Add functional C replacements for ?GEDMD?
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Martin Kroeker
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@ -509,3 +509,781 @@ static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integ
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/* -- translated by f2c (version 20000121).
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You must link the resulting object file with the libraries:
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-lf2c -lm (in that order)
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*/
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/* Table of constant values */
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static integer c_n1 = -1;
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/* Subroutine */ int cgedmdq_(char *jobs, char *jobz, char *jobr, char *jobq,
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char *jobt, char *jobf, integer *whtsvd, integer *m, integer *n,
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complex *f, integer *ldf, complex *x, integer *ldx, complex *y,
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integer *ldy, integer *nrnk, real *tol, integer *k, complex *eigs,
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complex *z__, integer *ldz, real *res, complex *b, integer *ldb,
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complex *v, integer *ldv, complex *s, integer *lds, complex *zwork,
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integer *lzwork, real *work, integer *lwork, integer *iwork, integer *
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liwork, integer *info)
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{
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/* System generated locals */
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integer f_dim1, f_offset, x_dim1, x_offset, y_dim1, y_offset, z_dim1,
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z_offset, b_dim1, b_offset, v_dim1, v_offset, s_dim1, s_offset,
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i__1, i__2;
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/* Local variables */
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real zero;
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integer info1;
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extern logical lsame_(char *, char *);
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char jobvl[1];
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integer minmn;
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logical wantq;
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integer mlwqr, olwqr;
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logical wntex;
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complex zzero;
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extern /* Subroutine */ int cgedmd_(char *, char *, char *, char *,
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integer *, integer *, integer *, complex *, integer *, complex *,
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integer *, integer *, real *, integer *, complex *, complex *,
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integer *, real *, complex *, integer *, complex *, integer *,
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complex *, integer *, complex *, integer *, real *, integer *,
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integer *, integer *, integer *),
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cgeqrf_(integer *, integer *, complex *, integer *, complex *,
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complex *, integer *, integer *), clacpy_(char *, integer *,
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integer *, complex *, integer *, complex *, integer *),
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claset_(char *, integer *, integer *, complex *, complex *,
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complex *, integer *), xerbla_(char *, integer *);
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integer mlwdmd, olwdmd;
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logical sccolx, sccoly;
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extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
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complex *, integer *, complex *, complex *, integer *, integer *);
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integer iminwr;
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logical wntvec, wntvcf;
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integer mlwgqr;
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logical wntref;
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integer mlwork, olwgqr, olwork, mlrwrk, mlwmqr, olwmqr;
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logical lquery, wntres, wnttrf, wntvcq;
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extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
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integer *, complex *, integer *, complex *, complex *, integer *,
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complex *, integer *, integer *);
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real one;
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/* March 2023 */
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/* ..... */
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/* USE iso_fortran_env */
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/* INTEGER, PARAMETER :: WP = real32 */
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/* ..... */
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/* Scalar arguments */
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/* Array arguments */
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/* ..... */
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/* Purpose */
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/* ======= */
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/* CGEDMDQ computes the Dynamic Mode Decomposition (DMD) for */
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/* a pair of data snapshot matrices, using a QR factorization */
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/* based compression of the data. For the input matrices */
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/* X and Y such that Y = A*X with an unaccessible matrix */
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/* A, CGEDMDQ computes a certain number of Ritz pairs of A using */
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/* the standard Rayleigh-Ritz extraction from a subspace of */
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/* range(X) that is determined using the leading left singular */
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/* vectors of X. Optionally, CGEDMDQ returns the residuals */
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/* of the computed Ritz pairs, the information needed for */
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/* a refinement of the Ritz vectors, or the eigenvectors of */
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/* the Exact DMD. */
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/* For further details see the references listed */
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/* below. For more details of the implementation see [3]. */
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/* References */
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/* ========== */
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/* [1] P. Schmid: Dynamic mode decomposition of numerical */
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/* and experimental data, */
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/* Journal of Fluid Mechanics 656, 5-28, 2010. */
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/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */
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/* decompositions: analysis and enhancements, */
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/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */
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/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */
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/* Mode Decomposition I. Technical report. AIMDyn Inc. */
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/* and LAPACK Working Note 298. */
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/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */
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/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */
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/* Theory and Applications, Journal of Computational */
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/* Dynamics 1(2), 391 -421, 2014. */
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/* Developed and supported by: */
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/* =========================== */
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/* Developed and coded by Zlatko Drmac, Faculty of Science, */
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/* University of Zagreb; drmac@math.hr */
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/* In cooperation with */
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/* AIMdyn Inc., Santa Barbara, CA. */
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/* and supported by */
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/* - DARPA SBIR project "Koopman Operator-Based Forecasting */
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/* for Nonstationary Processes from Near-Term, Limited */
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/* Observational Data" Contract No: W31P4Q-21-C-0007 */
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/* - DARPA PAI project "Physics-Informed Machine Learning */
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/* Methodologies" Contract No: HR0011-18-9-0033 */
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/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */
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/* Framework for Space-Time Analysis of Process Dynamics" */
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/* Contract No: HR0011-16-C-0116 */
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/* Any opinions, findings and conclusions or recommendations */
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/* expressed in this material are those of the author and */
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/* do not necessarily reflect the views of the DARPA SBIR */
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/* Program Office. */
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/* ============================================================ */
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/* Distribution Statement A: */
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/* Approved for Public Release, Distribution Unlimited. */
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/* Cleared by DARPA on September 29, 2022 */
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/* ============================================================ */
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/* ...................................................................... */
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/* Arguments */
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/* ========= */
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/* JOBS (input) CHARACTER*1 */
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/* Determines whether the initial data snapshots are scaled */
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/* by a diagonal matrix. The data snapshots are the columns */
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/* of F. The leading N-1 columns of F are denoted X and the */
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/* trailing N-1 columns are denoted Y. */
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/* 'S' :: The data snapshots matrices X and Y are multiplied */
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/* with a diagonal matrix D so that X*D has unit */
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/* nonzero columns (in the Euclidean 2-norm) */
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/* 'C' :: The snapshots are scaled as with the 'S' option. */
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/* If it is found that an i-th column of X is zero */
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/* vector and the corresponding i-th column of Y is */
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/* non-zero, then the i-th column of Y is set to */
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/* zero and a warning flag is raised. */
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/* 'Y' :: The data snapshots matrices X and Y are multiplied */
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/* by a diagonal matrix D so that Y*D has unit */
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/* nonzero columns (in the Euclidean 2-norm) */
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/* 'N' :: No data scaling. */
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/* ..... */
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/* JOBZ (input) CHARACTER*1 */
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/* Determines whether the eigenvectors (Koopman modes) will */
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/* be computed. */
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/* 'V' :: The eigenvectors (Koopman modes) will be computed */
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/* and returned in the matrix Z. */
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/* See the description of Z. */
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/* 'F' :: The eigenvectors (Koopman modes) will be returned */
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/* in factored form as the product Z*V, where Z */
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/* is orthonormal and V contains the eigenvectors */
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/* of the corresponding Rayleigh quotient. */
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/* See the descriptions of F, V, Z. */
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/* 'Q' :: The eigenvectors (Koopman modes) will be returned */
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/* in factored form as the product Q*Z, where Z */
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/* contains the eigenvectors of the compression of the */
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/* underlying discretised operator onto the span of */
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/* the data snapshots. See the descriptions of F, V, Z. */
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/* Q is from the inital QR facorization. */
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/* 'N' :: The eigenvectors are not computed. */
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/* ..... */
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/* JOBR (input) CHARACTER*1 */
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/* Determines whether to compute the residuals. */
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/* 'R' :: The residuals for the computed eigenpairs will */
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/* be computed and stored in the array RES. */
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/* See the description of RES. */
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/* For this option to be legal, JOBZ must be 'V'. */
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/* 'N' :: The residuals are not computed. */
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/* ..... */
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/* JOBQ (input) CHARACTER*1 */
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/* Specifies whether to explicitly compute and return the */
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/* unitary matrix from the QR factorization. */
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/* 'Q' :: The matrix Q of the QR factorization of the data */
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/* snapshot matrix is computed and stored in the */
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/* array F. See the description of F. */
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/* 'N' :: The matrix Q is not explicitly computed. */
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/* ..... */
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/* JOBT (input) CHARACTER*1 */
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/* Specifies whether to return the upper triangular factor */
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/* from the QR factorization. */
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/* 'R' :: The matrix R of the QR factorization of the data */
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/* snapshot matrix F is returned in the array Y. */
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/* See the description of Y and Further details. */
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/* 'N' :: The matrix R is not returned. */
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/* ..... */
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/* JOBF (input) CHARACTER*1 */
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/* Specifies whether to store information needed for post- */
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/* processing (e.g. computing refined Ritz vectors) */
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/* 'R' :: The matrix needed for the refinement of the Ritz */
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/* vectors is computed and stored in the array B. */
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/* See the description of B. */
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/* 'E' :: The unscaled eigenvectors of the Exact DMD are */
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/* computed and returned in the array B. See the */
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/* description of B. */
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/* 'N' :: No eigenvector refinement data is computed. */
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/* To be useful on exit, this option needs JOBQ='Q'. */
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/* ..... */
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/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */
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/* Allows for a selection of the SVD algorithm from the */
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/* LAPACK library. */
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/* 1 :: CGESVD (the QR SVD algorithm) */
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/* 2 :: CGESDD (the Divide and Conquer algorithm; if enough */
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/* workspace available, this is the fastest option) */
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/* 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 */
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/* are the most accurate options) */
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/* 4 :: CGEJSV (the preconditioned Jacobi SVD; this and 3 */
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/* are the most accurate options) */
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/* For the four methods above, a significant difference in */
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/* the accuracy of small singular values is possible if */
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/* the snapshots vary in norm so that X is severely */
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/* ill-conditioned. If small (smaller than EPS*||X||) */
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/* singular values are of interest and JOBS=='N', then */
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/* the options (3, 4) give the most accurate results, where */
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/* the option 4 is slightly better and with stronger */
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/* theoretical background. */
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/* If JOBS=='S', i.e. the columns of X will be normalized, */
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/* then all methods give nearly equally accurate results. */
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/* ..... */
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/* M (input) INTEGER, M >= 0 */
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/* The state space dimension (the number of rows of F). */
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/* ..... */
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/* N (input) INTEGER, 0 <= N <= M */
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/* The number of data snapshots from a single trajectory, */
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/* taken at equidistant discrete times. This is the */
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/* number of columns of F. */
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/* ..... */
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/* F (input/output) COMPLEX(KIND=WP) M-by-N array */
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/* > On entry, */
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/* the columns of F are the sequence of data snapshots */
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/* from a single trajectory, taken at equidistant discrete */
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/* times. It is assumed that the column norms of F are */
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/* in the range of the normalized floating point numbers. */
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/* < On exit, */
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/* If JOBQ == 'Q', the array F contains the orthogonal */
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/* matrix/factor of the QR factorization of the initial */
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/* data snapshots matrix F. See the description of JOBQ. */
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/* If JOBQ == 'N', the entries in F strictly below the main */
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/* diagonal contain, column-wise, the information on the */
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/* Householder vectors, as returned by CGEQRF. The */
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/* remaining information to restore the orthogonal matrix */
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/* of the initial QR factorization is stored in ZWORK(1:MIN(M,N)). */
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/* See the description of ZWORK. */
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/* ..... */
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/* LDF (input) INTEGER, LDF >= M */
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/* The leading dimension of the array F. */
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/* ..... */
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/* X (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array */
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/* X is used as workspace to hold representations of the */
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/* leading N-1 snapshots in the orthonormal basis computed */
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/* in the QR factorization of F. */
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/* On exit, the leading K columns of X contain the leading */
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/* K left singular vectors of the above described content */
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/* of X. To lift them to the space of the left singular */
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/* vectors U(:,1:K) of the input data, pre-multiply with the */
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/* Q factor from the initial QR factorization. */
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/* See the descriptions of F, K, V and Z. */
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/* ..... */
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/* LDX (input) INTEGER, LDX >= N */
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/* The leading dimension of the array X. */
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/* ..... */
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/* Y (workspace/output) COMPLEX(KIND=WP) MIN(M,N)-by-(N) array */
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/* Y is used as workspace to hold representations of the */
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/* trailing N-1 snapshots in the orthonormal basis computed */
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/* in the QR factorization of F. */
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/* On exit, */
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/* If JOBT == 'R', Y contains the MIN(M,N)-by-N upper */
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/* triangular factor from the QR factorization of the data */
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/* snapshot matrix F. */
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/* ..... */
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/* LDY (input) INTEGER , LDY >= N */
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/* The leading dimension of the array Y. */
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/* ..... */
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/* NRNK (input) INTEGER */
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/* Determines the mode how to compute the numerical rank, */
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/* i.e. how to truncate small singular values of the input */
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/* matrix X. On input, if */
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/* NRNK = -1 :: i-th singular value sigma(i) is truncated */
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/* if sigma(i) <= TOL*sigma(1) */
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/* This option is recommended. */
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/* NRNK = -2 :: i-th singular value sigma(i) is truncated */
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/* if sigma(i) <= TOL*sigma(i-1) */
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/* This option is included for R&D purposes. */
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/* It requires highly accurate SVD, which */
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/* may not be feasible. */
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/* The numerical rank can be enforced by using positive */
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/* value of NRNK as follows: */
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/* 0 < NRNK <= N-1 :: at most NRNK largest singular values */
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/* will be used. If the number of the computed nonzero */
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/* singular values is less than NRNK, then only those */
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/* nonzero values will be used and the actually used */
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/* dimension is less than NRNK. The actual number of */
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/* the nonzero singular values is returned in the variable */
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/* K. See the description of K. */
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/* ..... */
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/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */
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/* The tolerance for truncating small singular values. */
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/* See the description of NRNK. */
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/* ..... */
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/* K (output) INTEGER, 0 <= K <= N */
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/* The dimension of the SVD/POD basis for the leading N-1 */
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/* data snapshots (columns of F) and the number of the */
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/* computed Ritz pairs. The value of K is determined */
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/* according to the rule set by the parameters NRNK and */
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/* TOL. See the descriptions of NRNK and TOL. */
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/* ..... */
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/* EIGS (output) COMPLEX(KIND=WP) (N-1)-by-1 array */
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/* The leading K (K<=N-1) entries of EIGS contain */
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/* the computed eigenvalues (Ritz values). */
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/* See the descriptions of K, and Z. */
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/* ..... */
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/* Z (workspace/output) COMPLEX(KIND=WP) M-by-(N-1) array */
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/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */
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/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */
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/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */
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/* Z*V, where Z contains orthonormal matrix (the product of */
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/* Q from the initial QR factorization and the SVD/POD_basis */
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/* returned by CGEDMD in X) and the second factor (the */
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/* eigenvectors of the Rayleigh quotient) is in the array V, */
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/* as returned by CGEDMD. That is, X(:,1:K)*V(:,i) */
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/* is an eigenvector corresponding to EIGS(i). The columns */
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/* of V(1:K,1:K) are the computed eigenvectors of the */
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/* K-by-K Rayleigh quotient. */
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/* See the descriptions of EIGS, X and V. */
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/* ..... */
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/* LDZ (input) INTEGER , LDZ >= M */
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/* The leading dimension of the array Z. */
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/* ..... */
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/* RES (output) REAL(KIND=WP) (N-1)-by-1 array */
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/* RES(1:K) contains the residuals for the K computed */
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/* Ritz pairs, */
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/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */
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/* See the description of EIGS and Z. */
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/* ..... */
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/* B (output) COMPLEX(KIND=WP) MIN(M,N)-by-(N-1) array. */
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/* IF JOBF =='R', B(1:N,1:K) contains A*U(:,1:K), and can */
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/* be used for computing the refined vectors; see further */
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/* details in the provided references. */
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/* If JOBF == 'E', B(1:N,1;K) contains */
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/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */
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/* Exact DMD, up to scaling by the inverse eigenvalues. */
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/* In both cases, the content of B can be lifted to the */
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/* original dimension of the input data by pre-multiplying */
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/* with the Q factor from the initial QR factorization. */
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/* Here A denotes a compression of the underlying operator. */
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/* See the descriptions of F and X. */
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/* If JOBF =='N', then B is not referenced. */
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/* ..... */
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/* LDB (input) INTEGER, LDB >= MIN(M,N) */
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/* The leading dimension of the array B. */
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/* ..... */
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/* V (workspace/output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */
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/* On exit, V(1:K,1:K) V contains the K eigenvectors of */
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/* the Rayleigh quotient. The Ritz vectors */
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/* (returned in Z) are the product of Q from the initial QR */
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/* factorization (see the description of F) X (see the */
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/* description of X) and V. */
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/* ..... */
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/* LDV (input) INTEGER, LDV >= N-1 */
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/* The leading dimension of the array V. */
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/* ..... */
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/* S (output) COMPLEX(KIND=WP) (N-1)-by-(N-1) array */
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/* The array S(1:K,1:K) is used for the matrix Rayleigh */
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/* quotient. This content is overwritten during */
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/* the eigenvalue decomposition by CGEEV. */
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/* See the description of K. */
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/* ..... */
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/* LDS (input) INTEGER, LDS >= N-1 */
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/* The leading dimension of the array S. */
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/* ..... */
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/* ZWORK (workspace/output) COMPLEX(KIND=WP) LWORK-by-1 array */
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/* On exit, */
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/* ZWORK(1:MIN(M,N)) contains the scalar factors of the */
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/* elementary reflectors as returned by CGEQRF of the */
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/* M-by-N input matrix F. */
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/* If the call to CGEDMDQ is only workspace query, then */
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/* 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_ */
|
||||
|
||||
|
|
|
|||
File diff suppressed because it is too large
Load Diff
|
|
@ -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 (K<N) entries of REIG 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 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_ */
|
||||
|
||||
|
|
|
|||
File diff suppressed because it is too large
Load Diff
|
|
@ -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 (K<N) entries of REIG 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, 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_ */
|
||||
|
||||
|
|
|
|||
File diff suppressed because it is too large
Load Diff
|
|
@ -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_ */
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue