1301 lines
46 KiB
C
1301 lines
46 KiB
C
#include <math.h>
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#include <stdlib.h>
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#include <string.h>
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#include <stdio.h>
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#include <complex.h>
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#ifdef complex
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#undef complex
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#endif
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#ifdef I
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#undef I
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#endif
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#if defined(_WIN64)
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typedef long long BLASLONG;
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typedef unsigned long long BLASULONG;
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#else
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typedef long BLASLONG;
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typedef unsigned long BLASULONG;
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#endif
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#ifdef LAPACK_ILP64
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typedef BLASLONG blasint;
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#if defined(_WIN64)
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#define blasabs(x) llabs(x)
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#else
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#define blasabs(x) labs(x)
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#endif
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#else
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typedef int blasint;
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#define blasabs(x) abs(x)
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#endif
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typedef blasint integer;
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typedef unsigned int uinteger;
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typedef char *address;
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typedef short int shortint;
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typedef float real;
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typedef double doublereal;
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typedef struct { real r, i; } complex;
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typedef struct { doublereal r, i; } doublecomplex;
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#ifdef _MSC_VER
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static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
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static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
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static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
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static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
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#else
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static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
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static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
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static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
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static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
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#endif
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#define pCf(z) (*_pCf(z))
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#define pCd(z) (*_pCd(z))
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typedef blasint logical;
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typedef char logical1;
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typedef char integer1;
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#define TRUE_ (1)
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#define FALSE_ (0)
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/* Extern is for use with -E */
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#ifndef Extern
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#define Extern extern
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#endif
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/* I/O stuff */
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typedef int flag;
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typedef int ftnlen;
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typedef int ftnint;
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/*external read, write*/
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typedef struct
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{ flag cierr;
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ftnint ciunit;
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flag ciend;
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char *cifmt;
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ftnint cirec;
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} cilist;
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/*internal read, write*/
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typedef struct
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{ flag icierr;
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char *iciunit;
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flag iciend;
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char *icifmt;
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ftnint icirlen;
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ftnint icirnum;
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} icilist;
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/*open*/
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typedef struct
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{ flag oerr;
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ftnint ounit;
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char *ofnm;
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ftnlen ofnmlen;
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char *osta;
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char *oacc;
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char *ofm;
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ftnint orl;
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char *oblnk;
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} olist;
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/*close*/
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typedef struct
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{ flag cerr;
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ftnint cunit;
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char *csta;
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} cllist;
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/*rewind, backspace, endfile*/
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typedef struct
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{ flag aerr;
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ftnint aunit;
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} alist;
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/* inquire */
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typedef struct
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{ flag inerr;
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ftnint inunit;
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char *infile;
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ftnlen infilen;
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ftnint *inex; /*parameters in standard's order*/
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ftnint *inopen;
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ftnint *innum;
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ftnint *innamed;
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char *inname;
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ftnlen innamlen;
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char *inacc;
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ftnlen inacclen;
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char *inseq;
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ftnlen inseqlen;
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char *indir;
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ftnlen indirlen;
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char *infmt;
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ftnlen infmtlen;
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char *inform;
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ftnint informlen;
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char *inunf;
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ftnlen inunflen;
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ftnint *inrecl;
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ftnint *innrec;
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char *inblank;
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ftnlen inblanklen;
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} inlist;
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#define VOID void
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union Multitype { /* for multiple entry points */
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integer1 g;
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shortint h;
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integer i;
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/* longint j; */
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real r;
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doublereal d;
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complex c;
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doublecomplex z;
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};
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typedef union Multitype Multitype;
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struct Vardesc { /* for Namelist */
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char *name;
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char *addr;
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ftnlen *dims;
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int type;
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};
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typedef struct Vardesc Vardesc;
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struct Namelist {
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char *name;
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Vardesc **vars;
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int nvars;
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};
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typedef struct Namelist Namelist;
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#define abs(x) ((x) >= 0 ? (x) : -(x))
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#define dabs(x) (fabs(x))
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#define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
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#define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
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#define dmin(a,b) (f2cmin(a,b))
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#define dmax(a,b) (f2cmax(a,b))
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#define bit_test(a,b) ((a) >> (b) & 1)
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#define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
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#define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
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#define abort_() { sig_die("Fortran abort routine called", 1); }
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#define c_abs(z) (cabsf(Cf(z)))
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#define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
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#ifdef _MSC_VER
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#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]);}
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#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]);}
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#else
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#define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
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#define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
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#endif
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#define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
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#define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
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#define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
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//#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
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#define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
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#define d_abs(x) (fabs(*(x)))
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#define d_acos(x) (acos(*(x)))
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#define d_asin(x) (asin(*(x)))
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#define d_atan(x) (atan(*(x)))
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#define d_atn2(x, y) (atan2(*(x),*(y)))
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#define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
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#define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
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#define d_cos(x) (cos(*(x)))
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#define d_cosh(x) (cosh(*(x)))
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#define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
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#define d_exp(x) (exp(*(x)))
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#define d_imag(z) (cimag(Cd(z)))
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#define r_imag(z) (cimagf(Cf(z)))
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#define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
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#define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
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#define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
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#define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
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#define d_log(x) (log(*(x)))
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#define d_mod(x, y) (fmod(*(x), *(y)))
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#define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
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#define d_nint(x) u_nint(*(x))
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#define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
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#define d_sign(a,b) u_sign(*(a),*(b))
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#define r_sign(a,b) u_sign(*(a),*(b))
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#define d_sin(x) (sin(*(x)))
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#define d_sinh(x) (sinh(*(x)))
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#define d_sqrt(x) (sqrt(*(x)))
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#define d_tan(x) (tan(*(x)))
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#define d_tanh(x) (tanh(*(x)))
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#define i_abs(x) abs(*(x))
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#define i_dnnt(x) ((integer)u_nint(*(x)))
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#define i_len(s, n) (n)
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#define i_nint(x) ((integer)u_nint(*(x)))
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#define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
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#define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
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#define pow_si(B,E) spow_ui(*(B),*(E))
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#define pow_ri(B,E) spow_ui(*(B),*(E))
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#define pow_di(B,E) dpow_ui(*(B),*(E))
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#define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
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#define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
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#define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
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#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++ = ' '; }
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#define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
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#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]; }
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#define sig_die(s, kill) { exit(1); }
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#define s_stop(s, n) {exit(0);}
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static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
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#define z_abs(z) (cabs(Cd(z)))
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#define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
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#define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
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#define myexit_() break;
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#define mycycle_() continue;
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#define myceiling_(w) {ceil(w)}
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#define myhuge_(w) {HUGE_VAL}
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//#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
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#define mymaxloc_(w,s,e,n) dmaxloc_(w,*(s),*(e),n)
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/* procedure parameter types for -A and -C++ */
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#ifdef __cplusplus
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typedef logical (*L_fp)(...);
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#else
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typedef logical (*L_fp)();
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#endif
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static float spow_ui(float x, integer n) {
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float pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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static double dpow_ui(double x, integer n) {
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double pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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#ifdef _MSC_VER
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static _Fcomplex cpow_ui(complex x, integer n) {
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complex pow={1.0,0.0}; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
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for(u = n; ; ) {
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if(u & 01) pow.r *= x.r, pow.i *= x.i;
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if(u >>= 1) x.r *= x.r, x.i *= x.i;
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else break;
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}
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}
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_Fcomplex p={pow.r, pow.i};
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return p;
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}
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#else
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static _Complex float cpow_ui(_Complex float x, integer n) {
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_Complex float pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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#endif
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#ifdef _MSC_VER
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static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
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_Dcomplex pow={1.0,0.0}; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
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for(u = n; ; ) {
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if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
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if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
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else break;
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}
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}
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_Dcomplex p = {pow._Val[0], pow._Val[1]};
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return p;
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}
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#else
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static _Complex double zpow_ui(_Complex double x, integer n) {
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_Complex double pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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#endif
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static integer pow_ii(integer x, integer n) {
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integer pow; unsigned long int u;
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if (n <= 0) {
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if (n == 0 || x == 1) pow = 1;
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else if (x != -1) pow = x == 0 ? 1/x : 0;
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else n = -n;
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}
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if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
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u = n;
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for(pow = 1; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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static integer dmaxloc_(double *w, integer s, integer e, integer *n)
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{
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double m; integer i, mi;
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for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
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if (w[i-1]>m) mi=i ,m=w[i-1];
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return mi-s+1;
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}
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static integer smaxloc_(float *w, integer s, integer e, integer *n)
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{
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float m; integer i, mi;
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for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
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if (w[i-1]>m) mi=i ,m=w[i-1];
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return mi-s+1;
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}
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static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
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integer n = *n_, incx = *incx_, incy = *incy_, i;
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#ifdef _MSC_VER
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_Fcomplex zdotc = {0.0, 0.0};
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if (incx == 1 && incy == 1) {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
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zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
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}
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} else {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
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zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
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}
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}
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pCf(z) = zdotc;
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}
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#else
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_Complex float zdotc = 0.0;
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if (incx == 1 && incy == 1) {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
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}
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} else {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
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}
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}
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pCf(z) = zdotc;
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}
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#endif
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static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
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integer n = *n_, incx = *incx_, incy = *incy_, i;
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#ifdef _MSC_VER
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_Dcomplex zdotc = {0.0, 0.0};
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if (incx == 1 && incy == 1) {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
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zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
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}
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} else {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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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;
|
|
|
|
/* 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_ */
|
|
|