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OpenBLAS/lapack-netlib/SRC/cgedmd.c

1671 lines
58 KiB
C
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#include <math.h>
#include <stdlib.h>
#include <string.h>
#include <stdio.h>
#include <complex.h>
#ifdef complex
#undef complex
#endif
#ifdef I
#undef I
#endif
#if defined(_WIN64)
typedef long long BLASLONG;
typedef unsigned long long BLASULONG;
#else
typedef long BLASLONG;
typedef unsigned long BLASULONG;
#endif
#ifdef LAPACK_ILP64
typedef BLASLONG blasint;
#if defined(_WIN64)
#define blasabs(x) llabs(x)
#else
#define blasabs(x) labs(x)
#endif
#else
typedef int blasint;
#define blasabs(x) abs(x)
#endif
typedef blasint integer;
typedef unsigned int uinteger;
typedef char *address;
typedef short int shortint;
typedef float real;
typedef double doublereal;
typedef struct { real r, i; } complex;
typedef struct { doublereal r, i; } doublecomplex;
#ifdef _MSC_VER
static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
#else
static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
#endif
#define pCf(z) (*_pCf(z))
#define pCd(z) (*_pCd(z))
typedef int logical;
typedef short int shortlogical;
typedef char logical1;
typedef char integer1;
#define TRUE_ (1)
#define FALSE_ (0)
/* Extern is for use with -E */
#ifndef Extern
#define Extern extern
#endif
/* I/O stuff */
typedef int flag;
typedef int ftnlen;
typedef int ftnint;
/*external read, write*/
typedef struct
{ flag cierr;
ftnint ciunit;
flag ciend;
char *cifmt;
ftnint cirec;
} cilist;
/*internal read, write*/
typedef struct
{ flag icierr;
char *iciunit;
flag iciend;
char *icifmt;
ftnint icirlen;
ftnint icirnum;
} icilist;
/*open*/
typedef struct
{ flag oerr;
ftnint ounit;
char *ofnm;
ftnlen ofnmlen;
char *osta;
char *oacc;
char *ofm;
ftnint orl;
char *oblnk;
} olist;
/*close*/
typedef struct
{ flag cerr;
ftnint cunit;
char *csta;
} cllist;
/*rewind, backspace, endfile*/
typedef struct
{ flag aerr;
ftnint aunit;
} alist;
/* inquire */
typedef struct
{ flag inerr;
ftnint inunit;
char *infile;
ftnlen infilen;
ftnint *inex; /*parameters in standard's order*/
ftnint *inopen;
ftnint *innum;
ftnint *innamed;
char *inname;
ftnlen innamlen;
char *inacc;
ftnlen inacclen;
char *inseq;
ftnlen inseqlen;
char *indir;
ftnlen indirlen;
char *infmt;
ftnlen infmtlen;
char *inform;
ftnint informlen;
char *inunf;
ftnlen inunflen;
ftnint *inrecl;
ftnint *innrec;
char *inblank;
ftnlen inblanklen;
} inlist;
#define VOID void
union Multitype { /* for multiple entry points */
integer1 g;
shortint h;
integer i;
/* longint j; */
real r;
doublereal d;
complex c;
doublecomplex z;
};
typedef union Multitype Multitype;
struct Vardesc { /* for Namelist */
char *name;
char *addr;
ftnlen *dims;
int type;
};
typedef struct Vardesc Vardesc;
struct Namelist {
char *name;
Vardesc **vars;
int nvars;
};
typedef struct Namelist Namelist;
#define abs(x) ((x) >= 0 ? (x) : -(x))
#define dabs(x) (fabs(x))
#define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
#define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
#define dmin(a,b) (f2cmin(a,b))
#define dmax(a,b) (f2cmax(a,b))
#define bit_test(a,b) ((a) >> (b) & 1)
#define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
#define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
#define abort_() { sig_die("Fortran abort routine called", 1); }
#define c_abs(z) (cabsf(Cf(z)))
#define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
#ifdef _MSC_VER
#define c_div(c, a, b) {Cf(c)._Val[0] = (Cf(a)._Val[0]/Cf(b)._Val[0]); Cf(c)._Val[1]=(Cf(a)._Val[1]/Cf(b)._Val[1]);}
#define z_div(c, a, b) {Cd(c)._Val[0] = (Cd(a)._Val[0]/Cd(b)._Val[0]); Cd(c)._Val[1]=(Cd(a)._Val[1]/Cd(b)._Val[1]);}
#else
#define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
#define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
#endif
#define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
#define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
#define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
//#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
#define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
#define d_abs(x) (fabs(*(x)))
#define d_acos(x) (acos(*(x)))
#define d_asin(x) (asin(*(x)))
#define d_atan(x) (atan(*(x)))
#define d_atn2(x, y) (atan2(*(x),*(y)))
#define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
#define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
#define d_cos(x) (cos(*(x)))
#define d_cosh(x) (cosh(*(x)))
#define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
#define d_exp(x) (exp(*(x)))
#define d_imag(z) (cimag(Cd(z)))
#define r_imag(z) (cimagf(Cf(z)))
#define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
#define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
#define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
#define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
#define d_log(x) (log(*(x)))
#define d_mod(x, y) (fmod(*(x), *(y)))
#define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
#define d_nint(x) u_nint(*(x))
#define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
#define d_sign(a,b) u_sign(*(a),*(b))
#define r_sign(a,b) u_sign(*(a),*(b))
#define d_sin(x) (sin(*(x)))
#define d_sinh(x) (sinh(*(x)))
#define d_sqrt(x) (sqrt(*(x)))
#define d_tan(x) (tan(*(x)))
#define d_tanh(x) (tanh(*(x)))
#define i_abs(x) abs(*(x))
#define i_dnnt(x) ((integer)u_nint(*(x)))
#define i_len(s, n) (n)
#define i_nint(x) ((integer)u_nint(*(x)))
#define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
#define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
#define pow_si(B,E) spow_ui(*(B),*(E))
#define pow_ri(B,E) spow_ui(*(B),*(E))
#define pow_di(B,E) dpow_ui(*(B),*(E))
#define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
#define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
#define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
#define s_cat(lpp, rpp, rnp, np, llp) { ftnlen i, nc, ll; char *f__rp, *lp; ll = (llp); lp = (lpp); for(i=0; i < (int)*(np); ++i) { nc = ll; if((rnp)[i] < nc) nc = (rnp)[i]; ll -= nc; f__rp = (rpp)[i]; while(--nc >= 0) *lp++ = *(f__rp)++; } while(--ll >= 0) *lp++ = ' '; }
#define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
#define s_copy(A,B,C,D) { int __i,__m; for (__i=0, __m=f2cmin((C),(D)); __i<__m && (B)[__i] != 0; ++__i) (A)[__i] = (B)[__i]; }
#define sig_die(s, kill) { exit(1); }
#define s_stop(s, n) {exit(0);}
static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
#define z_abs(z) (cabs(Cd(z)))
#define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
#define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
#define myexit_() break;
#define mycycle_() continue;
#define myceiling_(w) {ceil(w)}
#define myhuge_(w) {HUGE_VAL}
//#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
#define mymaxloc_(w,s,e,n) dmaxloc_(w,*(s),*(e),n)
/* procedure parameter types for -A and -C++ */
#define F2C_proc_par_types 1
#ifdef __cplusplus
typedef logical (*L_fp)(...);
#else
typedef logical (*L_fp)();
#endif
static float spow_ui(float x, integer n) {
float pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
static double dpow_ui(double x, integer n) {
double pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
#ifdef _MSC_VER
static _Fcomplex cpow_ui(complex x, integer n) {
complex pow={1.0,0.0}; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
for(u = n; ; ) {
if(u & 01) pow.r *= x.r, pow.i *= x.i;
if(u >>= 1) x.r *= x.r, x.i *= x.i;
else break;
}
}
_Fcomplex p={pow.r, pow.i};
return p;
}
#else
static _Complex float cpow_ui(_Complex float x, integer n) {
_Complex float pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
#endif
#ifdef _MSC_VER
static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
_Dcomplex pow={1.0,0.0}; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
for(u = n; ; ) {
if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
else break;
}
}
_Dcomplex p = {pow._Val[0], pow._Val[1]};
return p;
}
#else
static _Complex double zpow_ui(_Complex double x, integer n) {
_Complex double pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
#endif
static integer pow_ii(integer x, integer n) {
integer pow; unsigned long int u;
if (n <= 0) {
if (n == 0 || x == 1) pow = 1;
else if (x != -1) pow = x == 0 ? 1/x : 0;
else n = -n;
}
if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
u = n;
for(pow = 1; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
static integer dmaxloc_(double *w, integer s, integer e, integer *n)
{
double m; integer i, mi;
for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
if (w[i-1]>m) mi=i ,m=w[i-1];
return mi-s+1;
}
static integer smaxloc_(float *w, integer s, integer e, integer *n)
{
float m; integer i, mi;
for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
if (w[i-1]>m) mi=i ,m=w[i-1];
return mi-s+1;
}
static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Fcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
}
}
pCf(z) = zdotc;
}
#else
_Complex float zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
}
}
pCf(z) = zdotc;
}
#endif
static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Dcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
}
}
pCd(z) = zdotc;
}
#else
_Complex double zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
}
}
pCd(z) = zdotc;
}
#endif
static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Fcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
}
}
pCf(z) = zdotc;
}
#else
_Complex float zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cf(&x[i]) * Cf(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
}
}
pCf(z) = zdotc;
}
#endif
static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Dcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
}
}
pCd(z) = zdotc;
}
#else
_Complex double zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cd(&x[i]) * Cd(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
}
}
pCd(z) = zdotc;
}
#endif
/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/
/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/
/* Table of constant values */
static integer c_n1 = -1;
static integer c__1 = 1;
static integer c__0 = 0;
/* Subroutine */ int cgedmd_(char *jobs, char *jobz, char *jobr, char *jobf,
integer *whtsvd, integer *m, integer *n, complex *x, integer *ldx,
complex *y, integer *ldy, integer *nrnk, real *tol, integer *k,
complex *eigs, complex *z__, integer *ldz, real *res, complex *b,
integer *ldb, complex *w, integer *ldw, complex *s, integer *lds,
complex *zwork, integer *lzwork, real *rwork, integer *lrwork,
integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer x_dim1, x_offset, y_dim1, y_offset, z_dim1, z_offset, b_dim1,
b_offset, w_dim1, w_offset, s_dim1, s_offset, i__1, i__2, i__3,
i__4, i__5;
real r__1, r__2;
complex q__1, q__2;
/* Local variables */
complex zone;
real zero, ssum;
integer info1, info2;
real xscl1, xscl2;
integer i__, j;
real scale;
extern /* Subroutine */ int cgemm_(char *, char *, integer *, integer *,
integer *, complex *, complex *, integer *, complex *, integer *,
complex *, complex *, integer *), cgeev_(char *,
char *, integer *, complex *, integer *, complex *, complex *,
integer *, complex *, integer *, complex *, integer *, real *,
integer *);
extern logical lsame_(char *, char *);
logical badxy;
real small;
char jobzl[1];
extern /* Subroutine */ int caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *);
logical wntex;
complex zzero;
extern real scnrm2_(integer *, complex *, integer *);
extern /* Subroutine */ int cgesdd_(char *, integer *, integer *, complex
*, integer *, real *, complex *, integer *, complex *, integer *,
complex *, integer *, real *, integer *, integer *),
clascl_(char *, integer *, integer *, real *, real *, integer *,
integer *, complex *, integer *, integer *);
extern integer icamax_(integer *, complex *, integer *);
extern real slamch_(char *);
extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer
*), cgesvd_(char *, char *, integer *, integer *, complex *,
integer *, real *, complex *, integer *, complex *, integer *,
complex *, integer *, real *, integer *), clacpy_(
char *, integer *, integer *, complex *, integer *, complex *,
integer *), xerbla_(char *, integer *);
char t_or_n__[1];
extern /* Subroutine */ int cgejsv_(char *, char *, char *, char *, char *
, char *, integer *, integer *, complex *, integer *, real *,
complex *, integer *, complex *, integer *, complex *, integer *,
real *, integer *, integer *, integer *), classq_(integer *, complex *, integer *,
real *, real *);
logical sccolx, sccoly;
extern logical sisnan_(real *);
integer lwrsdd, mwrsdd, iminwr;
logical wntref, wntvec;
real rootsc;
integer lwrkev, mlwork, mwrkev, numrnk, olwork, lwrsvd, mwrsvd, mlrwrk;
logical lquery, wntres;
char jsvopt[1];
integer lwrsvj, mwrsvj;
real rdummy[2];
extern /* Subroutine */ int mecago_();
integer lwrsvq, mwrsvq;
real ofl, one;
extern /* Subroutine */ int cgesvdq_(char *, char *, char *, char *, char
*, integer *, integer *, complex *, integer *, real *, complex *,
integer *, complex *, integer *, integer *, integer *, integer *,
complex *, integer *, real *, integer *, integer *);
/* March 2023 */
/* ..... */
/* USE iso_fortran_env */
/* INTEGER, PARAMETER :: WP = real32 */
/* ..... */
/* Scalar arguments */
/* Array arguments */
/* ............................................................ */
/* Purpose */
/* ======= */
/* CGEDMD computes the Dynamic Mode Decomposition (DMD) for */
/* a pair of data snapshot matrices. For the input matrices */
/* X and Y such that Y = A*X with an unaccessible matrix */
/* A, CGEDMD computes a certain number of Ritz pairs of A using */
/* the standard Rayleigh-Ritz extraction from a subspace of */
/* range(X) that is determined using the leading left singular */
/* vectors of X. Optionally, CGEDMD returns the residuals */
/* of the computed Ritz pairs, the information needed for */
/* a refinement of the Ritz vectors, or the eigenvectors of */
/* the Exact DMD. */
/* For further details see the references listed */
/* below. For more details of the implementation see [3]. */
/* References */
/* ========== */
/* [1] P. Schmid: Dynamic mode decomposition of numerical */
/* and experimental data, */
/* Journal of Fluid Mechanics 656, 5-28, 2010. */
/* [2] Z. Drmac, I. Mezic, R. Mohr: Data driven modal */
/* decompositions: analysis and enhancements, */
/* SIAM J. on Sci. Comp. 40 (4), A2253-A2285, 2018. */
/* [3] Z. Drmac: A LAPACK implementation of the Dynamic */
/* Mode Decomposition I. Technical report. AIMDyn Inc. */
/* and LAPACK Working Note 298. */
/* [4] J. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. */
/* Brunton, N. Kutz: On Dynamic Mode Decomposition: */
/* Theory and Applications, Journal of Computational */
/* Dynamics 1(2), 391 -421, 2014. */
/* ...................................................................... */
/* Developed and supported by: */
/* =========================== */
/* Developed and coded by Zlatko Drmac, Faculty of Science, */
/* University of Zagreb; drmac@math.hr */
/* In cooperation with */
/* AIMdyn Inc., Santa Barbara, CA. */
/* and supported by */
/* - DARPA SBIR project "Koopman Operator-Based Forecasting */
/* for Nonstationary Processes from Near-Term, Limited */
/* Observational Data" Contract No: W31P4Q-21-C-0007 */
/* - DARPA PAI project "Physics-Informed Machine Learning */
/* Methodologies" Contract No: HR0011-18-9-0033 */
/* - DARPA MoDyL project "A Data-Driven, Operator-Theoretic */
/* Framework for Space-Time Analysis of Process Dynamics" */
/* Contract No: HR0011-16-C-0116 */
/* Any opinions, findings and conclusions or recommendations */
/* expressed in this material are those of the author and */
/* do not necessarily reflect the views of the DARPA SBIR */
/* Program Office */
/* ============================================================ */
/* Distribution Statement A: */
/* Approved for Public Release, Distribution Unlimited. */
/* Cleared by DARPA on September 29, 2022 */
/* ============================================================ */
/* ...................................................................... */
/* Arguments */
/* ========= */
/* JOBS (input) CHARACTER*1 */
/* Determines whether the initial data snapshots are scaled */
/* by a diagonal matrix. */
/* 'S' :: The data snapshots matrices X and Y are multiplied */
/* with a diagonal matrix D so that X*D has unit */
/* nonzero columns (in the Euclidean 2-norm) */
/* 'C' :: The snapshots are scaled as with the 'S' option. */
/* If it is found that an i-th column of X is zero */
/* vector and the corresponding i-th column of Y is */
/* non-zero, then the i-th column of Y is set to */
/* zero and a warning flag is raised. */
/* 'Y' :: The data snapshots matrices X and Y are multiplied */
/* by a diagonal matrix D so that Y*D has unit */
/* nonzero columns (in the Euclidean 2-norm) */
/* 'N' :: No data scaling. */
/* ..... */
/* JOBZ (input) CHARACTER*1 */
/* Determines whether the eigenvectors (Koopman modes) will */
/* be computed. */
/* 'V' :: The eigenvectors (Koopman modes) will be computed */
/* and returned in the matrix Z. */
/* See the description of Z. */
/* 'F' :: The eigenvectors (Koopman modes) will be returned */
/* in factored form as the product X(:,1:K)*W, where X */
/* contains a POD basis (leading left singular vectors */
/* of the data matrix X) and W contains the eigenvectors */
/* of the corresponding Rayleigh quotient. */
/* See the descriptions of K, X, W, Z. */
/* 'N' :: The eigenvectors are not computed. */
/* ..... */
/* JOBR (input) CHARACTER*1 */
/* Determines whether to compute the residuals. */
/* 'R' :: The residuals for the computed eigenpairs will be */
/* computed and stored in the array RES. */
/* See the description of RES. */
/* For this option to be legal, JOBZ must be 'V'. */
/* 'N' :: The residuals are not computed. */
/* ..... */
/* JOBF (input) CHARACTER*1 */
/* Specifies whether to store information needed for post- */
/* processing (e.g. computing refined Ritz vectors) */
/* 'R' :: The matrix needed for the refinement of the Ritz */
/* vectors is computed and stored in the array B. */
/* See the description of B. */
/* 'E' :: The unscaled eigenvectors of the Exact DMD are */
/* computed and returned in the array B. See the */
/* description of B. */
/* 'N' :: No eigenvector refinement data is computed. */
/* ..... */
/* WHTSVD (input) INTEGER, WHSTVD in { 1, 2, 3, 4 } */
/* Allows for a selection of the SVD algorithm from the */
/* LAPACK library. */
/* 1 :: CGESVD (the QR SVD algorithm) */
/* 2 :: CGESDD (the Divide and Conquer algorithm; if enough */
/* workspace available, this is the fastest option) */
/* 3 :: CGESVDQ (the preconditioned QR SVD ; this and 4 */
/* are the most accurate options) */
/* 4 :: CGEJSV (the preconditioned Jacobi SVD; this and 3 */
/* are the most accurate options) */
/* For the four methods above, a significant difference in */
/* the accuracy of small singular values is possible if */
/* the snapshots vary in norm so that X is severely */
/* ill-conditioned. If small (smaller than EPS*||X||) */
/* singular values are of interest and JOBS=='N', then */
/* the options (3, 4) give the most accurate results, where */
/* the option 4 is slightly better and with stronger */
/* theoretical background. */
/* If JOBS=='S', i.e. the columns of X will be normalized, */
/* then all methods give nearly equally accurate results. */
/* ..... */
/* M (input) INTEGER, M>= 0 */
/* The state space dimension (the row dimension of X, Y). */
/* ..... */
/* N (input) INTEGER, 0 <= N <= M */
/* The number of data snapshot pairs */
/* (the number of columns of X and Y). */
/* ..... */
/* X (input/output) COMPLEX(KIND=WP) M-by-N array */
/* > On entry, X contains the data snapshot matrix X. It is */
/* assumed that the column norms of X are in the range of */
/* the normalized floating point numbers. */
/* < On exit, the leading K columns of X contain a POD basis, */
/* i.e. the leading K left singular vectors of the input */
/* data matrix X, U(:,1:K). All N columns of X contain all */
/* left singular vectors of the input matrix X. */
/* See the descriptions of K, Z and W. */
/* ..... */
/* LDX (input) INTEGER, LDX >= M */
/* The leading dimension of the array X. */
/* ..... */
/* Y (input/workspace/output) COMPLEX(KIND=WP) M-by-N array */
/* > On entry, Y contains the data snapshot matrix Y */
/* < On exit, */
/* If JOBR == 'R', the leading K columns of Y contain */
/* the residual vectors for the computed Ritz pairs. */
/* See the description of RES. */
/* If JOBR == 'N', Y contains the original input data, */
/* scaled according to the value of JOBS. */
/* ..... */
/* LDY (input) INTEGER , LDY >= M */
/* The leading dimension of the array Y. */
/* ..... */
/* NRNK (input) INTEGER */
/* Determines the mode how to compute the numerical rank, */
/* i.e. how to truncate small singular values of the input */
/* matrix X. On input, if */
/* NRNK = -1 :: i-th singular value sigma(i) is truncated */
/* if sigma(i) <= TOL*sigma(1) */
/* This option is recommended. */
/* NRNK = -2 :: i-th singular value sigma(i) is truncated */
/* if sigma(i) <= TOL*sigma(i-1) */
/* This option is included for R&D purposes. */
/* It requires highly accurate SVD, which */
/* may not be feasible. */
/* The numerical rank can be enforced by using positive */
/* value of NRNK as follows: */
/* 0 < NRNK <= N :: at most NRNK largest singular values */
/* will be used. If the number of the computed nonzero */
/* singular values is less than NRNK, then only those */
/* nonzero values will be used and the actually used */
/* dimension is less than NRNK. The actual number of */
/* the nonzero singular values is returned in the variable */
/* K. See the descriptions of TOL and K. */
/* ..... */
/* TOL (input) REAL(KIND=WP), 0 <= TOL < 1 */
/* The tolerance for truncating small singular values. */
/* See the description of NRNK. */
/* ..... */
/* K (output) INTEGER, 0 <= K <= N */
/* The dimension of the POD basis for the data snapshot */
/* matrix X and the number of the computed Ritz pairs. */
/* The value of K is determined according to the rule set */
/* by the parameters NRNK and TOL. */
/* See the descriptions of NRNK and TOL. */
/* ..... */
/* EIGS (output) COMPLEX(KIND=WP) N-by-1 array */
/* The leading K (K<=N) entries of EIGS contain */
/* the computed eigenvalues (Ritz values). */
/* See the descriptions of K, and Z. */
/* ..... */
/* Z (workspace/output) COMPLEX(KIND=WP) M-by-N array */
/* If JOBZ =='V' then Z contains the Ritz vectors. Z(:,i) */
/* is an eigenvector of the i-th Ritz value; ||Z(:,i)||_2=1. */
/* If JOBZ == 'F', then the Z(:,i)'s are given implicitly as */
/* the columns of X(:,1:K)*W(1:K,1:K), i.e. X(:,1:K)*W(:,i) */
/* is an eigenvector corresponding to EIGS(i). The columns */
/* of W(1:k,1:K) are the computed eigenvectors of the */
/* K-by-K Rayleigh quotient. */
/* See the descriptions of EIGS, X and W. */
/* ..... */
/* LDZ (input) INTEGER , LDZ >= M */
/* The leading dimension of the array Z. */
/* ..... */
/* RES (output) REAL(KIND=WP) N-by-1 array */
/* RES(1:K) contains the residuals for the K computed */
/* Ritz pairs, */
/* RES(i) = || A * Z(:,i) - EIGS(i)*Z(:,i))||_2. */
/* See the description of EIGS and Z. */
/* ..... */
/* B (output) COMPLEX(KIND=WP) M-by-N array. */
/* IF JOBF =='R', B(1:M,1:K) contains A*U(:,1:K), and can */
/* be used for computing the refined vectors; see further */
/* details in the provided references. */
/* If JOBF == 'E', B(1:M,1:K) contains */
/* A*U(:,1:K)*W(1:K,1:K), which are the vectors from the */
/* Exact DMD, up to scaling by the inverse eigenvalues. */
/* If JOBF =='N', then B is not referenced. */
/* See the descriptions of X, W, K. */
/* ..... */
/* LDB (input) INTEGER, LDB >= M */
/* The leading dimension of the array B. */
/* ..... */
/* W (workspace/output) COMPLEX(KIND=WP) N-by-N array */
/* On exit, W(1:K,1:K) contains the K computed */
/* eigenvectors of the matrix Rayleigh quotient. */
/* The Ritz vectors (returned in Z) are the */
/* product of X (containing a POD basis for the input */
/* matrix X) and W. See the descriptions of K, S, X and Z. */
/* W is also used as a workspace to temporarily store the */
/* right singular vectors of X. */
/* ..... */
/* LDW (input) INTEGER, LDW >= N */
/* The leading dimension of the array W. */
/* ..... */
/* S (workspace/output) COMPLEX(KIND=WP) N-by-N array */
/* The array S(1:K,1:K) is used for the matrix Rayleigh */
/* quotient. This content is overwritten during */
/* the eigenvalue decomposition by CGEEV. */
/* See the description of K. */
/* ..... */
/* LDS (input) INTEGER, LDS >= N */
/* The leading dimension of the array S. */
/* ..... */
/* ZWORK (workspace/output) COMPLEX(KIND=WP) LZWORK-by-1 array */
/* ZWORK is used as complex workspace in the complex SVD, as */
/* specified by WHTSVD (1,2, 3 or 4) and for CGEEV for computing */
/* the eigenvalues of a Rayleigh quotient. */
/* If the call to CGEDMD is only workspace query, then */
/* ZWORK(1) contains the minimal complex workspace length and */
/* ZWORK(2) is the optimal complex workspace length. */
/* Hence, the length of work is at least 2. */
/* See the description of LZWORK. */
/* ..... */
/* LZWORK (input) INTEGER */
/* The minimal length of the workspace vector ZWORK. */
/* LZWORK is calculated as MAX(LZWORK_SVD, LZWORK_CGEEV), */
/* where LZWORK_CGEEV = MAX( 1, 2*N ) and the minimal */
/* LZWORK_SVD is calculated as follows */
/* If WHTSVD == 1 :: CGESVD :: */
/* LZWORK_SVD = MAX(1,2*MIN(M,N)+MAX(M,N)) */
/* If WHTSVD == 2 :: CGESDD :: */
/* LZWORK_SVD = 2*MIN(M,N)*MIN(M,N)+2*MIN(M,N)+MAX(M,N) */
/* If WHTSVD == 3 :: CGESVDQ :: */
/* LZWORK_SVD = obtainable by a query */
/* If WHTSVD == 4 :: CGEJSV :: */
/* LZWORK_SVD = obtainable by a query */
/* If on entry LZWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* and the optimal workspace lengths and returns them in */
/* LZWORK(1) and LZWORK(2), respectively. */
/* ..... */
/* RWORK (workspace/output) REAL(KIND=WP) LRWORK-by-1 array */
/* On exit, RWORK(1:N) contains the singular values of */
/* X (for JOBS=='N') or column scaled X (JOBS=='S', 'C'). */
/* If WHTSVD==4, then RWORK(N+1) and RWORK(N+2) contain */
/* scaling factor RWORK(N+2)/RWORK(N+1) used to scale X */
/* and Y to avoid overflow in the SVD of X. */
/* This may be of interest if the scaling option is off */
/* and as many as possible smallest eigenvalues are */
/* desired to the highest feasible accuracy. */
/* If the call to CGEDMD is only workspace query, then */
/* RWORK(1) contains the minimal workspace length. */
/* See the description of LRWORK. */
/* ..... */
/* LRWORK (input) INTEGER */
/* The minimal length of the workspace vector RWORK. */
/* LRWORK is calculated as follows: */
/* LRWORK = MAX(1, N+LRWORK_SVD,N+LRWORK_CGEEV), where */
/* LRWORK_CGEEV = MAX(1,2*N) and RWORK_SVD is the real workspace */
/* for the SVD subroutine determined by the input parameter */
/* WHTSVD. */
/* If WHTSVD == 1 :: CGESVD :: */
/* LRWORK_SVD = 5*MIN(M,N) */
/* If WHTSVD == 2 :: CGESDD :: */
/* LRWORK_SVD = MAX(5*MIN(M,N)*MIN(M,N)+7*MIN(M,N), */
/* 2*MAX(M,N)*MIN(M,N)+2*MIN(M,N)*MIN(M,N)+MIN(M,N) ) ) */
/* If WHTSVD == 3 :: CGESVDQ :: */
/* LRWORK_SVD = obtainable by a query */
/* If WHTSVD == 4 :: CGEJSV :: */
/* LRWORK_SVD = obtainable by a query */
/* If on entry LRWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* real workspace length and returns it in RWORK(1). */
/* ..... */
/* IWORK (workspace/output) INTEGER LIWORK-by-1 array */
/* Workspace that is required only if WHTSVD equals */
/* 2 , 3 or 4. (See the description of WHTSVD). */
/* If on entry LWORK =-1 or LIWORK=-1, then the */
/* minimal length of IWORK is computed and returned in */
/* IWORK(1). See the description of LIWORK. */
/* ..... */
/* LIWORK (input) INTEGER */
/* The minimal length of the workspace vector IWORK. */
/* If WHTSVD == 1, then only IWORK(1) is used; LIWORK >=1 */
/* If WHTSVD == 2, then LIWORK >= MAX(1,8*MIN(M,N)) */
/* If WHTSVD == 3, then LIWORK >= MAX(1,M+N-1) */
/* If WHTSVD == 4, then LIWORK >= MAX(3,M+3*N) */
/* If on entry LIWORK = -1, then a workspace query is */
/* assumed and the procedure only computes the minimal */
/* and the optimal workspace lengths for ZWORK, RWORK and */
/* IWORK. See the descriptions of ZWORK, RWORK and IWORK. */
/* ..... */
/* INFO (output) INTEGER */
/* -i < 0 :: On entry, the i-th argument had an */
/* illegal value */
/* = 0 :: Successful return. */
/* = 1 :: Void input. Quick exit (M=0 or N=0). */
/* = 2 :: The SVD computation of X did not converge. */
/* Suggestion: Check the input data and/or */
/* repeat with different WHTSVD. */
/* = 3 :: The computation of the eigenvalues did not */
/* converge. */
/* = 4 :: If data scaling was requested on input and */
/* the procedure found inconsistency in the data */
/* such that for some column index i, */
/* X(:,i) = 0 but Y(:,i) /= 0, then Y(:,i) is set */
/* to zero if JOBS=='C'. The computation proceeds */
/* with original or modified data and warning */
/* flag is set with INFO=4. */
/* ............................................................. */
/* ............................................................. */
/* Parameters */
/* ~~~~~~~~~~ */
/* Local scalars */
/* ~~~~~~~~~~~~~ */
/* Local arrays */
/* ~~~~~~~~~~~~ */
/* External functions (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~ */
/* External subroutines (BLAS and LAPACK) */
/* ~~~~~~~~~~~~~~~~~~~~ */
/* Intrinsic functions */
/* ~~~~~~~~~~~~~~~~~~~ */
/* ............................................................ */
/* Parameter adjustments */
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1 * 1;
y -= y_offset;
--eigs;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--res;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
w_dim1 = *ldw;
w_offset = 1 + w_dim1 * 1;
w -= w_offset;
s_dim1 = *lds;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
--zwork;
--rwork;
--iwork;
/* Function Body */
zero = 0.f;
one = 1.f;
zzero.r = 0.f, zzero.i = 0.f;
zone.r = 1.f, zone.i = 0.f;
/* Test the input arguments */
wntres = lsame_(jobr, "R");
sccolx = lsame_(jobs, "S") || lsame_(jobs, "C");
sccoly = lsame_(jobs, "Y");
wntvec = lsame_(jobz, "V");
wntref = lsame_(jobf, "R");
wntex = lsame_(jobf, "E");
*info = 0;
lquery = *lzwork == -1 || *liwork == -1 || *lrwork == -1;
if (! (sccolx || sccoly || lsame_(jobs, "N"))) {
*info = -1;
} else if (! (wntvec || lsame_(jobz, "N") || lsame_(
jobz, "F"))) {
*info = -2;
} else if (! (wntres || lsame_(jobr, "N")) ||
wntres && ! wntvec) {
*info = -3;
} else if (! (wntref || wntex || lsame_(jobf, "N")))
{
*info = -4;
} else if (! (*whtsvd == 1 || *whtsvd == 2 || *whtsvd == 3 || *whtsvd ==
4)) {
*info = -5;
} else if (*m < 0) {
*info = -6;
} else if (*n < 0 || *n > *m) {
*info = -7;
} else if (*ldx < *m) {
*info = -9;
} else if (*ldy < *m) {
*info = -11;
} else if (! (*nrnk == -2 || *nrnk == -1 || *nrnk >= 1 && *nrnk <= *n)) {
*info = -12;
} else if (*tol < zero || *tol >= one) {
*info = -13;
} else if (*ldz < *m) {
*info = -17;
} else if ((wntref || wntex) && *ldb < *m) {
*info = -20;
} else if (*ldw < *n) {
*info = -22;
} else if (*lds < *n) {
*info = -24;
}
if (*info == 0) {
/* Compute the minimal and the optimal workspace */
/* requirements. Simulate running the code and */
/* determine minimal and optimal sizes of the */
/* workspace at any moment of the run. */
if (*n == 0) {
/* Quick return. All output except K is void. */
/* INFO=1 signals the void input. */
/* In case of a workspace query, the default */
/* minimal workspace lengths are returned. */
if (lquery) {
iwork[1] = 1;
rwork[1] = 1.f;
zwork[1].r = 2.f, zwork[1].i = 0.f;
zwork[2].r = 2.f, zwork[2].i = 0.f;
} else {
*k = 0;
}
*info = 1;
return 0;
}
iminwr = 1;
mlrwrk = f2cmax(1,*n);
mlwork = 2;
olwork = 2;
/* SELECT CASE ( WHTSVD ) */
if (*whtsvd == 1) {
/* The following is specified as the minimal */
/* length of WORK in the definition of CGESVD: */
/* MWRSVD = MAX(1,2*MIN(M,N)+MAX(M,N)) */
/* Computing MAX */
i__1 = 1, i__2 = (f2cmin(*m,*n) << 1) + f2cmax(*m,*n);
mwrsvd = f2cmax(i__1,i__2);
mlwork = f2cmax(mlwork,mwrsvd);
/* Computing MAX */
i__1 = mlrwrk, i__2 = *n + f2cmin(*m,*n) * 5;
mlrwrk = f2cmax(i__1,i__2);
if (lquery) {
cgesvd_("O", "S", m, n, &x[x_offset], ldx, &rwork[1], &b[
b_offset], ldb, &w[w_offset], ldw, &zwork[1], &c_n1,
rdummy, &info1);
lwrsvd = (integer) zwork[1].r;
olwork = f2cmax(olwork,lwrsvd);
}
} else if (*whtsvd == 2) {
/* The following is specified as the minimal */
/* length of WORK in the definition of CGESDD: */
/* MWRSDD = 2*f2cmin(M,N)*f2cmin(M,N)+2*f2cmin(M,N)+f2cmax(M,N). */
/* RWORK length: 5*MIN(M,N)*MIN(M,N)+7*MIN(M,N) */
/* In LAPACK 3.10.1 RWORK is defined differently. */
/* Below we take f2cmax over the two versions. */
/* IMINWR = 8*MIN(M,N) */
mwrsdd = (f2cmin(*m,*n) << 1) * f2cmin(*m,*n) + (f2cmin(*m,*n) << 1) + f2cmax(
*m,*n);
mlwork = f2cmax(mlwork,mwrsdd);
iminwr = f2cmin(*m,*n) << 3;
/* Computing MAX */
/* Computing MAX */
i__3 = f2cmin(*m,*n) * 5 * f2cmin(*m,*n) + f2cmin(*m,*n) * 7, i__4 = f2cmin(*
m,*n) * 5 * f2cmin(*m,*n) + f2cmin(*m,*n) * 5, i__3 = f2cmax(i__3,
i__4), i__4 = (f2cmax(*m,*n) << 1) * f2cmin(*m,*n) + (f2cmin(*m,*n)
<< 1) * f2cmin(*m,*n) + f2cmin(*m,*n);
i__1 = mlrwrk, i__2 = *n + f2cmax(i__3,i__4);
mlrwrk = f2cmax(i__1,i__2);
if (lquery) {
cgesdd_("O", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset],
ldb, &w[w_offset], ldw, &zwork[1], &c_n1, rdummy, &
iwork[1], &info1);
/* Computing MAX */
i__1 = mwrsdd, i__2 = (integer) zwork[1].r;
lwrsdd = f2cmax(i__1,i__2);
olwork = f2cmax(olwork,lwrsdd);
}
} else if (*whtsvd == 3) {
cgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &rwork[
1], &z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, &
iwork[1], &c_n1, &zwork[1], &c_n1, rdummy, &c_n1, &info1);
iminwr = iwork[1];
mwrsvq = (integer) zwork[2].r;
mlwork = f2cmax(mlwork,mwrsvq);
/* Computing MAX */
i__1 = mlrwrk, i__2 = *n + (integer) rdummy[0];
mlrwrk = f2cmax(i__1,i__2);
if (lquery) {
lwrsvq = (integer) zwork[1].r;
olwork = f2cmax(olwork,lwrsvq);
}
} else if (*whtsvd == 4) {
*(unsigned char *)jsvopt = 'J';
cgejsv_("F", "U", jsvopt, "N", "N", "P", m, n, &x[x_offset], ldx,
&rwork[1], &z__[z_offset], ldz, &w[w_offset], ldw, &zwork[
1], &c_n1, rdummy, &c_n1, &iwork[1], &info1);
iminwr = iwork[1];
mwrsvj = (integer) zwork[2].r;
mlwork = f2cmax(mlwork,mwrsvj);
/* Computing MAX */
/* Computing MAX */
i__3 = 7, i__4 = (integer) rdummy[0];
i__1 = mlrwrk, i__2 = *n + f2cmax(i__3,i__4);
mlrwrk = f2cmax(i__1,i__2);
if (lquery) {
lwrsvj = (integer) zwork[1].r;
olwork = f2cmax(olwork,lwrsvj);
}
/* END SELECT */
}
if (wntvec || wntex || lsame_(jobz, "F")) {
*(unsigned char *)jobzl = 'V';
} else {
*(unsigned char *)jobzl = 'N';
}
/* Workspace calculation to the CGEEV call */
/* Computing MAX */
i__1 = 1, i__2 = *n << 1;
mwrkev = f2cmax(i__1,i__2);
mlwork = f2cmax(mlwork,mwrkev);
/* Computing MAX */
i__1 = mlrwrk, i__2 = *n + (*n << 1);
mlrwrk = f2cmax(i__1,i__2);
if (lquery) {
cgeev_("N", jobzl, n, &s[s_offset], lds, &eigs[1], &w[w_offset],
ldw, &w[w_offset], ldw, &zwork[1], &c_n1, &rwork[1], &
info1);
/* LAPACK CALL */
lwrkev = (integer) zwork[1].r;
olwork = f2cmax(olwork,lwrkev);
olwork = f2cmax(2,olwork);
}
if (*liwork < iminwr && ! lquery) {
*info = -30;
}
if (*lrwork < mlrwrk && ! lquery) {
*info = -28;
}
if (*lzwork < mlwork && ! lquery) {
*info = -26;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGEDMD", &i__1);
return 0;
} else if (lquery) {
/* Return minimal and optimal workspace sizes */
iwork[1] = iminwr;
rwork[1] = (real) mlrwrk;
zwork[1].r = (real) mlwork, zwork[1].i = 0.f;
zwork[2].r = (real) olwork, zwork[2].i = 0.f;
return 0;
}
/* ............................................................ */
ofl = slamch_("O") * slamch_("P");
small = slamch_("S");
badxy = FALSE_;
/* <1> Optional scaling of the snapshots (columns of X, Y) */
/* ========================================================== */
if (sccolx) {
/* The columns of X will be normalized. */
/* To prevent overflows, the column norms of X are */
/* carefully computed using CLASSQ. */
*k = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* WORK(i) = SCNRM2( M, X(1,i), 1 ) */
scale = zero;
classq_(m, &x[i__ * x_dim1 + 1], &c__1, &scale, &ssum);
if (sisnan_(&scale) || sisnan_(&ssum)) {
*k = 0;
*info = -8;
i__2 = -(*info);
xerbla_("CGEDMD", &i__2);
}
if (scale != zero && ssum != zero) {
rootsc = sqrt(ssum);
if (scale >= ofl / rootsc) {
/* Norm of X(:,i) overflows. First, X(:,i) */
/* is scaled by */
/* ( ONE / ROOTSC ) / SCALE = 1/||X(:,i)||_2. */
/* Next, the norm of X(:,i) is stored without */
/* overflow as WORK(i) = - SCALE * (ROOTSC/M), */
/* the minus sign indicating the 1/M factor. */
/* Scaling is performed without overflow, and */
/* underflow may occur in the smallest entries */
/* of X(:,i). The relative backward and forward */
/* errors are small in the ell_2 norm. */
r__1 = one / rootsc;
clascl_("G", &c__0, &c__0, &scale, &r__1, m, &c__1, &x[
i__ * x_dim1 + 1], ldx, &info2);
rwork[i__] = -scale * (rootsc / (real) (*m));
} else {
/* X(:,i) will be scaled to unit 2-norm */
rwork[i__] = scale * rootsc;
clascl_("G", &c__0, &c__0, &rwork[i__], &one, m, &c__1, &
x[i__ * x_dim1 + 1], ldx, &info2);
/* X(1:M,i) = (ONE/RWORK(i)) * X(1:M,i) ! INTRINSIC */
/* LAPAC */
}
} else {
rwork[i__] = zero;
++(*k);
}
}
if (*k == *n) {
/* All columns of X are zero. Return error code -8. */
/* (the 8th input variable had an illegal value) */
*k = 0;
*info = -8;
i__1 = -(*info);
xerbla_("CGEDMD", &i__1);
return 0;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Now, apply the same scaling to the columns of Y. */
if (rwork[i__] > zero) {
r__1 = one / rwork[i__];
csscal_(m, &r__1, &y[i__ * y_dim1 + 1], &c__1);
/* Y(1:M,i) = (ONE/RWORK(i)) * Y(1:M,i) ! INTRINSIC */
/* BLAS CALL */
} else if (rwork[i__] < zero) {
r__1 = -rwork[i__];
r__2 = one / (real) (*m);
clascl_("G", &c__0, &c__0, &r__1, &r__2, m, &c__1, &y[i__ *
y_dim1 + 1], ldy, &info2);
/* LAPACK C */
} else if (c_abs(&y[icamax_(m, &y[i__ * y_dim1 + 1], &c__1) + i__
* y_dim1]) != zero) {
/* X(:,i) is zero vector. For consistency, */
/* Y(:,i) should also be zero. If Y(:,i) is not */
/* zero, then the data might be inconsistent or */
/* corrupted. If JOBS == 'C', Y(:,i) is set to */
/* zero and a warning flag is raised. */
/* The computation continues but the */
/* situation will be reported in the output. */
badxy = TRUE_;
if (lsame_(jobs, "C")) {
csscal_(m, &zero, &y[i__ * y_dim1 + 1], &c__1);
}
/* BLAS CALL */
}
}
}
if (sccoly) {
/* The columns of Y will be normalized. */
/* To prevent overflows, the column norms of Y are */
/* carefully computed using CLASSQ. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* RWORK(i) = SCNRM2( M, Y(1,i), 1 ) */
scale = zero;
classq_(m, &y[i__ * y_dim1 + 1], &c__1, &scale, &ssum);
if (sisnan_(&scale) || sisnan_(&ssum)) {
*k = 0;
*info = -10;
i__2 = -(*info);
xerbla_("CGEDMD", &i__2);
}
if (scale != zero && ssum != zero) {
rootsc = sqrt(ssum);
if (scale >= ofl / rootsc) {
/* Norm of Y(:,i) overflows. First, Y(:,i) */
/* is scaled by */
/* ( ONE / ROOTSC ) / SCALE = 1/||Y(:,i)||_2. */
/* Next, the norm of Y(:,i) is stored without */
/* overflow as RWORK(i) = - SCALE * (ROOTSC/M), */
/* the minus sign indicating the 1/M factor. */
/* Scaling is performed without overflow, and */
/* underflow may occur in the smallest entries */
/* of Y(:,i). The relative backward and forward */
/* errors are small in the ell_2 norm. */
r__1 = one / rootsc;
clascl_("G", &c__0, &c__0, &scale, &r__1, m, &c__1, &y[
i__ * y_dim1 + 1], ldy, &info2);
rwork[i__] = -scale * (rootsc / (real) (*m));
} else {
/* Y(:,i) will be scaled to unit 2-norm */
rwork[i__] = scale * rootsc;
clascl_("G", &c__0, &c__0, &rwork[i__], &one, m, &c__1, &
y[i__ * y_dim1 + 1], ldy, &info2);
/* Y(1:M,i) = (ONE/RWORK(i)) * Y(1:M,i) ! INTRINSIC */
/* LAPA */
}
} else {
rwork[i__] = zero;
}
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Now, apply the same scaling to the columns of X. */
if (rwork[i__] > zero) {
r__1 = one / rwork[i__];
csscal_(m, &r__1, &x[i__ * x_dim1 + 1], &c__1);
/* X(1:M,i) = (ONE/RWORK(i)) * X(1:M,i) ! INTRINSIC */
/* BLAS CALL */
} else if (rwork[i__] < zero) {
r__1 = -rwork[i__];
r__2 = one / (real) (*m);
clascl_("G", &c__0, &c__0, &r__1, &r__2, m, &c__1, &x[i__ *
x_dim1 + 1], ldx, &info2);
/* LAPACK */
} else if (c_abs(&x[icamax_(m, &x[i__ * x_dim1 + 1], &c__1) + i__
* x_dim1]) != zero) {
/* Y(:,i) is zero vector. If X(:,i) is not */
/* zero, then a warning flag is raised. */
/* The computation continues but the */
/* situation will be reported in the output. */
badxy = TRUE_;
}
}
}
/* <2> SVD of the data snapshot matrix X. */
/* ===================================== */
/* The left singular vectors are stored in the array X. */
/* The right singular vectors are in the array W. */
/* The array W will later on contain the eigenvectors */
/* of a Rayleigh quotient. */
numrnk = *n;
/* SELECT CASE ( WHTSVD ) */
if (*whtsvd == 1) {
cgesvd_("O", "S", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset],
ldb, &w[w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &
info1);
/* LA */
*(unsigned char *)t_or_n__ = 'C';
} else if (*whtsvd == 2) {
cgesdd_("O", m, n, &x[x_offset], ldx, &rwork[1], &b[b_offset], ldb, &
w[w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &iwork[1]
, &info1);
/* LAP */
*(unsigned char *)t_or_n__ = 'C';
} else if (*whtsvd == 3) {
i__1 = *lrwork - *n;
cgesvdq_("H", "P", "N", "R", "R", m, n, &x[x_offset], ldx, &rwork[1],
&z__[z_offset], ldz, &w[w_offset], ldw, &numrnk, &iwork[1],
liwork, &zwork[1], lzwork, &rwork[*n + 1], &i__1, &info1);
/* LAPACK CA */
clacpy_("A", m, &numrnk, &z__[z_offset], ldz, &x[x_offset], ldx);
/* LAPACK C */
*(unsigned char *)t_or_n__ = 'C';
} else if (*whtsvd == 4) {
i__1 = *lrwork - *n;
cgejsv_("F", "U", jsvopt, "N", "N", "P", m, n, &x[x_offset], ldx, &
rwork[1], &z__[z_offset], ldz, &w[w_offset], ldw, &zwork[1],
lzwork, &rwork[*n + 1], &i__1, &iwork[1], &info1);
clacpy_("A", m, n, &z__[z_offset], ldz, &x[x_offset], ldx);
/* LAPACK CALL */
*(unsigned char *)t_or_n__ = 'N';
xscl1 = rwork[*n + 1];
xscl2 = rwork[*n + 2];
if (xscl1 != xscl2) {
/* This is an exceptional situation. If the */
/* data matrices are not scaled and the */
/* largest singular value of X overflows. */
/* In that case CGEJSV can return the SVD */
/* in scaled form. The scaling factor can be used */
/* to rescale the data (X and Y). */
clascl_("G", &c__0, &c__0, &xscl1, &xscl2, m, n, &y[y_offset],
ldy, &info2);
}
/* END SELECT */
}
if (info1 > 0) {
/* The SVD selected subroutine did not converge. */
/* Return with an error code. */
*info = 2;
return 0;
}
if (rwork[1] == zero) {
/* The largest computed singular value of (scaled) */
/* X is zero. Return error code -8 */
/* (the 8th input variable had an illegal value). */
*k = 0;
*info = -8;
i__1 = -(*info);
xerbla_("CGEDMD", &i__1);
return 0;
}
/* <3> Determine the numerical rank of the data */
/* snapshots matrix X. This depends on the */
/* parameters NRNK and TOL. */
/* SELECT CASE ( NRNK ) */
if (*nrnk == -1) {
*k = 1;
i__1 = numrnk;
for (i__ = 2; i__ <= i__1; ++i__) {
if (rwork[i__] <= rwork[1] * *tol || rwork[i__] <= small) {
myexit_();
}
++(*k);
}
} else if (*nrnk == -2) {
*k = 1;
i__1 = numrnk - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if (rwork[i__ + 1] <= rwork[i__] * *tol || rwork[i__] <= small) {
myexit_();
}
++(*k);
}
} else {
*k = 1;
i__1 = *nrnk;
for (i__ = 2; i__ <= i__1; ++i__) {
if (rwork[i__] <= small) {
myexit_();
}
++(*k);
}
/* END SELECT */
}
/* Now, U = X(1:M,1:K) is the SVD/POD basis for the */
/* snapshot data in the input matrix X. */
/* <4> Compute the Rayleigh quotient S = U^H * A * U. */
/* Depending on the requested outputs, the computation */
/* is organized to compute additional auxiliary */
/* matrices (for the residuals and refinements). */
/* In all formulas below, we need V_k*Sigma_k^(-1) */
/* where either V_k is in W(1:N,1:K), or V_k^H is in */
/* W(1:K,1:N). Here Sigma_k=diag(WORK(1:K)). */
if (lsame_(t_or_n__, "N")) {
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__1 = one / rwork[i__];
csscal_(n, &r__1, &w[i__ * w_dim1 + 1], &c__1);
/* W(1:N,i) = (ONE/RWORK(i)) * W(1:N,i) ! INTRINSIC */
/* BLAS CALL */
}
} else {
/* This non-unit stride access is due to the fact */
/* that CGESVD, CGESVDQ and CGESDD return the */
/* adjoint matrix of the right singular vectors. */
/* DO i = 1, K */
/* CALL DSCAL( N, ONE/RWORK(i), W(i,1), LDW ) ! BLAS CALL */
/* ! W(i,1:N) = (ONE/RWORK(i)) * W(i,1:N) ! INTRINSIC */
/* END DO */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
rwork[*n + i__] = one / rwork[i__];
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * w_dim1;
i__4 = *n + i__;
q__2.r = rwork[i__4], q__2.i = zero;
i__5 = i__ + j * w_dim1;
q__1.r = q__2.r * w[i__5].r - q__2.i * w[i__5].i, q__1.i =
q__2.r * w[i__5].i + q__2.i * w[i__5].r;
w[i__3].r = q__1.r, w[i__3].i = q__1.i;
}
}
}
if (wntref) {
/* Need A*U(:,1:K)=Y*V_k*inv(diag(WORK(1:K))) */
/* for computing the refined Ritz vectors */
/* (optionally, outside CGEDMD). */
cgemm_("N", t_or_n__, m, k, n, &zone, &y[y_offset], ldy, &w[w_offset],
ldw, &zzero, &z__[z_offset], ldz);
/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRI */
/* Z(1:M,1:K)=MATMUL(Y(1:M,1:N),W(1:N,1:K)) ! INTRI */
/* At this point Z contains */
/* A * U(:,1:K) = Y * V_k * Sigma_k^(-1), and */
/* this is needed for computing the residuals. */
/* This matrix is returned in the array B and */
/* it can be used to compute refined Ritz vectors. */
/* BLAS */
clacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb);
/* B(1:M,1:K) = Z(1:M,1:K) ! INTRINSIC */
/* BLAS CALL */
cgemm_("C", "N", k, k, m, &zone, &x[x_offset], ldx, &z__[z_offset],
ldz, &zzero, &s[s_offset], lds);
/* S(1:K,1:K) = MATMUL(TANSPOSE(X(1:M,1:K)),Z(1:M,1:K)) ! INTRI */
/* At this point S = U^H * A * U is the Rayleigh quotient. */
/* BLAS */
} else {
/* A * U(:,1:K) is not explicitly needed and the */
/* computation is organized differently. The Rayleigh */
/* quotient is computed more efficiently. */
cgemm_("C", "N", k, n, m, &zone, &x[x_offset], ldx, &y[y_offset], ldy,
&zzero, &z__[z_offset], ldz);
/* Z(1:K,1:N) = MATMUL( TRANSPOSE(X(1:M,1:K)), Y(1:M,1:N) ) ! IN */
/* B */
cgemm_("N", t_or_n__, k, k, n, &zone, &z__[z_offset], ldz, &w[
w_offset], ldw, &zzero, &s[s_offset], lds);
/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),TRANSPOSE(W(1:K,1:N))) ! INTRIN */
/* S(1:K,1:K) = MATMUL(Z(1:K,1:N),(W(1:N,1:K))) ! INTRIN */
/* At this point S = U^H * A * U is the Rayleigh quotient. */
/* If the residuals are requested, save scaled V_k into Z. */
/* Recall that V_k or V_k^H is stored in W. */
/* BLAS */
if (wntres || wntex) {
if (lsame_(t_or_n__, "N")) {
clacpy_("A", n, k, &w[w_offset], ldw, &z__[z_offset], ldz);
} else {
clacpy_("A", k, n, &w[w_offset], ldw, &z__[z_offset], ldz);
}
}
}
/* <5> Compute the Ritz values and (if requested) the */
/* right eigenvectors of the Rayleigh quotient. */
cgeev_("N", jobzl, k, &s[s_offset], lds, &eigs[1], &w[w_offset], ldw, &w[
w_offset], ldw, &zwork[1], lzwork, &rwork[*n + 1], &info1);
/* W(1:K,1:K) contains the eigenvectors of the Rayleigh */
/* quotient. See the description of Z. */
/* Also, see the description of CGEEV. */
/* LAPACK CA */
if (info1 > 0) {
/* CGEEV failed to compute the eigenvalues and */
/* eigenvectors of the Rayleigh quotient. */
*info = 3;
return 0;
}
/* <6> Compute the eigenvectors (if requested) and, */
/* the residuals (if requested). */
if (wntvec || wntex) {
if (wntres) {
if (wntref) {
/* Here, if the refinement is requested, we have */
/* A*U(:,1:K) already computed and stored in Z. */
/* For the residuals, need Y = A * U(:,1;K) * W. */
cgemm_("N", "N", m, k, k, &zone, &z__[z_offset], ldz, &w[
w_offset], ldw, &zzero, &y[y_offset], ldy);
/* Y(1:M,1:K) = Z(1:M,1:K) * W(1:K,1:K) ! INTRINSIC */
/* This frees Z; Y contains A * U(:,1:K) * W. */
/* BLAS CALL */
} else {
/* Compute S = V_k * Sigma_k^(-1) * W, where */
/* V_k * Sigma_k^(-1) (or its adjoint) is stored in Z */
cgemm_(t_or_n__, "N", n, k, k, &zone, &z__[z_offset], ldz, &w[
w_offset], ldw, &zzero, &s[s_offset], lds);
/* Then, compute Z = Y * S = */
/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */
/* = A * U(:,1:K) * W(1:K,1:K) */
cgemm_("N", "N", m, k, n, &zone, &y[y_offset], ldy, &s[
s_offset], lds, &zzero, &z__[z_offset], ldz);
/* Save a copy of Z into Y and free Z for holding */
/* the Ritz vectors. */
clacpy_("A", m, k, &z__[z_offset], ldz, &y[y_offset], ldy);
if (wntex) {
clacpy_("A", m, k, &z__[z_offset], ldz, &b[b_offset], ldb);
}
}
} else if (wntex) {
/* Compute S = V_k * Sigma_k^(-1) * W, where */
/* V_k * Sigma_k^(-1) is stored in Z */
cgemm_(t_or_n__, "N", n, k, k, &zone, &z__[z_offset], ldz, &w[
w_offset], ldw, &zzero, &s[s_offset], lds);
/* Then, compute Z = Y * S = */
/* = Y * V_k * Sigma_k^(-1) * W(1:K,1:K) = */
/* = A * U(:,1:K) * W(1:K,1:K) */
cgemm_("N", "N", m, k, n, &zone, &y[y_offset], ldy, &s[s_offset],
lds, &zzero, &b[b_offset], ldb);
/* The above call replaces the following two calls */
/* that were used in the developing-testing phase. */
/* CALL CGEMM( 'N', 'N', M, K, N, ZONE, Y, LDY, S, & */
/* LDS, ZZERO, Z, LDZ) */
/* Save a copy of Z into Y and free Z for holding */
/* the Ritz vectors. */
/* CALL CLACPY( 'A', M, K, Z, LDZ, B, LDB ) */
}
/* Compute the Ritz vectors */
if (wntvec) {
cgemm_("N", "N", m, k, k, &zone, &x[x_offset], ldx, &w[w_offset],
ldw, &zzero, &z__[z_offset], ldz);
}
/* Z(1:M,1:K) = MATMUL(X(1:M,1:K), W(1:K,1:K)) ! INTRIN */
/* BLAS CALL */
if (wntres) {
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
q__1.r = -eigs[i__2].r, q__1.i = -eigs[i__2].i;
caxpy_(m, &q__1, &z__[i__ * z_dim1 + 1], &c__1, &y[i__ *
y_dim1 + 1], &c__1);
/* Y(1:M,i) = Y(1:M,i) - EIGS(i) * Z(1:M,i) ! */
res[i__] = scnrm2_(m, &y[i__ * y_dim1 + 1], &c__1);
}
}
}
if (*whtsvd == 4) {
rwork[*n + 1] = xscl1;
rwork[*n + 2] = xscl2;
}
/* Successful exit. */
if (! badxy) {
*info = 0;
} else {
/* A warning on possible data inconsistency. */
/* This should be a rare event. */
*info = 4;
}
/* ............................................................ */
return 0;
/* ...... */
} /* cgedmd_ */
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