size. */ + olwork = f2cmax(olwork,lwrsdd); + } + } else if (*whtsvd == 3) { + zgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &rwork[ + 1], &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, & + iwork[1], &c_n1, &zwork[1], &c_n1, rdummy, &c_n1, &info1); + iminwr = iwork[1]; + mwrsvq = (integer) zwork[2].r; + mlwork = f2cmax(mlwork,mwrsvq); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = *n + (integer) rdummy[0]; + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + lwrsvq = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrsvq); + } + } else if (*whtsvd == 4) { + *(unsigned char *)jsvopt = 'J'; + zgejsv_("F", "U", jsvopt, "R", "N", "P", m, n, &x[x_offset], ldx, + &rwork[1], &z__[z_offset], ldz, &w[w_offset], ldw, &zwork[ + 1], &c_n1, rdummy, &c_n1, &iwork[1], &info1); + iminwr = iwork[1]; + mwrsvj = (integer) zwork[2].r; + mlwork = f2cmax(mlwork,mwrsvj); +/* Computing MAX */ +/* Computing MAX */ + i__3 = 7, i__4 = (integer) rdummy[0]; + i__1 = mlrwrk, i__2 = *n + f2cmax(i__3,i__4); + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + lwrsvj = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrsvj); + } +/* END SELECT */ + } + if (wntvec || wntex || lsame_(jobz, "F")) { + *(unsigned char *)jobzl = 'V'; + } else { + *(unsigned char *)jobzl = 'N'; + } +/* Workspace calculation to the ZGEEV call */ +/* Computing MAX */ + i__1 = 1, i__2 = *n << 1; + mwrkev = f2cmax(i__1,i__2); + mlwork = f2cmax(mlwork,mwrkev); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = *n + (*n << 1); + mlrwrk = f2cmax(i__1,i__2); + if (lquery) { + zgeev_("N", jobzl, n, &s[s_offset], lds, &eigs[1], &w[w_offset], + ldw, &w[w_offset], ldw, &zwork[1], &c_n1, &rwork[1], & + info1); + lwrkev = (integer) zwork[1].r; + olwork = f2cmax(olwork,lwrkev); + } + + if (*liwork < iminwr && ! lquery) { + *info = -30; + } + if (*lrwork < mlrwrk && ! lquery) { + *info = -28; + } + if (*lzwork < mlwork && ! lquery) { + *info = -26; + } + } + + if (*info != 0) { + i__1 = -(*info); + xerbla_("ZGEDMD", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + rwork[1] = (doublereal) mlrwrk; + zwork[1].r = (doublereal) mlwork, zwork[1].i = 0.; + zwork[2].r = (doublereal) olwork, zwork[2].i = 0.; + return 0; + } +/* ............................................................ */ + + ofl = dlamch_("O"); + small = dlamch_("S"); + badxy = FALSE_; + +/* <1> Optional scaling of the snapshots (columns of X, Y) */ +/* ========================================================== */ + if (sccolx) { +/* The columns of X will be normalized. */ +/* To prevent overflows, the column norms of X are */ +/* carefully computed using ZLASSQ. */ + *k = 0; + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* WORK(i) = DZNRM2( M, X(1,i), 1 ) */ + scale = zero; + zlassq_(m, &x[i__ * x_dim1 + 1], &c__1, &scale, &ssum); + if (disnan_(&scale) || disnan_(&ssum)) { + *k = 0; + *info = -8; + i__2 = -(*info); + xerbla_("ZGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of X(:,i) overflows. First, X(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||X(:,i)||_2. */ +/* Next, the norm of X(:,i) is stored without */ +/* overflow as RWORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of X(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + d__1 = one / rootsc; + zlascl_("G", &c__0, &c__0, &scale, &d__1, m, &c__1, &x[ + i__ * x_dim1 + 1], ldx, &info2); + rwork[i__] = -scale * (rootsc / (doublereal) (*m)); + } else { +/* X(:,i) will be scaled to unit 2-norm */ + rwork[i__] = scale * rootsc; + zlascl_("G", &c__0, &c__0, &rwork[i__], &one, m, &c__1, & + x[i__ * x_dim1 + 1], ldx, &info2); +/* X(1:M,i) = (ONE/RWORK(i)) * X(1:M,i) ! INTRINSIC */ +/* LAPACK CALL */ + } + } else { + rwork[i__] = zero; + ++(*k); + } + } + if (*k == *n) { +/* All columns of X are zero. Return error code -8. */ +/* (the 8th input variable had an illegal value) */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("ZGEDMD", &i__1); + return 0; + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of Y. */ + if (rwork[i__] > zero) { + d__1 = one / rwork[i__]; + zdscal_(m, &d__1, &y[i__ * y_dim1 + 1], &c__1); +/* Y(1:M,i) = (ONE/RWORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (rwork[i__] < zero) { + d__1 = -rwork[i__]; + d__2 = one / (doublereal) (*m); + zlascl_("G", &c__0, &c__0, &d__1, &d__2, m, &c__1, &y[i__ * + y_dim1 + 1], ldy, &info2); +/* LAPACK C */ + } else if (z_abs(&y[izamax_(m, &y[i__ * y_dim1 + 1], &c__1) + i__ + * y_dim1]) != zero) { +/* X(:,i) is zero vector. For consistency, */ +/* Y(:,i) should also be zero. If Y(:,i) is not */ +/* zero, then the data might be inconsistent or */ +/* corrupted. If JOBS == 'C', Y(:,i) is set to */ +/* zero and a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + if (lsame_(jobs, "C")) { + zdscal_(m, &zero, &y[i__ * y_dim1 + 1], &c__1); + } +/* BLAS CALL */ + } + } + } + + if (sccoly) { +/* The columns of Y will be normalized. */ +/* To prevent overflows, the column norms of Y are */ +/* carefully computed using ZLASSQ. */ + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* RWORK(i) = DZNRM2( M, Y(1,i), 1 ) */ + scale = zero; + zlassq_(m, &y[i__ * y_dim1 + 1], &c__1, &scale, &ssum); + if (disnan_(&scale) || disnan_(&ssum)) { + *k = 0; + *info = -10; + i__2 = -(*info); + xerbla_("ZGEDMD", &i__2); + } + if (scale != zero && ssum != zero) { + rootsc = sqrt(ssum); + if (scale >= ofl / rootsc) { +/* Norm of Y(:,i) overflows. First, Y(:,i) */ +/* is scaled by */ +/* ( ONE / ROOTSC ) / SCALE = 1/||Y(:,i)||_2. */ +/* Next, the norm of Y(:,i) is stored without */ +/* overflow as RWORK(i) = - SCALE * (ROOTSC/M), */ +/* the minus sign indicating the 1/M factor. */ +/* Scaling is performed without overflow, and */ +/* underflow may occur in the smallest entries */ +/* of Y(:,i). The relative backward and forward */ +/* errors are small in the ell_2 norm. */ + d__1 = one / rootsc; + zlascl_("G", &c__0, &c__0, &scale, &d__1, m, &c__1, &y[ + i__ * y_dim1 + 1], ldy, &info2); + rwork[i__] = -scale * (rootsc / (doublereal) (*m)); + } else { +/* Y(:,i) will be scaled to unit 2-norm */ + rwork[i__] = scale * rootsc; + zlascl_("G", &c__0, &c__0, &rwork[i__], &one, m, &c__1, & + y[i__ * y_dim1 + 1], ldy, &info2); +/* Y(1:M,i) = (ONE/RWORK(i)) * Y(1:M,i) ! INTRINSIC */ +/* LAPAC */ + } + } else { + rwork[i__] = zero; + } + } + i__1 = *n; + for (i__ = 1; i__ <= i__1; ++i__) { +/* Now, apply the same scaling to the columns of X. */ + if (rwork[i__] > zero) { + d__1 = one / rwork[i__]; + zdscal_(m, &d__1, &x[i__ * x_dim1 + 1], &c__1); +/* X(1:M,i) = (ONE/RWORK(i)) * X(1:M,i) ! INTRINSIC */ +/* BLAS CALL */ + } else if (rwork[i__] < zero) { + d__1 = -rwork[i__]; + d__2 = one / (doublereal) (*m); + zlascl_("G", &c__0, &c__0, &d__1, &d__2, m, &c__1, &x[i__ * + x_dim1 + 1], ldx, &info2); +/* LAPACK C */ + } else if (z_abs(&x[izamax_(m, &x[i__ * x_dim1 + 1], &c__1) + i__ + * x_dim1]) != zero) { +/* Y(:,i) is zero vector. If X(:,i) is not */ +/* zero, then a warning flag is raised. */ +/* The computation continues but the */ +/* situation will be reported in the output. */ + badxy = TRUE_; + } + } + } + +/* <2> SVD of the data snapshot matrix X. */ +/* ===================================== */ +/* The left singular vectors are stored in the array X. */ +/* The right singular vectors are in the array W. */ +/* The array W will later on contain the eigenvectors */ +/* of a Rayleigh quotient. */ + numrnk = *n; +/* SELECT CASE ( WHTSVD ) */ + if (*whtsvd == 1) { + zgesvd_("O", "S", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], + ldb, &w[w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], & + info1); +/* LA */ + *(unsigned char *)t_or_n__ = 'C'; + } else if (*whtsvd == 2) { + zgesdd_("O", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], ldb, & + w[w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &iwork[1] + , &info1); +/* LAP */ + *(unsigned char *)t_or_n__ = 'C'; + } else if (*whtsvd == 3) { + i__1 = *lrwork - *n; + zgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &rwork[1], + &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, &iwork[1], + liwork, &zwork[1], lzwork, &rwork[*n + 1], &i__1, &info1); +/* LAPACK CA */ + zlacpy_("A", m, &numrnk, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK C */ + *(unsigned char *)t_or_n__ = 'C'; + } else if (*whtsvd == 4) { + i__1 = *lrwork - *n; + zgejsv_("F", "U", jsvopt, "R", "N", "P", m, n, &x[x_offset], ldx, & + rwork[1], &z__[z_offset], ldz, &w[w_offset], ldw, &zwork[1], + lzwork, &rwork[*n + 1], &i__1, &iwork[1], &info1); + zlacpy_("A", m, n, &z__[z_offset], ldz, &x[x_offset], ldx); +/* LAPACK CALL */ + *(unsigned char *)t_or_n__ = 'N'; + xscl1 = rwork[*n + 1]; + xscl2 = rwork[*n + 2]; + if (xscl1 != xscl2) { +/* This is an exceptional situation. If the */ +/* data matrices are not scaled and the */ +/* largest singular value of X overflows. */ +/* In that case ZGEJSV can return the SVD */ +/* in scaled form. The scaling factor can be used */ +/* to rescale the data (X and Y). */ + zlascl_("G", &c__0, &c__0, &xscl1, &xscl2, m, n, &y[y_offset], + ldy, &info2); + } +/* END SELECT */ + } + + if (info1 > 0) { +/* The SVD selected subroutine did not converge. */ +/* Return with an error code. */ + *info = 2; + return 0; + } + + if (rwork[1] == zero) { +/* The largest computed singular value of (scaled) */ +/* X is zero. Return error code -8 */ +/* (the 8th input variable had an illegal value). */ + *k = 0; + *info = -8; + i__1 = -(*info); + xerbla_("ZGEDMD", &i__1); + return 0; + } + +/* <3> Determine the numerical rank of the data */ +/* snapshots matrix X. This depends on the */ +/* parameters NRNK and TOL. */ +/* SELECT CASE ( NRNK ) */ + if (*nrnk == -1) { + *k = 1; + i__1 = numrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (rwork[i__] <= rwork[1] * *tol || rwork[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else if (*nrnk == -2) { + *k = 1; + i__1 = numrnk - 1; + for (i__ = 1; i__ <= i__1; ++i__) { + if (rwork[i__ + 1] <= rwork[i__] * *tol || rwork[i__] <= small) { + myexit_(); + } + ++(*k); + } + } else { + *k = 1; + i__1 = *nrnk; + for (i__ = 2; i__ <= i__1; ++i__) { + if (rwork[i__] <= small) { + myexit_(); + } + ++(*k); + } +/* END SELECT */ + } +/* Now, U = X(1:M,1:K) is the SVD/POD basis for the */ +/* snapshot data in the input matrix X. */ +/* <4> Compute the Rayleigh quotient S = U^H * A * U. */ +/* Depending on the requested outputs, the computation */ +/* is organized to compute additional auxiliary */ +/* matrices (for the residuals and refinements). */ + +/* In all formulas below, we need V_k*Sigma_k^(-1) */ +/* where either V_k is in W(1:N,1:K), or V_k^H is in */ +/* W(1:K,1:N). Here Sigma_k=diag(WORK(1:K)). */ + if (lsame_(t_or_n__, "N")) { + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + d__1 = one / rwork[i__]; + zdscal_(n, &d__1, &w[i__ * w_dim1 + 1], &c__1); +/* W(1:N,i) = (ONE/RWORK(i)) * W(1:N,i) ! INTRINSIC */ +/* BLAS CALL */ + } + } else { +/* This non-unit stride access is due to the fact */ +/* that ZGESVD, ZGESVDQ and ZGESDD return the */ +/* adjoint matrix of the right singular vectors. */ +/* DO i = 1, K */ +/* CALL ZDSCAL( N, ONE/RWORK(i), W(i,1), LDW ) ! BLAS CALL */ +/* ! W(i,1:N) = (ONE/RWORK(i)) * W(i,1:N) ! INTRINSIC */ +/* END DO */ + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + rwork[*n + i__] = one / rwork[i__]; + } + i__1 = *n; + for (j = 1; j <= i__1; ++j) { + i__2 = *k; + for (i__ = 1; i__ <= i__2; ++i__) { + i__3 = i__ + j * w_dim1; + i__4 = *n + i__; + z__2.r = rwork[i__4], z__2.i = zero; + i__5 = i__ + j * w_dim1; + z__1.r = z__2.r * w[i__5].r - z__2.i * w[i__5].i, z__1.i = + z__2.r * w[i__5].i + z__2.i * w[i__5].r; + w[i__3].r = z__1.r, w[i__3].i = z__1.i; + } + } + } + + if (wntref) { + +/* Need A*U(:,1:K)=Y*V_k*inv(diag(WORK(1:K))) */ +/* for computing the refined Ritz vectors */ +/* (optionally, outside ZGEDMD). */ + zgemm_("N", t_or_n__, m, k, n, &zone, &y[y_offset], ldy, &w[w_offset], + ldw, &zzero, &z__[z_offset], ldz); +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),TRANSPOSE(CONJG(W(1:K,1:N)))) ! */ +/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),W(1:N,1:K)) ! */ + +/* At this point Z contains */ +/* A * U(:,1:K) = Y * V_k * Sigma_k^(-1), and */ +/* this is needed for computing the residuals. */ +/* This matrix is returned in the array B and */ +/* it can be used to compute refined Ritz vectors. */ +/* BLA */ + zlacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); +/* B(1:M,1:K) = Z(1:M,1:K) ! INTRINSIC */ +/* BLAS CALL */ + zgemm_("C", "N", k, k, m, &zone, &x[x_offset], ldx, &z__[z_offset], + ldz, &zzero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(TRANSPOSE(CONJG(X(1:M,1:K))),Z(1:M,1:K)) */ +/* At this point S = U^H * A * U is the Rayleigh quotient. */ +/* BLA */ + } else { +/* A * U(:,1:K) is not explicitly needed and the */ +/* computation is organized differently. The Rayleigh */ +/* quotient is computed more efficiently. */ + zgemm_("C", "N", k, n, m, &zone, &x[x_offset], ldx, &y[y_offset], ldy, + &zzero, &z__[z_offset], ldz); +/* Z(1:K,1:N) = MATMUL( TRANSPOSE(CONJG(X(1:M,1:K))), Y(1:M,1:N) */ + + zgemm_("N", t_or_n__, k, k, n, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &s[s_offset], lds); +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),TRANSPOSE(CONJG(W(1:K,1:N)))) ! */ +/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),(W(1:N,1:K))) ! */ +/* At this point S = U^H * A * U is the Rayleigh quotient. */ +/* If the residuals are requested, save scaled V_k into Z. */ +/* Recall that V_k or V_k^H is stored in W. */ +/* BLAS */ + if (wntres || wntex) { + if (lsame_(t_or_n__, "N")) { + zlacpy_("A", n, k, &w[w_offset], ldw, &z__[z_offset], ldz); + } else { + zlacpy_("A", k, n, &w[w_offset], ldw, &z__[z_offset], ldz); + } + } + } + +/* <5> Compute the Ritz values and (if requested) the */ +/* right eigenvectors of the Rayleigh quotient. */ + + zgeev_("N", jobzl, k, &s[s_offset], lds, &eigs[1], &w[w_offset], ldw, &w[ + w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &info1); + +/* W(1:K,1:K) contains the eigenvectors of the Rayleigh */ +/* quotient. See the description of Z. */ +/* Also, see the description of ZGEEV. */ +/* LAPACK CALL */ + if (info1 > 0) { +/* ZGEEV failed to compute the eigenvalues and */ +/* eigenvectors of the Rayleigh quotient. */ + *info = 3; + return 0; + } + +/* <6> Compute the eigenvectors (if requested) and, */ +/* the residuals (if requested). */ + + if (wntvec || wntex) { + if (wntres) { + if (wntref) { +/* Here, if the refinement is requested, we have */ +/* A*U(:,1:K) already computed and stored in Z. */ +/* For the residuals, need Y = A * U(:,1;K) * W. */ + zgemm_("N", "N", m, k, k, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &y[y_offset], ldy); +/* Y(1:M,1:K) = Z(1:M,1:K) * W(1:K,1:K) ! INTRINSIC */ +/* This frees Z; Y contains A * U(:,1:K) * W. */ +/* BLAS CALL */ + } else { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) (or its adjoint) is stored in Z */ + zgemm_(t_or_n__, "N", n, k, k, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + zgemm_("N", "N", m, k, n, &zone, &y[y_offset], ldy, &s[ + s_offset], lds, &zzero, &z__[z_offset], ldz); +/* Save a copy of Z into Y and free Z for holding */ +/* the Ritz vectors. */ + zlacpy_("A", m, k, &z__[z_offset], ldz, &y[y_offset], ldy); + if (wntex) { + zlacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb); + } + } + } else if (wntex) { +/* Compute S = V_k * Sigma_k^(-1) * W, where */ +/* V_k * Sigma_k^(-1) is stored in Z */ + zgemm_(t_or_n__, "N", n, k, k, &zone, &z__[z_offset], ldz, &w[ + w_offset], ldw, &zzero, &s[s_offset], lds); +/* Then, compute Z = Y * S = */ +/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */ +/* = A * U(:,1:K) * W(1:K,1:K) */ + zgemm_("N", "N", m, k, n, &zone, &y[y_offset], ldy, &s[s_offset], + lds, &zzero, &b[b_offset], ldb); +/* The above call replaces the following two calls */ +/* that were used in the developing-testing phase. */ +/* CALL ZGEMM( 'N', 'N', M, K, N, ZONE, Y, LDY, S, & */ +/* LDS, ZZERO, Z, LDZ) */ +/* Save a copy of Z into B and free Z for holding */ +/* the Ritz vectors. */ +/* CALL ZLACPY( 'A', M, K, Z, LDZ, B, LDB ) */ + } + +/* Compute the Ritz vectors */ + if (wntvec) { + zgemm_("N", "N", m, k, k, &zone, &x[x_offset], ldx, &w[w_offset], + ldw, &zzero, &z__[z_offset], ldz); + } +/* Z(1:M,1:K) = MATMUL(X(1:M,1:K), W(1:K,1:K)) ! INTRINSIC */ + +/* BLAS CALL */ + if (wntres) { + i__1 = *k; + for (i__ = 1; i__ <= i__1; ++i__) { + i__2 = i__; + z__1.r = -eigs[i__2].r, z__1.i = -eigs[i__2].i; + zaxpy_(m, &z__1, &z__[i__ * z_dim1 + 1], &c__1, &y[i__ * + y_dim1 + 1], &c__1); +/* Y(1:M,i) = Y(1:M,i) - EIGS(i) * Z(1:M,i) ! INTR */ +/* BLAS */ + res[i__] = dznrm2_(m, &y[i__ * y_dim1 + 1], &c__1); +/* BLAS */ + } + } + } + + if (*whtsvd == 4) { + rwork[*n + 1] = xscl1; + rwork[*n + 2] = xscl2; + } + +/* Successful exit. */ + if (! badxy) { + *info = 0; + } else { +/* A warning on possible data inconsistency. */ +/* This should be a rare event. */ + *info = 4; + } +/* ............................................................ */ + return 0; +/* ...... */ +} /* zgedmd_ */ + diff --git a/lapack-netlib/SRC/zgedmdq.c b/lapack-netlib/SRC/zgedmdq.c index 447b23014..1815f0814 100644 --- a/lapack-netlib/SRC/zgedmdq.c +++ b/lapack-netlib/SRC/zgedmdq.c @@ -509,3 +509,785 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ +/* -- translated by f2c (version 20000121). + You must link the resulting object file with the libraries: + -lf2c -lm (in that order) +*/ + + + +/* Table of constant values */ + +static integer c_n1 = -1; + +/* Subroutine */ int zgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq, + char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n, + doublecomplex *f, integer *ldf, doublecomplex *x, integer *ldx, + doublecomplex *y, integer *ldy, integer *nrnk, doublereal *tol, + integer *k, doublecomplex *eigs, doublecomplex *z__, integer *ldz, + doublereal *res, doublecomplex *b, integer *ldb, doublecomplex *v, + integer *ldv, doublecomplex *s, integer *lds, doublecomplex *zwork, + integer *lzwork, doublereal *work, integer *lwork, integer *iwork, + integer *liwork, integer *info) +{ + /* System generated locals */ + integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1, + z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset, + i__1, i__2; + + /* Local variables */ + doublereal zero; + integer info1; + extern logical lsame_(char *, char *); + char jobvl[1]; + integer minmn; + logical wantq; + integer mlwqr, olwqr; + logical wntex; + doublecomplex zzero; + extern /* Subroutine */ int zgedmd_(char *, char *, char *, char *, + integer *, integer *, integer *, doublecomplex *, integer *, + doublecomplex *, integer *, integer *, doublereal *, integer *, + doublecomplex *, doublecomplex *, integer *, doublereal *, + doublecomplex *, integer *, doublecomplex *, integer *, + doublecomplex *, integer *, doublecomplex *, integer *, + doublereal *, integer *, integer *, integer *, integer *), xerbla_(char *, integer *); + integer mlwdmd, olwdmd; + logical sccolx, sccoly; + extern /* Subroutine */ int zgeqrf_(integer *, integer *, doublecomplex *, + integer *, doublecomplex *, doublecomplex *, integer *, integer * + ), zlacpy_(char *, integer *, integer *, doublecomplex *, integer + *, doublecomplex *, integer *), zlaset_(char *, integer *, + integer *, doublecomplex *, doublecomplex *, doublecomplex *, + integer *); + integer iminwr; + logical wntvec, wntvcf; + integer mlwgqr; + logical wntref; + integer mlwork, olwgqr, olwork, mlrwrk, mlwmqr, olwmqr; + logical lquery, wntres, wnttrf, wntvcq; + extern /* Subroutine */ int zungqr_(integer *, integer *, integer *, + doublecomplex *, integer *, doublecomplex *, doublecomplex *, + integer *, integer *), zunmqr_(char *, char *, integer *, integer + *, integer *, doublecomplex *, integer *, doublecomplex *, + doublecomplex *, integer *, doublecomplex *, integer *, integer *); + doublereal one; + +/* March 2023 */ +/* ..... */ +/* USE iso_fortran_env */ +/* INTEGER, PARAMETER :: WP = real64 */ +/* ..... */ +/* Scalar arguments */ +/* Array arguments */ +/* ..... */ +/* Purpose */ +/* ======= */ +/* ZGEDMDQ computes the Dynamic Mode Decomposition (DMD) for */ +/* a pair of data snapshot matrices, using a QR factorization */ +/* based compression of the data. For the input matrices */ +/* X and Y such that Y = A*X with an unaccessible matrix */ +/* A, ZGEDMDQ computes a certain number of Ritz pairs of A using */ +/* the standard Rayleigh-Ritz extraction from a subspace of */ +/* range(X) that is determined using the leading left singular */ +/* vectors of X. Optionally, ZGEDMDQ returns the residuals */ +/* of the computed Ritz pairs, the information needed for */ +/* a refinement of the Ritz vectors, or the eigenvectors of */ +/* the Exact DMD. */ +/* For further details see the references listed */ +/* below. For more details of the implementation see [3]. */ + +/* References */ +/* ========== */ +/* [1] P. Schmid: Dynamic mode decomposition of numerical */ +/* and experimental data, */ +/* Journal of Fluid Mechanics 656, 5-28, 2010. */ +/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */ +/* decompositions: analysis and enhancements, */ +/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */ +/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */ +/* Mode Decomposition I. Technical report. AIMDyn Inc. */ +/* and LAPACK Working Note 298. */ +/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */ +/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */ +/* Theory and Applications, Journal of Computational */ +/* Dynamics 1(2), 391 -421, 2014. */ + +/* Developed and supported by: */ +/* =========================== */ +/* Developed and coded by Zlatko Drmac, Faculty of Science, */ +/* University of Zagreb; drmac@math.hr */ +/* In cooperation with */ +/* AIMdyn Inc., Santa Barbara, CA. */ +/* and supported by */ +/* - DARPA SBIR project "Koopman Operator-Based Forecasting */ +/* for Nonstationary Processes from Near-Term, Limited */ +/* Observational Data" Contract No: W31P4Q-21-C-0007 */ +/* - DARPA PAI project "Physics-Informed Machine Learning */ +/* Methodologies" Contract No: HR0011-18-9-0033 */ +/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */ +/* Framework for Space-Time Analysis of Process Dynamics" */ +/* Contract No: HR0011-16-C-0116 */ +/* Any opinions, findings and conclusions or recommendations */ +/* expressed in this material are those of the author and */ +/* do not necessarily reflect the views of the DARPA SBIR */ +/* Program Office. */ +/* ============================================================ */ +/* Distribution Statement A: */ +/* Approved for Public Release, Distribution Unlimited. */ +/* Cleared by DARPA on September 29, 2022 */ +/* ============================================================ */ +/* ...................................................................... */ +/* Arguments */ +/* ========= */ +/* JOBS (input) CHARACTER*1 */ +/* Determines whether the initial data snapshots are scaled */ +/* by a diagonal matrix. The data snapshots are the columns */ +/* of F. The leading N-1 columns of F are denoted X and the */ +/* trailing N-1 columns are denoted Y. */ +/* 'S' :: The data snapshots matrices X and Y are multiplied */ +/* with a diagonal matrix D so that X*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'C' :: The snapshots are scaled as with the 'S' option. */ +/* If it is found that an i-th column of X is zero */ +/* vector and the corresponding i-th column of Y is */ +/* non-zero, then the i-th column of Y is set to */ +/* zero and a warning flag is raised. */ +/* 'Y' :: The data snapshots matrices X and Y are multiplied */ +/* by a diagonal matrix D so that Y*D has unit */ +/* nonzero columns (in the Euclidean 2-norm) */ +/* 'N' :: No data scaling. */ +/* ..... */ +/* JOBZ (input) CHARACTER*1 */ +/* Determines whether the eigenvectors (Koopman modes) will */ +/* be computed. */ +/* 'V' :: The eigenvectors (Koopman modes) will be computed */ +/* and returned in the matrix Z. */ +/* See the description of Z. */ +/* 'F' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Z*V, where Z */ +/* is orthonormal and V contains the eigenvectors */ +/* of the corresponding Rayleigh quotient. */ +/* See the descriptions of F, V, Z. */ +/* 'Q' :: The eigenvectors (Koopman modes) will be returned */ +/* in factored form as the product Q*Z, where Z */ +/* contains the eigenvectors of the compression of the */ +/* underlying discretized operator onto the span of */ +/* the data snapshots. See the descriptions of F, V, Z. */ +/* Q is from the initial QR factorization. */ +/* 'N' :: The eigenvectors are not computed. */ +/* ..... */ +/* JOBR (input) CHARACTER*1 */ +/* Determines whether to compute the residuals. */ +/* 'R' :: The residuals for the computed eigenpairs will */ +/* be computed and stored in the array RES. */ +/* See the description of RES. */ +/* For this option to be legal, JOBZ must be 'V'. */ +/* 'N' :: The residuals are not computed. */ +/* ..... */ +/* JOBQ (input) CHARACTER*1 */ +/* Specifies whether to explicitly compute and return the */ +/* unitary matrix from the QR factorization. */ +/* 'Q' :: The matrix Q of the QR factorization of the data */ +/* snapshot matrix is computed and stored in the */ +/* array F. See the description of F. */ +/* 'N' :: The matrix Q is not explicitly computed. */ +/* ..... */ +/* JOBT (input) CHARACTER*1 */ +/* Specifies whether to return the upper triangular factor */ +/* from the QR factorization. */ +/* 'R' :: The matrix R of the QR factorization of the data */ +/* snapshot matrix F is returned in the array Y. */ +/* See the description of Y and Further details. */ +/* 'N' :: The matrix R is not returned. */ +/* ..... */ +/* JOBF (input) CHARACTER*1 */ +/* Specifies whether to store information needed for post- */ +/* processing (e.g. computing refined Ritz vectors) */ +/* 'R' :: The matrix needed for the refinement of the Ritz */ +/* vectors is computed and stored in the array B. */ +/* See the description of B. */ +/* 'E' :: The unscaled eigenvectors of the Exact DMD are */ +/* computed and returned in the array B. See the */ +/* description of B. */ +/* 'N' :: No eigenvector refinement data is computed. */ +/* To be useful on exit, this option needs JOBQ='Q'. */ +/* ..... */ +/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */ +/* Allows for a selection of the SVD algorithm from the */ +/* LAPACK library. */ +/* 1 :: ZGESVD (the QR SVD algorithm) */ +/* 2 :: ZGESDD (the Divide and Conquer algorithm; if enough */ +/* workspace available, this is the fastest option) */ +/* 3 :: ZGESVDQ (the preconditioned QR SVD ; this and 4 */ +/* are the most accurate options) */ +/* 4 :: ZGEJSV (the preconditioned Jacobi SVD; this and 3 */ +/* are the most accurate options) */ +/* For the four methods above, a significant difference in */ +/* the accuracy of small singular values is possible if */ +/* the snapshots vary in norm so that X is severely */ +/* ill-conditioned. If small (smaller than EPS*||X||) */ +/* singular values are of interest and JOBS=='N', then */ +/* the options (3, 4) give the most accurate results, where */ +/* the option 4 is slightly better and with stronger */ +/* theoretical background. */ +/* If JOBS=='S', i.e. the columns of X will be normalized, */ +/* then all methods give nearly equally accurate results. */ +/* ..... */ +/* M (input) INTEGER, M >= 0 */ +/* The state space dimension (the number of rows of F). */ +/* ..... */ +/* N (input) INTEGER, 0 <= N <= M */ +/* The number of data snapshots from a single trajectory, */ +/* taken at equidistant discrete times. This is the */ +/* number of columns of F. */ +/* ..... */ +/* F (input/output) COMPLEX(KIND=WP) M-by-N array */ +/* > On entry, */ +/* the columns of F are the sequence of data snapshots */ +/* from a single trajectory, taken at equidistant discrete */ +/* times. It is assumed that the column norms of F are */ +/* in the range of the normalized floating point numbers. */ +/* < On exit, */ +/* If JOBQ == 'Q', the array F contains the orthogonal */ +/* matrix/factor of the QR factorization of the initial */ +/* data snapshots matrix F. See the description of JOBQ. */ +/* If JOBQ == 'N', the entries in F strictly below the main */ +/* diagonal contain, column-wise, the information on the */ +/* Householder vectors, as returned by ZGEQRF. The */ +/* remaining information to restore the orthogonal matrix */ +/* of the initial QR factorization is stored in ZWORK(1:MIN(M,N)). */ +/* See the description of ZWORK. */ +/* ..... */ +/* LDF (input) INTEGER, LDF >= M */ +/* The leading dimension of the array F. */ +/* ..... */ +/* X (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array */ +/* X is used as workspace to hold representations of the */ +/* leading N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, the leading K columns of X contain the leading */ +/* K left singular vectors of the above described content */ +/* of X. To lift them to the space of the left singular */ +/* vectors U(:,1:K) of the input data, pre-multiply with the */ +/* Q factor from the initial QR factorization. */ +/* See the descriptions of F, K, V and Z. */ +/* ..... */ +/* LDX (input) INTEGER, LDX >= N */ +/* The leading dimension of the array X. */ +/* ..... */ +/* Y (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N) array */ +/* Y is used as workspace to hold representations of the */ +/* trailing N-1 snapshots in the orthonormal basis computed */ +/* in the QR factorization of F. */ +/* On exit, */ +/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */ +/* triangular factor from the QR factorization of the data */ +/* snapshot matrix F. */ +/* ..... */ +/* LDY (input) INTEGER , LDY >= N */ +/* The leading dimension of the array Y. */ +/* ..... */ +/* NRNK (input) INTEGER */ +/* Determines the mode how to compute the numerical rank, */ +/* i.e. how to truncate small singular values of the input */ +/* matrix X. On input, if */ +/* NRNK = -1 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(1) */ +/* This option is recommended. */ +/* NRNK = -2 :: i-th singular value sigma(i) is truncated */ +/* if sigma(i) <= TOL*sigma(i-1) */ +/* This option is included for R&D purposes. */ +/* It requires highly accurate SVD, which */ +/* may not be feasible. */ +/* The numerical rank can be enforced by using positive */ +/* value of NRNK as follows: */ +/* 0 < NRNK <= N-1 :: at most NRNK largest singular values */ +/* will be used. If the number of the computed nonzero */ +/* singular values is less than NRNK, then only those */ +/* nonzero values will be used and the actually used */ +/* dimension is less than NRNK. The actual number of */ +/* the nonzero singular values is returned in the variable */ +/* K. See the description of K. */ +/* ..... */ +/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */ +/* The tolerance for truncating small singular values. */ +/* See the description of NRNK. */ +/* ..... */ +/* K (output) INTEGER, 0 <= K <= N */ +/* The dimension of the SVD/POD basis for the leading N-1 */ +/* data snapshots (columns of F) and the number of the */ +/* computed Ritz pairs. The value of K is determined */ +/* according to the rule set by the parameters NRNK and */ +/* TOL. See the descriptions of NRNK and TOL. */ +/* ..... */ +/* EIGS (output) COMPLEX(KIND=WP) (N-1)-by-1 array */ +/* The leading K (K<=N-1) entries of EIGS contain */ +/* the computed eigenvalues (Ritz values). */ +/* See the descriptions of K, and Z. */ +/* ..... */ +/* Z (workspace/output) COMPLEX(KIND=WP) M-by-(N-1) array */ +/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */ +/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */ +/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */ +/* Z*V, where Z contains orthonormal matrix (the product of */ +/* Q from the initial QR factorization and the SVD/POD_basis */ +/* returned by ZGEDMD in X) and the second factor (the */ +/* eigenvectors of the Rayleigh quotient) is in the array V, */ +/* as returned by ZGEDMD. That is, X(:,1:K)*V(:,i) */ +/* is an eigenvector corresponding to EIGS(i). The columns */ +/* of V(1:K,1:K) are the computed eigenvectors of the */ +/* K-by-K Rayleigh quotient. */ +/* See the descriptions of EIGS, X and V. */ +/* ..... */ +/* LDZ (input) INTEGER , LDZ >= M */ +/* The leading dimension of the array Z. */ +/* ..... */ +/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */ +/* RES(1:K) contains the residuals for the K computed */ +/* Ritz pairs, */ +/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */ +/* See the description of EIGS and Z. */ +/* ..... */ +/* B (output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array. */ +/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */ +/* be used for computing the refined vectors; see further */ +/* details in the provided references. */ +/* If JOBF == 'E', B(1:N,1;K) contains */ +/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */ +/* Exact DMD, up to scaling by the inverse eigenvalues. */ +/* In both cases, the content of B can be lifted to the */ +/* original dimension of the input data by pre-multiplying */ +/* with the Q factor from the initial QR factorization. */ +/* Here A denotes a compression of the underlying operator. */ +/* See the descriptions of F and X. */ +/* If JOBF =='N', then B is not referenced. */ +/* ..... */ +/* LDB (input) INTEGER, LDB >= MIN(M,N) */ +/* The leading dimension of the array B. */ +/* ..... */ +/* V (workspace/output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */ +/* On exit, V(1:K,1:K) V contains the K eigenvectors of */ +/* the Rayleigh quotient. The Ritz vectors */ +/* (returned in Z) are the product of Q from the initial QR */ +/* factorization (see the description of F) X (see the */ +/* description of X) and V. */ +/* ..... */ +/* LDV (input) INTEGER, LDV >= N-1 */ +/* The leading dimension of the array V. */ +/* ..... */ +/* S (output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */ +/* The array S(1:K,1:K) is used for the matrix Rayleigh */ +/* quotient. This content is overwritten during */ +/* the eigenvalue decomposition by ZGEEV. */ +/* See the description of K. */ +/* ..... */ +/* LDS (input) INTEGER, LDS >= N-1 */ +/* The leading dimension of the array S. */ +/* ..... */ +/* ZWORK (workspace/output) COMPLEX(KIND=WP) LWORK-by-1 array */ +/* On exit, */ +/* ZWORK(1:MIN(M,N)) contains the scalar factors of the */ +/* elementary reflectors as returned by ZGEQRF of the */ +/* M-by-N input matrix F. */ +/* If the call to ZGEDMDQ is only workspace query, then */ +/* ZWORK(1) contains the minimal complex workspace length and */ +/* ZWORK(2) is the optimal complex workspace length. */ +/* Hence, the length of work is at least 2. */ +/* See the description of LZWORK. */ +/* ..... */ +/* LZWORK (input) INTEGER */ +/* The minimal length of the workspace vector ZWORK. */ +/* LZWORK is calculated as follows: */ +/* Let MLWQR = N (minimal workspace for ZGEQRF[M,N]) */ +/* MLWDMD = minimal workspace for ZGEDMD (see the */ +/* description of LWORK in ZGEDMD) */ +/* MLWMQR = N (minimal workspace for */ +/* ZUNMQR['L','N',M,N,N]) */ +/* MLWGQR = N (minimal workspace for ZUNGQR[M,N,N]) */ +/* MINMN = MIN(M,N) */ +/* Then */ +/* LZWORK = MAX(2, MIN(M,N)+MLWQR, MINMN+MLWDMD) */ +/* is further updated as follows: */ +/* if JOBZ == 'V' or JOBZ == 'F' THEN */ +/* LZWORK = MAX(LZWORK, MINMN+MLWMQR) */ +/* if JOBQ == 'Q' THEN */ +/* LZWORK = MAX(ZLWORK, MINMN+MLWGQR) */ + +/* ..... */ +/* WORK (workspace/output) REAL(KIND=WP) LWORK-by-1 array */ +/* On exit, */ +/* WORK(1:N-1) contains the singular values of */ +/* the input submatrix F(1:M,1:N-1). */ +/* If the call to ZGEDMDQ is only workspace query, then */ +/* WORK(1) contains the minimal workspace length and */ +/* WORK(2) is the optimal workspace length. hence, the */ +/* length of work is at least 2. */ +/* See the description of LWORK. */ +/* ..... */ +/* LWORK (input) INTEGER */ +/* The minimal length of the workspace vector WORK. */ +/* LWORK is the same as in ZGEDMD, because in ZGEDMDQ */ +/* only ZGEDMD requires real workspace for snapshots */ +/* of dimensions MIN(M,N)-by-(N-1). */ +/* If on entry LWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace length for WORK. */ +/* ..... */ +/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */ +/* Workspace that is required only if WHTSVD equals */ +/* 2 , 3 or 4. (See the description of WHTSVD). */ +/* If on entry LWORK =-1 or LIWORK=-1, then the */ +/* minimal length of IWORK is computed and returned in */ +/* IWORK(1). See the description of LIWORK. */ +/* ..... */ +/* LIWORK (input) INTEGER */ +/* The minimal length of the workspace vector IWORK. */ +/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */ +/* Let M1=MIN(M,N), N1=N-1. Then */ +/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M1,N1)) */ +/* If WHTSVD == 3, then LIWORK >= MAX(1,M1+N1-1) */ +/* If WHTSVD == 4, then LIWORK >= MAX(3,M1+3*N1) */ +/* If on entry LIWORK = -1, then a workspace query is */ +/* assumed and the procedure only computes the minimal */ +/* and the optimal workspace lengths for both WORK and */ +/* IWORK. See the descriptions of WORK and IWORK. */ +/* ..... */ +/* INFO (output) INTEGER */ +/* -i < 0 :: On entry, the i-th argument had an */ +/* illegal value */ +/* = 0 :: Successful return. */ +/* = 1 :: Void input. Quick exit (M=0 or N=0). */ +/* = 2 :: The SVD computation of X did not converge. */ +/* Suggestion: Check the input data and/or */ +/* repeat with different WHTSVD. */ +/* = 3 :: The computation of the eigenvalues did not */ +/* converge. */ +/* = 4 :: If data scaling was requested on input and */ +/* the procedure found inconsistency in the data */ +/* such that for some column index i, */ +/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */ +/* to zero if JOBS=='C'. The computation proceeds */ +/* with original or modified data and warning */ +/* flag is set with INFO=4. */ +/* ............................................................. */ +/* ............................................................. */ +/* Parameters */ +/* ~~~~~~~~~~ */ +/* COMPLEX(KIND=WP), PARAMETER :: ZONE = ( 1.0_WP, 0.0_WP ) */ + +/* Local scalars */ +/* ~~~~~~~~~~~~~ */ + +/* External functions (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~ */ + +/* External subroutines (BLAS and LAPACK) */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* External subroutines */ +/* ~~~~~~~~~~~~~~~~~~~~ */ +/* Intrinsic functions */ +/* ~~~~~~~~~~~~~~~~~~~ */ +/* .......................................................... */ + /* Parameter adjustments */ + f_dim1 = *ldf; + f_offset = 1 + f_dim1 * 1; + f -= f_offset; + x_dim1 = *ldx; + x_offset = 1 + x_dim1 * 1; + x -= x_offset; + y_dim1 = *ldy; + y_offset = 1 + y_dim1 * 1; + y -= y_offset; + --eigs; + z_dim1 = *ldz; + z_offset = 1 + z_dim1 * 1; + z__ -= z_offset; + --res; + b_dim1 = *ldb; + b_offset = 1 + b_dim1 * 1; + b -= b_offset; + v_dim1 = *ldv; + v_offset = 1 + v_dim1 * 1; + v -= v_offset; + s_dim1 = *lds; + s_offset = 1 + s_dim1 * 1; + s -= s_offset; + --zwork; + --work; + --iwork; + + /* Function Body */ + one = 1.f; + zero = 0.f; + zzero.r = 0.f, zzero.i = 0.f; + +/* Test the input arguments */ + wntres = lsame_(jobr, "R"); + sccolx = lsame_(jobs, "S") || lsame_(jobs, "C"); + sccoly = lsame_(jobs, "Y"); + wntvec = lsame_(jobz, "V"); + wntvcf = lsame_(jobz, "F"); + wntvcq = lsame_(jobz, "Q"); + wntref = lsame_(jobf, "R"); + wntex = lsame_(jobf, "E"); + wantq = lsame_(jobq, "Q"); + wnttrf = lsame_(jobt, "R"); + minmn = f2cmin(*m,*n); + *info = 0; + lquery = *lzwork == -1 || *lwork == -1 || *liwork == -1; + + if (! (sccolx || sccoly || lsame_(jobs, "N"))) { + *info = -1; + } else if (! (wntvec || wntvcf || wntvcq || lsame_(jobz, "N"))) { + *info = -2; + } else if (! (wntres || lsame_(jobr, "N")) || + wntres && lsame_(jobz, "N")) { + *info = -3; + } else if (! (wantq || lsame_(jobq, "N"))) { + *info = -4; + } else if (! (wnttrf || lsame_(jobt, "N"))) { + *info = -5; + } else if (! (wntref || wntex || lsame_(jobf, "N"))) + { + *info = -6; + } else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd == + 4)) { + *info = -7; + } else if (*m < 0) { + *info = -8; + } else if (*n < 0 || *n > *m + 1) { + *info = -9; + } else if (*ldf < *m) { + *info = -11; + } else if (*ldx < minmn) { + *info = -13; + } else if (*ldy < minmn) { + *info = -15; + } else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) { + *info = -16; + } else if (*tol < zero || *tol >= one) { + *info = -17; + } else if (*ldz < *m) { + *info = -21; + } else if ((wntref || wntex) && *ldb < minmn) { + *info = -24; + } else if (*ldv < *n - 1) { + *info = -26; + } else if (*lds < *n - 1) { + *info = -28; + } + + if (wntvec || wntvcf || wntvcq) { + *(unsigned char *)jobvl = 'V'; + } else { + *(unsigned char *)jobvl = 'N'; + } + if (*info == 0) { +/* Compute the minimal and the optimal workspace */ +/* requirements. Simulate running the code and */ +/* determine minimal and optimal sizes of the */ +/* workspace at any moment of the run. */ + if (*n == 0 || *n == 1) { +/* All output except K is void. INFO=1 signals */ +/* the void input. In case of a workspace query, */ +/* the minimal workspace lengths are returned. */ + if (lquery) { + iwork[1] = 1; + zwork[1].r = 2., zwork[1].i = 0.; + zwork[2].r = 2., zwork[2].i = 0.; + work[1] = 2.; + work[2] = 2.; + } else { + *k = 0; + } + *info = 1; + return 0; + } + mlrwrk = 2; + mlwork = 2; + olwork = 2; + iminwr = 1; + mlwqr = f2cmax(1,*n); +/* Minimal workspace length for ZGEQRF. */ +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + zgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[1], &c_n1, & + info1); + olwqr = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwqr; + olwork = f2cmax(i__1,i__2); + } + i__1 = *n - 1; + zgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], + ldx, &y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset] + , ldz, &res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[ + s_offset], lds, &zwork[1], &c_n1, &work[1], &c_n1, &iwork[1], + &c_n1, &info1); + mlwdmd = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwdmd; + mlwork = f2cmax(i__1,i__2); +/* Computing MAX */ + i__1 = mlrwrk, i__2 = (integer) work[1]; + mlrwrk = f2cmax(i__1,i__2); + iminwr = f2cmax(iminwr,iwork[1]); + if (lquery) { + olwdmd = (integer) zwork[2].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwdmd; + olwork = f2cmax(i__1,i__2); + } + if (wntvec || wntvcf) { + mlwmqr = f2cmax(1,*n); +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwmqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + zunmqr_("L", "N", m, n, &minmn, &f[f_offset], ldf, &zwork[1], + &z__[z_offset], ldz, &zwork[1], &c_n1, &info1); + olwmqr = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwmqr; + olwork = f2cmax(i__1,i__2); + } + } + if (wantq) { + mlwgqr = f2cmax(1,*n); +/* Computing MAX */ + i__1 = mlwork, i__2 = minmn + mlwgqr; + mlwork = f2cmax(i__1,i__2); + if (lquery) { + zungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], & + zwork[1], &c_n1, &info1); + olwgqr = (integer) zwork[1].r; +/* Computing MAX */ + i__1 = olwork, i__2 = minmn + olwgqr; + olwork = f2cmax(i__1,i__2); + } + } + if (*liwork < iminwr && ! lquery) { + *info = -34; + } + if (*lwork < mlrwrk && ! lquery) { + *info = -32; + } + if (*lzwork < mlwork && ! lquery) { + *info = -30; + } + } + if (*info != 0) { + i__1 = -(*info); + xerbla_("ZGEDMDQ", &i__1); + return 0; + } else if (lquery) { +/* Return minimal and optimal workspace sizes */ + iwork[1] = iminwr; + zwork[1].r = (doublereal) mlwork, zwork[1].i = 0.; + zwork[2].r = (doublereal) olwork, zwork[2].i = 0.; + work[1] = (doublereal) mlrwrk; + work[2] = (doublereal) mlrwrk; + return 0; + } +/* ..... */ +/* Initial QR factorization that is used to represent the */ +/* snapshots as elements of lower dimensional subspace. */ +/* For large scale computation with M >> N, at this place */ +/* one can use an out of core QRF. */ + + i__1 = *lzwork - minmn; + zgeqrf_(m, n, &f[f_offset], ldf, &zwork[1], &zwork[minmn + 1], &i__1, & + info1); + +/* Define X and Y as the snapshots representations in the */ +/* orthogonal basis computed in the QR factorization. */ +/* X corresponds to the leading N-1 and Y to the trailing */ +/* N-1 snapshots. */ + i__1 = *n - 1; + zlaset_("L", &minmn, &i__1, &zzero, &zzero, &x[x_offset], ldx); + i__1 = *n - 1; + zlacpy_("U", &minmn, &i__1, &f[f_offset], ldf, &x[x_offset], ldx); + i__1 = *n - 1; + zlacpy_("A", &minmn, &i__1, &f[(f_dim1 << 1) + 1], ldf, &y[y_offset], ldy); + if (*m >= 3) { + i__1 = minmn - 2; + i__2 = *n - 2; + zlaset_("L", &i__1, &i__2, &zzero, &zzero, &y[y_dim1 + 3], ldy); + } + +/* Compute the DMD of the projected snapshot pairs (X,Y) */ + i__1 = *n - 1; + i__2 = *lzwork - minmn; + zgedmd_(jobs, jobvl, jobr, jobf, whtsvd, &minmn, &i__1, &x[x_offset], ldx, + &y[y_offset], ldy, nrnk, tol, k, &eigs[1], &z__[z_offset], ldz, & + res[1], &b[b_offset], ldb, &v[v_offset], ldv, &s[s_offset], lds, & + zwork[minmn + 1], &i__2, &work[1], lwork, &iwork[1], liwork, & + info1); + if (info1 == 2 || info1 == 3) { +/* Return with error code. See ZGEDMD for details. */ + *info = info1; + return 0; + } else { + *info = info1; + } + +/* The Ritz vectors (Koopman modes) can be explicitly */ +/* formed or returned in factored form. */ + if (wntvec) { +/* Compute the eigenvectors explicitly. */ + if (*m > minmn) { + i__1 = *m - minmn; + zlaset_("A", &i__1, k, &zzero, &zzero, &z__[minmn + 1 + z_dim1], + ldz); + } + i__1 = *lzwork - minmn; + zunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[ + z_offset], ldz, &zwork[minmn + 1], &i__1, &info1); + } else if (wntvcf) { +/* Return the Ritz vectors (eigenvectors) in factored */ +/* form Z*V, where Z contains orthonormal matrix (the */ +/* product of Q from the initial QR factorization and */ +/* the SVD/POD_basis returned by ZGEDMD in X) and the */ +/* second factor (the eigenvectors of the Rayleigh */ +/* quotient) is in the array V, as returned by ZGEDMD. */ + zlacpy_("A", n, k, &x[x_offset], ldx, &z__[z_offset], ldz); + if (*m > *n) { + i__1 = *m - *n; + zlaset_("A", &i__1, k, &zzero, &zzero, &z__[*n + 1 + z_dim1], ldz); + } + i__1 = *lzwork - minmn; + zunmqr_("L", "N", m, k, &minmn, &f[f_offset], ldf, &zwork[1], &z__[ + z_offset], ldz, &zwork[minmn + 1], &i__1, &info1); + } + +/* Some optional output variables: */ + +/* The upper triangular factor R in the initial QR */ +/* factorization is optionally returned in the array Y. */ +/* This is useful if this call to ZGEDMDQ is to be */ +/* followed by a streaming DMD that is implemented in a */ +/* QR compressed form. */ + if (wnttrf) { +/* Return the upper triangular R in Y */ + zlaset_("A", &minmn, n, &zzero, &zzero, &y[y_offset], ldy); + zlacpy_("U", &minmn, n, &f[f_offset], ldf, &y[y_offset], ldy); + } + +/* The orthonormal/unitary factor Q in the initial QR */ +/* factorization is optionally returned in the array F. */ +/* Same as with the triangular factor above, this is */ +/* useful in a streaming DMD. */ + if (wantq) { +/* Q overwrites F */ + i__1 = *lzwork - minmn; + zungqr_(m, &minmn, &minmn, &f[f_offset], ldf, &zwork[1], &zwork[minmn + + 1], &i__1, &info1); + } + + return 0; + +} /* zgedmdq_ */ +