2809 lines
92 KiB
C
2809 lines
92 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]/df(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) */
|
|
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)
|
|
*/
|
|
|
|
|
|
|
|
|
|
/* Table of constant values */
|
|
|
|
static integer c__1 = 1;
|
|
static doublereal c_b34 = 0.;
|
|
static doublereal c_b35 = 1.;
|
|
static integer c__0 = 0;
|
|
static integer c_n1 = -1;
|
|
|
|
/* > \brief \b DGEJSV */
|
|
|
|
/* =========== DOCUMENTATION =========== */
|
|
|
|
/* Online html documentation available at */
|
|
/* http://www.netlib.org/lapack/explore-html/ */
|
|
|
|
/* > \htmlonly */
|
|
/* > Download DGEJSV + dependencies */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgejsv.
|
|
f"> */
|
|
/* > [TGZ]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgejsv.
|
|
f"> */
|
|
/* > [ZIP]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgejsv.
|
|
f"> */
|
|
/* > [TXT]</a> */
|
|
/* > \endhtmlonly */
|
|
|
|
/* Definition: */
|
|
/* =========== */
|
|
|
|
/* SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, */
|
|
/* M, N, A, LDA, SVA, U, LDU, V, LDV, */
|
|
/* WORK, LWORK, IWORK, INFO ) */
|
|
|
|
/* IMPLICIT NONE */
|
|
/* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N */
|
|
/* DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ), */
|
|
/* $ WORK( LWORK ) */
|
|
/* INTEGER IWORK( * ) */
|
|
/* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV */
|
|
|
|
|
|
/* > \par Purpose: */
|
|
/* ============= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > DGEJSV computes the singular value decomposition (SVD) of a real M-by-N */
|
|
/* > matrix [A], where M >= N. The SVD of [A] is written as */
|
|
/* > */
|
|
/* > [A] = [U] * [SIGMA] * [V]^t, */
|
|
/* > */
|
|
/* > where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N */
|
|
/* > diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and */
|
|
/* > [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are */
|
|
/* > the singular values of [A]. The columns of [U] and [V] are the left and */
|
|
/* > the right singular vectors of [A], respectively. The matrices [U] and [V] */
|
|
/* > are computed and stored in the arrays U and V, respectively. The diagonal */
|
|
/* > of [SIGMA] is computed and stored in the array SVA. */
|
|
/* > DGEJSV can sometimes compute tiny singular values and their singular vectors much */
|
|
/* > more accurately than other SVD routines, see below under Further Details. */
|
|
/* > \endverbatim */
|
|
|
|
/* Arguments: */
|
|
/* ========== */
|
|
|
|
/* > \param[in] JOBA */
|
|
/* > \verbatim */
|
|
/* > JOBA is CHARACTER*1 */
|
|
/* > Specifies the level of accuracy: */
|
|
/* > = 'C': This option works well (high relative accuracy) if A = B * D, */
|
|
/* > with well-conditioned B and arbitrary diagonal matrix D. */
|
|
/* > The accuracy cannot be spoiled by COLUMN scaling. The */
|
|
/* > accuracy of the computed output depends on the condition of */
|
|
/* > B, and the procedure aims at the best theoretical accuracy. */
|
|
/* > The relative error max_{i=1:N}|d sigma_i| / sigma_i is */
|
|
/* > bounded by f(M,N)*epsilon* cond(B), independent of D. */
|
|
/* > The input matrix is preprocessed with the QRF with column */
|
|
/* > pivoting. This initial preprocessing and preconditioning by */
|
|
/* > a rank revealing QR factorization is common for all values of */
|
|
/* > JOBA. Additional actions are specified as follows: */
|
|
/* > = 'E': Computation as with 'C' with an additional estimate of the */
|
|
/* > condition number of B. It provides a realistic error bound. */
|
|
/* > = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings */
|
|
/* > D1, D2, and well-conditioned matrix C, this option gives */
|
|
/* > higher accuracy than the 'C' option. If the structure of the */
|
|
/* > input matrix is not known, and relative accuracy is */
|
|
/* > desirable, then this option is advisable. The input matrix A */
|
|
/* > is preprocessed with QR factorization with FULL (row and */
|
|
/* > column) pivoting. */
|
|
/* > = 'G': Computation as with 'F' with an additional estimate of the */
|
|
/* > condition number of B, where A=D*B. If A has heavily weighted */
|
|
/* > rows, then using this condition number gives too pessimistic */
|
|
/* > error bound. */
|
|
/* > = 'A': Small singular values are the noise and the matrix is treated */
|
|
/* > as numerically rank deficient. The error in the computed */
|
|
/* > singular values is bounded by f(m,n)*epsilon*||A||. */
|
|
/* > The computed SVD A = U * S * V^t restores A up to */
|
|
/* > f(m,n)*epsilon*||A||. */
|
|
/* > This gives the procedure the licence to discard (set to zero) */
|
|
/* > all singular values below N*epsilon*||A||. */
|
|
/* > = 'R': Similar as in 'A'. Rank revealing property of the initial */
|
|
/* > QR factorization is used do reveal (using triangular factor) */
|
|
/* > a gap sigma_{r+1} < epsilon * sigma_r in which case the */
|
|
/* > numerical RANK is declared to be r. The SVD is computed with */
|
|
/* > absolute error bounds, but more accurately than with 'A'. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] JOBU */
|
|
/* > \verbatim */
|
|
/* > JOBU is CHARACTER*1 */
|
|
/* > Specifies whether to compute the columns of U: */
|
|
/* > = 'U': N columns of U are returned in the array U. */
|
|
/* > = 'F': full set of M left sing. vectors is returned in the array U. */
|
|
/* > = 'W': U may be used as workspace of length M*N. See the description */
|
|
/* > of U. */
|
|
/* > = 'N': U is not computed. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] JOBV */
|
|
/* > \verbatim */
|
|
/* > JOBV is CHARACTER*1 */
|
|
/* > Specifies whether to compute the matrix V: */
|
|
/* > = 'V': N columns of V are returned in the array V; Jacobi rotations */
|
|
/* > are not explicitly accumulated. */
|
|
/* > = 'J': N columns of V are returned in the array V, but they are */
|
|
/* > computed as the product of Jacobi rotations. This option is */
|
|
/* > allowed only if JOBU .NE. 'N', i.e. in computing the full SVD. */
|
|
/* > = 'W': V may be used as workspace of length N*N. See the description */
|
|
/* > of V. */
|
|
/* > = 'N': V is not computed. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] JOBR */
|
|
/* > \verbatim */
|
|
/* > JOBR is CHARACTER*1 */
|
|
/* > Specifies the RANGE for the singular values. Issues the licence to */
|
|
/* > set to zero small positive singular values if they are outside */
|
|
/* > specified range. If A .NE. 0 is scaled so that the largest singular */
|
|
/* > value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues */
|
|
/* > the licence to kill columns of A whose norm in c*A is less than */
|
|
/* > DSQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN, */
|
|
/* > where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E'). */
|
|
/* > = 'N': Do not kill small columns of c*A. This option assumes that */
|
|
/* > BLAS and QR factorizations and triangular solvers are */
|
|
/* > implemented to work in that range. If the condition of A */
|
|
/* > is greater than BIG, use DGESVJ. */
|
|
/* > = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)] */
|
|
/* > (roughly, as described above). This option is recommended. */
|
|
/* > ~~~~~~~~~~~~~~~~~~~~~~~~~~~ */
|
|
/* > For computing the singular values in the FULL range [SFMIN,BIG] */
|
|
/* > use DGESVJ. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] JOBT */
|
|
/* > \verbatim */
|
|
/* > JOBT is CHARACTER*1 */
|
|
/* > If the matrix is square then the procedure may determine to use */
|
|
/* > transposed A if A^t seems to be better with respect to convergence. */
|
|
/* > If the matrix is not square, JOBT is ignored. This is subject to */
|
|
/* > changes in the future. */
|
|
/* > The decision is based on two values of entropy over the adjoint */
|
|
/* > orbit of A^t * A. See the descriptions of WORK(6) and WORK(7). */
|
|
/* > = 'T': transpose if entropy test indicates possibly faster */
|
|
/* > convergence of Jacobi process if A^t is taken as input. If A is */
|
|
/* > replaced with A^t, then the row pivoting is included automatically. */
|
|
/* > = 'N': do not speculate. */
|
|
/* > This option can be used to compute only the singular values, or the */
|
|
/* > full SVD (U, SIGMA and V). For only one set of singular vectors */
|
|
/* > (U or V), the caller should provide both U and V, as one of the */
|
|
/* > matrices is used as workspace if the matrix A is transposed. */
|
|
/* > The implementer can easily remove this constraint and make the */
|
|
/* > code more complicated. See the descriptions of U and V. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] JOBP */
|
|
/* > \verbatim */
|
|
/* > JOBP is CHARACTER*1 */
|
|
/* > Issues the licence to introduce structured perturbations to drown */
|
|
/* > denormalized numbers. This licence should be active if the */
|
|
/* > denormals are poorly implemented, causing slow computation, */
|
|
/* > especially in cases of fast convergence (!). For details see [1,2]. */
|
|
/* > For the sake of simplicity, this perturbations are included only */
|
|
/* > when the full SVD or only the singular values are requested. The */
|
|
/* > implementer/user can easily add the perturbation for the cases of */
|
|
/* > computing one set of singular vectors. */
|
|
/* > = 'P': introduce perturbation */
|
|
/* > = 'N': do not perturb */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] M */
|
|
/* > \verbatim */
|
|
/* > M is INTEGER */
|
|
/* > The number of rows of the input matrix A. M >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] N */
|
|
/* > \verbatim */
|
|
/* > N is INTEGER */
|
|
/* > The number of columns of the input matrix A. M >= N >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] A */
|
|
/* > \verbatim */
|
|
/* > A is DOUBLE PRECISION array, dimension (LDA,N) */
|
|
/* > On entry, the M-by-N matrix A. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDA */
|
|
/* > \verbatim */
|
|
/* > LDA is INTEGER */
|
|
/* > The leading dimension of the array A. LDA >= f2cmax(1,M). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] SVA */
|
|
/* > \verbatim */
|
|
/* > SVA is DOUBLE PRECISION array, dimension (N) */
|
|
/* > On exit, */
|
|
/* > - For WORK(1)/WORK(2) = ONE: The singular values of A. During the */
|
|
/* > computation SVA contains Euclidean column norms of the */
|
|
/* > iterated matrices in the array A. */
|
|
/* > - For WORK(1) .NE. WORK(2): The singular values of A are */
|
|
/* > (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if */
|
|
/* > sigma_max(A) overflows or if small singular values have been */
|
|
/* > saved from underflow by scaling the input matrix A. */
|
|
/* > - If JOBR='R' then some of the singular values may be returned */
|
|
/* > as exact zeros obtained by "set to zero" because they are */
|
|
/* > below the numerical rank threshold or are denormalized numbers. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] U */
|
|
/* > \verbatim */
|
|
/* > U is DOUBLE PRECISION array, dimension ( LDU, N ) */
|
|
/* > If JOBU = 'U', then U contains on exit the M-by-N matrix of */
|
|
/* > the left singular vectors. */
|
|
/* > If JOBU = 'F', then U contains on exit the M-by-M matrix of */
|
|
/* > the left singular vectors, including an ONB */
|
|
/* > of the orthogonal complement of the Range(A). */
|
|
/* > If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N), */
|
|
/* > then U is used as workspace if the procedure */
|
|
/* > replaces A with A^t. In that case, [V] is computed */
|
|
/* > in U as left singular vectors of A^t and then */
|
|
/* > copied back to the V array. This 'W' option is just */
|
|
/* > a reminder to the caller that in this case U is */
|
|
/* > reserved as workspace of length N*N. */
|
|
/* > If JOBU = 'N' U is not referenced, unless JOBT='T'. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDU */
|
|
/* > \verbatim */
|
|
/* > LDU is INTEGER */
|
|
/* > The leading dimension of the array U, LDU >= 1. */
|
|
/* > IF JOBU = 'U' or 'F' or 'W', then LDU >= M. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] V */
|
|
/* > \verbatim */
|
|
/* > V is DOUBLE PRECISION array, dimension ( LDV, N ) */
|
|
/* > If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of */
|
|
/* > the right singular vectors; */
|
|
/* > If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N), */
|
|
/* > then V is used as workspace if the pprocedure */
|
|
/* > replaces A with A^t. In that case, [U] is computed */
|
|
/* > in V as right singular vectors of A^t and then */
|
|
/* > copied back to the U array. This 'W' option is just */
|
|
/* > a reminder to the caller that in this case V is */
|
|
/* > reserved as workspace of length N*N. */
|
|
/* > If JOBV = 'N' V is not referenced, unless JOBT='T'. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDV */
|
|
/* > \verbatim */
|
|
/* > LDV is INTEGER */
|
|
/* > The leading dimension of the array V, LDV >= 1. */
|
|
/* > If JOBV = 'V' or 'J' or 'W', then LDV >= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WORK */
|
|
/* > \verbatim */
|
|
/* > WORK is DOUBLE PRECISION array, dimension (LWORK) */
|
|
/* > On exit, if N > 0 .AND. M > 0 (else not referenced), */
|
|
/* > WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such */
|
|
/* > that SCALE*SVA(1:N) are the computed singular values */
|
|
/* > of A. (See the description of SVA().) */
|
|
/* > WORK(2) = See the description of WORK(1). */
|
|
/* > WORK(3) = SCONDA is an estimate for the condition number of */
|
|
/* > column equilibrated A. (If JOBA = 'E' or 'G') */
|
|
/* > SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). */
|
|
/* > It is computed using DPOCON. It holds */
|
|
/* > N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA */
|
|
/* > where R is the triangular factor from the QRF of A. */
|
|
/* > However, if R is truncated and the numerical rank is */
|
|
/* > determined to be strictly smaller than N, SCONDA is */
|
|
/* > returned as -1, thus indicating that the smallest */
|
|
/* > singular values might be lost. */
|
|
/* > */
|
|
/* > If full SVD is needed, the following two condition numbers are */
|
|
/* > useful for the analysis of the algorithm. They are provied for */
|
|
/* > a developer/implementer who is familiar with the details of */
|
|
/* > the method. */
|
|
/* > */
|
|
/* > WORK(4) = an estimate of the scaled condition number of the */
|
|
/* > triangular factor in the first QR factorization. */
|
|
/* > WORK(5) = an estimate of the scaled condition number of the */
|
|
/* > triangular factor in the second QR factorization. */
|
|
/* > The following two parameters are computed if JOBT = 'T'. */
|
|
/* > They are provided for a developer/implementer who is familiar */
|
|
/* > with the details of the method. */
|
|
/* > */
|
|
/* > WORK(6) = the entropy of A^t*A :: this is the Shannon entropy */
|
|
/* > of diag(A^t*A) / Trace(A^t*A) taken as point in the */
|
|
/* > probability simplex. */
|
|
/* > WORK(7) = the entropy of A*A^t. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LWORK */
|
|
/* > \verbatim */
|
|
/* > LWORK is INTEGER */
|
|
/* > Length of WORK to confirm proper allocation of work space. */
|
|
/* > LWORK depends on the job: */
|
|
/* > */
|
|
/* > If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and */
|
|
/* > -> .. no scaled condition estimate required (JOBE = 'N'): */
|
|
/* > LWORK >= f2cmax(2*M+N,4*N+1,7). This is the minimal requirement. */
|
|
/* > ->> For optimal performance (blocked code) the optimal value */
|
|
/* > is LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal */
|
|
/* > block size for DGEQP3 and DGEQRF. */
|
|
/* > In general, optimal LWORK is computed as */
|
|
/* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7). */
|
|
/* > -> .. an estimate of the scaled condition number of A is */
|
|
/* > required (JOBA='E', 'G'). In this case, LWORK is the maximum */
|
|
/* > of the above and N*N+4*N, i.e. LWORK >= f2cmax(2*M+N,N*N+4*N,7). */
|
|
/* > ->> For optimal performance (blocked code) the optimal value */
|
|
/* > is LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7). */
|
|
/* > In general, the optimal length LWORK is computed as */
|
|
/* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), */
|
|
/* > N+N*N+LWORK(DPOCON),7). */
|
|
/* > */
|
|
/* > If SIGMA and the right singular vectors are needed (JOBV = 'V'), */
|
|
/* > -> the minimal requirement is LWORK >= f2cmax(2*M+N,4*N+1,7). */
|
|
/* > -> For optimal performance, LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,7), */
|
|
/* > where NB is the optimal block size for DGEQP3, DGEQRF, DGELQF, */
|
|
/* > DORMLQ. In general, the optimal length LWORK is computed as */
|
|
/* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), */
|
|
/* > N+LWORK(DGELQF), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)). */
|
|
/* > */
|
|
/* > If SIGMA and the left singular vectors are needed */
|
|
/* > -> the minimal requirement is LWORK >= f2cmax(2*M+N,4*N+1,7). */
|
|
/* > -> For optimal performance: */
|
|
/* > if JOBU = 'U' :: LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,7), */
|
|
/* > if JOBU = 'F' :: LWORK >= f2cmax(2*M+N,3*N+(N+1)*NB,N+M*NB,7), */
|
|
/* > where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR. */
|
|
/* > In general, the optimal length LWORK is computed as */
|
|
/* > LWORK >= f2cmax(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON), */
|
|
/* > 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). */
|
|
/* > Here LWORK(DORMQR) equals N*NB (for JOBU = 'U') or */
|
|
/* > M*NB (for JOBU = 'F'). */
|
|
/* > */
|
|
/* > If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and */
|
|
/* > -> if JOBV = 'V' */
|
|
/* > the minimal requirement is LWORK >= f2cmax(2*M+N,6*N+2*N*N). */
|
|
/* > -> if JOBV = 'J' the minimal requirement is */
|
|
/* > LWORK >= f2cmax(2*M+N, 4*N+N*N,2*N+N*N+6). */
|
|
/* > -> For optimal performance, LWORK should be additionally */
|
|
/* > larger than N+M*NB, where NB is the optimal block size */
|
|
/* > for DORMQR. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] IWORK */
|
|
/* > \verbatim */
|
|
/* > IWORK is INTEGER array, dimension (M+3*N). */
|
|
/* > On exit, */
|
|
/* > IWORK(1) = the numerical rank determined after the initial */
|
|
/* > QR factorization with pivoting. See the descriptions */
|
|
/* > of JOBA and JOBR. */
|
|
/* > IWORK(2) = the number of the computed nonzero singular values */
|
|
/* > IWORK(3) = if nonzero, a warning message: */
|
|
/* > If IWORK(3) = 1 then some of the column norms of A */
|
|
/* > were denormalized floats. The requested high accuracy */
|
|
/* > is not warranted by the data. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] INFO */
|
|
/* > \verbatim */
|
|
/* > INFO is INTEGER */
|
|
/* > < 0: if INFO = -i, then the i-th argument had an illegal value. */
|
|
/* > = 0: successful exit; */
|
|
/* > > 0: DGEJSV did not converge in the maximal allowed number */
|
|
/* > of sweeps. The computed values may be inaccurate. */
|
|
/* > \endverbatim */
|
|
|
|
/* Authors: */
|
|
/* ======== */
|
|
|
|
/* > \author Univ. of Tennessee */
|
|
/* > \author Univ. of California Berkeley */
|
|
/* > \author Univ. of Colorado Denver */
|
|
/* > \author NAG Ltd. */
|
|
|
|
/* > \date June 2016 */
|
|
|
|
/* > \ingroup doubleGEsing */
|
|
|
|
/* > \par Further Details: */
|
|
/* ===================== */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3, */
|
|
/* > DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an */
|
|
/* > additional row pivoting can be used as a preprocessor, which in some */
|
|
/* > cases results in much higher accuracy. An example is matrix A with the */
|
|
/* > structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned */
|
|
/* > diagonal matrices and C is well-conditioned matrix. In that case, complete */
|
|
/* > pivoting in the first QR factorizations provides accuracy dependent on the */
|
|
/* > condition number of C, and independent of D1, D2. Such higher accuracy is */
|
|
/* > not completely understood theoretically, but it works well in practice. */
|
|
/* > Further, if A can be written as A = B*D, with well-conditioned B and some */
|
|
/* > diagonal D, then the high accuracy is guaranteed, both theoretically and */
|
|
/* > in software, independent of D. For more details see [1], [2]. */
|
|
/* > The computational range for the singular values can be the full range */
|
|
/* > ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS */
|
|
/* > & LAPACK routines called by DGEJSV are implemented to work in that range. */
|
|
/* > If that is not the case, then the restriction for safe computation with */
|
|
/* > the singular values in the range of normalized IEEE numbers is that the */
|
|
/* > spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not */
|
|
/* > overflow. This code (DGEJSV) is best used in this restricted range, */
|
|
/* > meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are */
|
|
/* > returned as zeros. See JOBR for details on this. */
|
|
/* > Further, this implementation is somewhat slower than the one described */
|
|
/* > in [1,2] due to replacement of some non-LAPACK components, and because */
|
|
/* > the choice of some tuning parameters in the iterative part (DGESVJ) is */
|
|
/* > left to the implementer on a particular machine. */
|
|
/* > The rank revealing QR factorization (in this code: DGEQP3) should be */
|
|
/* > implemented as in [3]. We have a new version of DGEQP3 under development */
|
|
/* > that is more robust than the current one in LAPACK, with a cleaner cut in */
|
|
/* > rank deficient cases. It will be available in the SIGMA library [4]. */
|
|
/* > If M is much larger than N, it is obvious that the initial QRF with */
|
|
/* > column pivoting can be preprocessed by the QRF without pivoting. That */
|
|
/* > well known trick is not used in DGEJSV because in some cases heavy row */
|
|
/* > weighting can be treated with complete pivoting. The overhead in cases */
|
|
/* > M much larger than N is then only due to pivoting, but the benefits in */
|
|
/* > terms of accuracy have prevailed. The implementer/user can incorporate */
|
|
/* > this extra QRF step easily. The implementer can also improve data movement */
|
|
/* > (matrix transpose, matrix copy, matrix transposed copy) - this */
|
|
/* > implementation of DGEJSV uses only the simplest, naive data movement. */
|
|
/* > \endverbatim */
|
|
|
|
/* > \par Contributors: */
|
|
/* ================== */
|
|
/* > */
|
|
/* > Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) */
|
|
|
|
/* > \par References: */
|
|
/* ================ */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. */
|
|
/* > SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. */
|
|
/* > LAPACK Working note 169. */
|
|
/* > [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. */
|
|
/* > SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. */
|
|
/* > LAPACK Working note 170. */
|
|
/* > [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR */
|
|
/* > factorization software - a case study. */
|
|
/* > ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. */
|
|
/* > LAPACK Working note 176. */
|
|
/* > [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, */
|
|
/* > QSVD, (H,K)-SVD computations. */
|
|
/* > Department of Mathematics, University of Zagreb, 2008. */
|
|
/* > \endverbatim */
|
|
|
|
/* > \par Bugs, examples and comments: */
|
|
/* ================================= */
|
|
/* > */
|
|
/* > Please report all bugs and send interesting examples and/or comments to */
|
|
/* > drmac@math.hr. Thank you. */
|
|
/* > */
|
|
/* ===================================================================== */
|
|
/* Subroutine */ void dgejsv_(char *joba, char *jobu, char *jobv, char *jobr,
|
|
char *jobt, char *jobp, integer *m, integer *n, doublereal *a,
|
|
integer *lda, doublereal *sva, doublereal *u, integer *ldu,
|
|
doublereal *v, integer *ldv, doublereal *work, integer *lwork,
|
|
integer *iwork, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer a_dim1, a_offset, u_dim1, u_offset, v_dim1, v_offset, i__1, i__2,
|
|
i__3, i__4, i__5, i__6, i__7, i__8, i__9, i__10, i__11, i__12;
|
|
doublereal d__1, d__2, d__3, d__4;
|
|
|
|
/* Local variables */
|
|
logical defr;
|
|
doublereal aapp, aaqq;
|
|
logical kill;
|
|
integer ierr;
|
|
extern doublereal dnrm2_(integer *, doublereal *, integer *);
|
|
doublereal temp1;
|
|
integer p, q;
|
|
logical jracc;
|
|
extern /* Subroutine */ void dscal_(integer *, doublereal *, doublereal *,
|
|
integer *);
|
|
extern logical lsame_(char *, char *);
|
|
doublereal small, entra, sfmin;
|
|
logical lsvec;
|
|
extern /* Subroutine */ void dcopy_(integer *, doublereal *, integer *,
|
|
doublereal *, integer *), dswap_(integer *, doublereal *, integer
|
|
*, doublereal *, integer *);
|
|
doublereal epsln;
|
|
logical rsvec;
|
|
extern /* Subroutine */ void dtrsm_(char *, char *, char *, char *,
|
|
integer *, integer *, doublereal *, doublereal *, integer *,
|
|
doublereal *, integer *);
|
|
integer n1;
|
|
logical l2aber;
|
|
extern /* Subroutine */ void dgeqp3_(integer *, integer *, doublereal *,
|
|
integer *, integer *, doublereal *, doublereal *, integer *,
|
|
integer *);
|
|
doublereal condr1, condr2, uscal1, uscal2;
|
|
logical l2kill, l2rank, l2tran, l2pert;
|
|
extern doublereal dlamch_(char *);
|
|
integer nr;
|
|
extern /* Subroutine */ void dgelqf_(integer *, integer *, doublereal *,
|
|
integer *, doublereal *, doublereal *, integer *, integer *);
|
|
extern integer idamax_(integer *, doublereal *, integer *);
|
|
doublereal scalem;
|
|
extern /* Subroutine */ void dlascl_(char *, integer *, integer *,
|
|
doublereal *, doublereal *, integer *, integer *, doublereal *,
|
|
integer *, integer *);
|
|
doublereal sconda;
|
|
logical goscal;
|
|
doublereal aatmin;
|
|
extern /* Subroutine */ void dgeqrf_(integer *, integer *, doublereal *,
|
|
integer *, doublereal *, doublereal *, integer *, integer *);
|
|
doublereal aatmax;
|
|
extern /* Subroutine */ void dlacpy_(char *, integer *, integer *,
|
|
doublereal *, integer *, doublereal *, integer *),
|
|
dlaset_(char *, integer *, integer *, doublereal *, doublereal *,
|
|
doublereal *, integer *);
|
|
extern int xerbla_(char *, integer *, ftnlen);
|
|
logical noscal;
|
|
extern /* Subroutine */ void dpocon_(char *, integer *, doublereal *,
|
|
integer *, doublereal *, doublereal *, doublereal *, integer *,
|
|
integer *), dgesvj_(char *, char *, char *, integer *,
|
|
integer *, doublereal *, integer *, doublereal *, integer *,
|
|
doublereal *, integer *, doublereal *, integer *, integer *), dlassq_(integer *, doublereal *, integer
|
|
*, doublereal *, doublereal *);
|
|
extern int dlaswp_(integer *, doublereal *,
|
|
integer *, integer *, integer *, integer *, integer *);
|
|
doublereal entrat;
|
|
logical almort;
|
|
extern /* Subroutine */ void dorgqr_(integer *, integer *, integer *,
|
|
doublereal *, integer *, doublereal *, doublereal *, integer *,
|
|
integer *), dormlq_(char *, char *, integer *, integer *, integer
|
|
*, doublereal *, integer *, doublereal *, doublereal *, integer *,
|
|
doublereal *, integer *, integer *);
|
|
doublereal maxprj;
|
|
logical errest;
|
|
extern /* Subroutine */ void dormqr_(char *, char *, integer *, integer *,
|
|
integer *, doublereal *, integer *, doublereal *, doublereal *,
|
|
integer *, doublereal *, integer *, integer *);
|
|
logical transp, rowpiv;
|
|
doublereal big, cond_ok__, xsc, big1;
|
|
integer warning, numrank;
|
|
|
|
|
|
/* -- LAPACK computational routine (version 3.7.1) -- */
|
|
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
|
|
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
|
|
/* June 2016 */
|
|
|
|
|
|
/* =========================================================================== */
|
|
|
|
|
|
|
|
/* Test the input arguments */
|
|
|
|
/* Parameter adjustments */
|
|
--sva;
|
|
a_dim1 = *lda;
|
|
a_offset = 1 + a_dim1 * 1;
|
|
a -= a_offset;
|
|
u_dim1 = *ldu;
|
|
u_offset = 1 + u_dim1 * 1;
|
|
u -= u_offset;
|
|
v_dim1 = *ldv;
|
|
v_offset = 1 + v_dim1 * 1;
|
|
v -= v_offset;
|
|
--work;
|
|
--iwork;
|
|
|
|
/* Function Body */
|
|
lsvec = lsame_(jobu, "U") || lsame_(jobu, "F");
|
|
jracc = lsame_(jobv, "J");
|
|
rsvec = lsame_(jobv, "V") || jracc;
|
|
rowpiv = lsame_(joba, "F") || lsame_(joba, "G");
|
|
l2rank = lsame_(joba, "R");
|
|
l2aber = lsame_(joba, "A");
|
|
errest = lsame_(joba, "E") || lsame_(joba, "G");
|
|
l2tran = lsame_(jobt, "T");
|
|
l2kill = lsame_(jobr, "R");
|
|
defr = lsame_(jobr, "N");
|
|
l2pert = lsame_(jobp, "P");
|
|
|
|
if (! (rowpiv || l2rank || l2aber || errest || lsame_(joba, "C"))) {
|
|
*info = -1;
|
|
} else if (! (lsvec || lsame_(jobu, "N") || lsame_(
|
|
jobu, "W"))) {
|
|
*info = -2;
|
|
} else if (! (rsvec || lsame_(jobv, "N") || lsame_(
|
|
jobv, "W")) || jracc && ! lsvec) {
|
|
*info = -3;
|
|
} else if (! (l2kill || defr)) {
|
|
*info = -4;
|
|
} else if (! (l2tran || lsame_(jobt, "N"))) {
|
|
*info = -5;
|
|
} else if (! (l2pert || lsame_(jobp, "N"))) {
|
|
*info = -6;
|
|
} else if (*m < 0) {
|
|
*info = -7;
|
|
} else if (*n < 0 || *n > *m) {
|
|
*info = -8;
|
|
} else if (*lda < *m) {
|
|
*info = -10;
|
|
} else if (lsvec && *ldu < *m) {
|
|
*info = -13;
|
|
} else if (rsvec && *ldv < *n) {
|
|
*info = -15;
|
|
} else /* if(complicated condition) */ {
|
|
/* Computing MAX */
|
|
i__1 = 7, i__2 = (*n << 2) + 1, i__1 = f2cmax(i__1,i__2), i__2 = (*m <<
|
|
1) + *n;
|
|
/* Computing MAX */
|
|
i__3 = 7, i__4 = (*n << 2) + *n * *n, i__3 = f2cmax(i__3,i__4), i__4 = (*
|
|
m << 1) + *n;
|
|
/* Computing MAX */
|
|
i__5 = 7, i__6 = (*m << 1) + *n, i__5 = f2cmax(i__5,i__6), i__6 = (*n <<
|
|
2) + 1;
|
|
/* Computing MAX */
|
|
i__7 = 7, i__8 = (*m << 1) + *n, i__7 = f2cmax(i__7,i__8), i__8 = (*n <<
|
|
2) + 1;
|
|
/* Computing MAX */
|
|
i__9 = (*m << 1) + *n, i__10 = *n * 6 + (*n << 1) * *n;
|
|
/* Computing MAX */
|
|
i__11 = (*m << 1) + *n, i__12 = (*n << 2) + *n * *n, i__11 = f2cmax(
|
|
i__11,i__12), i__12 = (*n << 1) + *n * *n + 6;
|
|
if (! (lsvec || rsvec || errest) && *lwork < f2cmax(i__1,i__2) || ! (
|
|
lsvec || rsvec) && errest && *lwork < f2cmax(i__3,i__4) || lsvec
|
|
&& ! rsvec && *lwork < f2cmax(i__5,i__6) || rsvec && ! lsvec && *
|
|
lwork < f2cmax(i__7,i__8) || lsvec && rsvec && ! jracc && *lwork
|
|
< f2cmax(i__9,i__10) || lsvec && rsvec && jracc && *lwork < f2cmax(
|
|
i__11,i__12)) {
|
|
*info = -17;
|
|
} else {
|
|
/* #:) */
|
|
*info = 0;
|
|
}
|
|
}
|
|
|
|
if (*info != 0) {
|
|
/* #:( */
|
|
i__1 = -(*info);
|
|
xerbla_("DGEJSV", &i__1, (ftnlen)6);
|
|
return;
|
|
}
|
|
|
|
/* Quick return for void matrix (Y3K safe) */
|
|
/* #:) */
|
|
if (*m == 0 || *n == 0) {
|
|
iwork[1] = 0;
|
|
iwork[2] = 0;
|
|
iwork[3] = 0;
|
|
work[1] = 0.;
|
|
work[2] = 0.;
|
|
work[3] = 0.;
|
|
work[4] = 0.;
|
|
work[5] = 0.;
|
|
work[6] = 0.;
|
|
work[7] = 0.;
|
|
return;
|
|
}
|
|
|
|
/* Determine whether the matrix U should be M x N or M x M */
|
|
|
|
if (lsvec) {
|
|
n1 = *n;
|
|
if (lsame_(jobu, "F")) {
|
|
n1 = *m;
|
|
}
|
|
}
|
|
|
|
/* Set numerical parameters */
|
|
|
|
/* ! NOTE: Make sure DLAMCH() does not fail on the target architecture. */
|
|
|
|
epsln = dlamch_("Epsilon");
|
|
sfmin = dlamch_("SafeMinimum");
|
|
small = sfmin / epsln;
|
|
big = dlamch_("O");
|
|
/* BIG = ONE / SFMIN */
|
|
|
|
/* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N */
|
|
|
|
/* (!) If necessary, scale SVA() to protect the largest norm from */
|
|
/* overflow. It is possible that this scaling pushes the smallest */
|
|
/* column norm left from the underflow threshold (extreme case). */
|
|
|
|
scalem = 1. / sqrt((doublereal) (*m) * (doublereal) (*n));
|
|
noscal = TRUE_;
|
|
goscal = TRUE_;
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
aapp = 0.;
|
|
aaqq = 1.;
|
|
dlassq_(m, &a[p * a_dim1 + 1], &c__1, &aapp, &aaqq);
|
|
if (aapp > big) {
|
|
*info = -9;
|
|
i__2 = -(*info);
|
|
xerbla_("DGEJSV", &i__2, (ftnlen)6);
|
|
return;
|
|
}
|
|
aaqq = sqrt(aaqq);
|
|
if (aapp < big / aaqq && noscal) {
|
|
sva[p] = aapp * aaqq;
|
|
} else {
|
|
noscal = FALSE_;
|
|
sva[p] = aapp * (aaqq * scalem);
|
|
if (goscal) {
|
|
goscal = FALSE_;
|
|
i__2 = p - 1;
|
|
dscal_(&i__2, &scalem, &sva[1], &c__1);
|
|
}
|
|
}
|
|
/* L1874: */
|
|
}
|
|
|
|
if (noscal) {
|
|
scalem = 1.;
|
|
}
|
|
|
|
aapp = 0.;
|
|
aaqq = big;
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
/* Computing MAX */
|
|
d__1 = aapp, d__2 = sva[p];
|
|
aapp = f2cmax(d__1,d__2);
|
|
if (sva[p] != 0.) {
|
|
/* Computing MIN */
|
|
d__1 = aaqq, d__2 = sva[p];
|
|
aaqq = f2cmin(d__1,d__2);
|
|
}
|
|
/* L4781: */
|
|
}
|
|
|
|
/* Quick return for zero M x N matrix */
|
|
/* #:) */
|
|
if (aapp == 0.) {
|
|
if (lsvec) {
|
|
dlaset_("G", m, &n1, &c_b34, &c_b35, &u[u_offset], ldu)
|
|
;
|
|
}
|
|
if (rsvec) {
|
|
dlaset_("G", n, n, &c_b34, &c_b35, &v[v_offset], ldv);
|
|
}
|
|
work[1] = 1.;
|
|
work[2] = 1.;
|
|
if (errest) {
|
|
work[3] = 1.;
|
|
}
|
|
if (lsvec && rsvec) {
|
|
work[4] = 1.;
|
|
work[5] = 1.;
|
|
}
|
|
if (l2tran) {
|
|
work[6] = 0.;
|
|
work[7] = 0.;
|
|
}
|
|
iwork[1] = 0;
|
|
iwork[2] = 0;
|
|
iwork[3] = 0;
|
|
return;
|
|
}
|
|
|
|
/* Issue warning if denormalized column norms detected. Override the */
|
|
/* high relative accuracy request. Issue licence to kill columns */
|
|
/* (set them to zero) whose norm is less than sigma_max / BIG (roughly). */
|
|
/* #:( */
|
|
warning = 0;
|
|
if (aaqq <= sfmin) {
|
|
l2rank = TRUE_;
|
|
l2kill = TRUE_;
|
|
warning = 1;
|
|
}
|
|
|
|
/* Quick return for one-column matrix */
|
|
/* #:) */
|
|
if (*n == 1) {
|
|
|
|
if (lsvec) {
|
|
dlascl_("G", &c__0, &c__0, &sva[1], &scalem, m, &c__1, &a[a_dim1
|
|
+ 1], lda, &ierr);
|
|
dlacpy_("A", m, &c__1, &a[a_offset], lda, &u[u_offset], ldu);
|
|
/* computing all M left singular vectors of the M x 1 matrix */
|
|
if (n1 != *n) {
|
|
i__1 = *lwork - *n;
|
|
dgeqrf_(m, n, &u[u_offset], ldu, &work[1], &work[*n + 1], &
|
|
i__1, &ierr);
|
|
i__1 = *lwork - *n;
|
|
dorgqr_(m, &n1, &c__1, &u[u_offset], ldu, &work[1], &work[*n
|
|
+ 1], &i__1, &ierr);
|
|
dcopy_(m, &a[a_dim1 + 1], &c__1, &u[u_dim1 + 1], &c__1);
|
|
}
|
|
}
|
|
if (rsvec) {
|
|
v[v_dim1 + 1] = 1.;
|
|
}
|
|
if (sva[1] < big * scalem) {
|
|
sva[1] /= scalem;
|
|
scalem = 1.;
|
|
}
|
|
work[1] = 1. / scalem;
|
|
work[2] = 1.;
|
|
if (sva[1] != 0.) {
|
|
iwork[1] = 1;
|
|
if (sva[1] / scalem >= sfmin) {
|
|
iwork[2] = 1;
|
|
} else {
|
|
iwork[2] = 0;
|
|
}
|
|
} else {
|
|
iwork[1] = 0;
|
|
iwork[2] = 0;
|
|
}
|
|
iwork[3] = 0;
|
|
if (errest) {
|
|
work[3] = 1.;
|
|
}
|
|
if (lsvec && rsvec) {
|
|
work[4] = 1.;
|
|
work[5] = 1.;
|
|
}
|
|
if (l2tran) {
|
|
work[6] = 0.;
|
|
work[7] = 0.;
|
|
}
|
|
return;
|
|
|
|
}
|
|
|
|
transp = FALSE_;
|
|
l2tran = l2tran && *m == *n;
|
|
|
|
aatmax = -1.;
|
|
aatmin = big;
|
|
if (rowpiv || l2tran) {
|
|
|
|
/* Compute the row norms, needed to determine row pivoting sequence */
|
|
/* (in the case of heavily row weighted A, row pivoting is strongly */
|
|
/* advised) and to collect information needed to compare the */
|
|
/* structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.). */
|
|
|
|
if (l2tran) {
|
|
i__1 = *m;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
xsc = 0.;
|
|
temp1 = 1.;
|
|
dlassq_(n, &a[p + a_dim1], lda, &xsc, &temp1);
|
|
/* DLASSQ gets both the ell_2 and the ell_infinity norm */
|
|
/* in one pass through the vector */
|
|
work[*m + *n + p] = xsc * scalem;
|
|
work[*n + p] = xsc * (scalem * sqrt(temp1));
|
|
/* Computing MAX */
|
|
d__1 = aatmax, d__2 = work[*n + p];
|
|
aatmax = f2cmax(d__1,d__2);
|
|
if (work[*n + p] != 0.) {
|
|
/* Computing MIN */
|
|
d__1 = aatmin, d__2 = work[*n + p];
|
|
aatmin = f2cmin(d__1,d__2);
|
|
}
|
|
/* L1950: */
|
|
}
|
|
} else {
|
|
i__1 = *m;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
work[*m + *n + p] = scalem * (d__1 = a[p + idamax_(n, &a[p +
|
|
a_dim1], lda) * a_dim1], abs(d__1));
|
|
/* Computing MAX */
|
|
d__1 = aatmax, d__2 = work[*m + *n + p];
|
|
aatmax = f2cmax(d__1,d__2);
|
|
/* Computing MIN */
|
|
d__1 = aatmin, d__2 = work[*m + *n + p];
|
|
aatmin = f2cmin(d__1,d__2);
|
|
/* L1904: */
|
|
}
|
|
}
|
|
|
|
}
|
|
|
|
/* For square matrix A try to determine whether A^t would be better */
|
|
/* input for the preconditioned Jacobi SVD, with faster convergence. */
|
|
/* The decision is based on an O(N) function of the vector of column */
|
|
/* and row norms of A, based on the Shannon entropy. This should give */
|
|
/* the right choice in most cases when the difference actually matters. */
|
|
/* It may fail and pick the slower converging side. */
|
|
|
|
entra = 0.;
|
|
entrat = 0.;
|
|
if (l2tran) {
|
|
|
|
xsc = 0.;
|
|
temp1 = 1.;
|
|
dlassq_(n, &sva[1], &c__1, &xsc, &temp1);
|
|
temp1 = 1. / temp1;
|
|
|
|
entra = 0.;
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
/* Computing 2nd power */
|
|
d__1 = sva[p] / xsc;
|
|
big1 = d__1 * d__1 * temp1;
|
|
if (big1 != 0.) {
|
|
entra += big1 * log(big1);
|
|
}
|
|
/* L1113: */
|
|
}
|
|
entra = -entra / log((doublereal) (*n));
|
|
|
|
/* Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex. */
|
|
/* It is derived from the diagonal of A^t * A. Do the same with the */
|
|
/* diagonal of A * A^t, compute the entropy of the corresponding */
|
|
/* probability distribution. Note that A * A^t and A^t * A have the */
|
|
/* same trace. */
|
|
|
|
entrat = 0.;
|
|
i__1 = *n + *m;
|
|
for (p = *n + 1; p <= i__1; ++p) {
|
|
/* Computing 2nd power */
|
|
d__1 = work[p] / xsc;
|
|
big1 = d__1 * d__1 * temp1;
|
|
if (big1 != 0.) {
|
|
entrat += big1 * log(big1);
|
|
}
|
|
/* L1114: */
|
|
}
|
|
entrat = -entrat / log((doublereal) (*m));
|
|
|
|
/* Analyze the entropies and decide A or A^t. Smaller entropy */
|
|
/* usually means better input for the algorithm. */
|
|
|
|
transp = entrat < entra;
|
|
|
|
/* If A^t is better than A, transpose A. */
|
|
|
|
if (transp) {
|
|
/* In an optimal implementation, this trivial transpose */
|
|
/* should be replaced with faster transpose. */
|
|
i__1 = *n - 1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *n;
|
|
for (q = p + 1; q <= i__2; ++q) {
|
|
temp1 = a[q + p * a_dim1];
|
|
a[q + p * a_dim1] = a[p + q * a_dim1];
|
|
a[p + q * a_dim1] = temp1;
|
|
/* L1116: */
|
|
}
|
|
/* L1115: */
|
|
}
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
work[*m + *n + p] = sva[p];
|
|
sva[p] = work[*n + p];
|
|
/* L1117: */
|
|
}
|
|
temp1 = aapp;
|
|
aapp = aatmax;
|
|
aatmax = temp1;
|
|
temp1 = aaqq;
|
|
aaqq = aatmin;
|
|
aatmin = temp1;
|
|
kill = lsvec;
|
|
lsvec = rsvec;
|
|
rsvec = kill;
|
|
if (lsvec) {
|
|
n1 = *n;
|
|
}
|
|
|
|
rowpiv = TRUE_;
|
|
}
|
|
|
|
}
|
|
/* END IF L2TRAN */
|
|
|
|
/* Scale the matrix so that its maximal singular value remains less */
|
|
/* than DSQRT(BIG) -- the matrix is scaled so that its maximal column */
|
|
/* has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep */
|
|
/* DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and */
|
|
/* BLAS routines that, in some implementations, are not capable of */
|
|
/* working in the full interval [SFMIN,BIG] and that they may provoke */
|
|
/* overflows in the intermediate results. If the singular values spread */
|
|
/* from SFMIN to BIG, then DGESVJ will compute them. So, in that case, */
|
|
/* one should use DGESVJ instead of DGEJSV. */
|
|
|
|
big1 = sqrt(big);
|
|
temp1 = sqrt(big / (doublereal) (*n));
|
|
|
|
dlascl_("G", &c__0, &c__0, &aapp, &temp1, n, &c__1, &sva[1], n, &ierr);
|
|
if (aaqq > aapp * sfmin) {
|
|
aaqq = aaqq / aapp * temp1;
|
|
} else {
|
|
aaqq = aaqq * temp1 / aapp;
|
|
}
|
|
temp1 *= scalem;
|
|
dlascl_("G", &c__0, &c__0, &aapp, &temp1, m, n, &a[a_offset], lda, &ierr);
|
|
|
|
/* To undo scaling at the end of this procedure, multiply the */
|
|
/* computed singular values with USCAL2 / USCAL1. */
|
|
|
|
uscal1 = temp1;
|
|
uscal2 = aapp;
|
|
|
|
if (l2kill) {
|
|
/* L2KILL enforces computation of nonzero singular values in */
|
|
/* the restricted range of condition number of the initial A, */
|
|
/* sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN). */
|
|
xsc = sqrt(sfmin);
|
|
} else {
|
|
xsc = small;
|
|
|
|
/* Now, if the condition number of A is too big, */
|
|
/* sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN, */
|
|
/* as a precaution measure, the full SVD is computed using DGESVJ */
|
|
/* with accumulated Jacobi rotations. This provides numerically */
|
|
/* more robust computation, at the cost of slightly increased run */
|
|
/* time. Depending on the concrete implementation of BLAS and LAPACK */
|
|
/* (i.e. how they behave in presence of extreme ill-conditioning) the */
|
|
/* implementor may decide to remove this switch. */
|
|
if (aaqq < sqrt(sfmin) && lsvec && rsvec) {
|
|
jracc = TRUE_;
|
|
}
|
|
|
|
}
|
|
if (aaqq < xsc) {
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
if (sva[p] < xsc) {
|
|
dlaset_("A", m, &c__1, &c_b34, &c_b34, &a[p * a_dim1 + 1],
|
|
lda);
|
|
sva[p] = 0.;
|
|
}
|
|
/* L700: */
|
|
}
|
|
}
|
|
|
|
/* Preconditioning using QR factorization with pivoting */
|
|
|
|
if (rowpiv) {
|
|
/* Optional row permutation (Bjoerck row pivoting): */
|
|
/* A result by Cox and Higham shows that the Bjoerck's */
|
|
/* row pivoting combined with standard column pivoting */
|
|
/* has similar effect as Powell-Reid complete pivoting. */
|
|
/* The ell-infinity norms of A are made nonincreasing. */
|
|
i__1 = *m - 1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *m - p + 1;
|
|
q = idamax_(&i__2, &work[*m + *n + p], &c__1) + p - 1;
|
|
iwork[(*n << 1) + p] = q;
|
|
if (p != q) {
|
|
temp1 = work[*m + *n + p];
|
|
work[*m + *n + p] = work[*m + *n + q];
|
|
work[*m + *n + q] = temp1;
|
|
}
|
|
/* L1952: */
|
|
}
|
|
i__1 = *m - 1;
|
|
dlaswp_(n, &a[a_offset], lda, &c__1, &i__1, &iwork[(*n << 1) + 1], &
|
|
c__1);
|
|
}
|
|
|
|
/* End of the preparation phase (scaling, optional sorting and */
|
|
/* transposing, optional flushing of small columns). */
|
|
|
|
/* Preconditioning */
|
|
|
|
/* If the full SVD is needed, the right singular vectors are computed */
|
|
/* from a matrix equation, and for that we need theoretical analysis */
|
|
/* of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF. */
|
|
/* In all other cases the first RR QRF can be chosen by other criteria */
|
|
/* (eg speed by replacing global with restricted window pivoting, such */
|
|
/* as in SGEQPX from TOMS # 782). Good results will be obtained using */
|
|
/* SGEQPX with properly (!) chosen numerical parameters. */
|
|
/* Any improvement of DGEQP3 improves overal performance of DGEJSV. */
|
|
|
|
/* A * P1 = Q1 * [ R1^t 0]^t: */
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
iwork[p] = 0;
|
|
/* L1963: */
|
|
}
|
|
i__1 = *lwork - *n;
|
|
dgeqp3_(m, n, &a[a_offset], lda, &iwork[1], &work[1], &work[*n + 1], &
|
|
i__1, &ierr);
|
|
|
|
/* The upper triangular matrix R1 from the first QRF is inspected for */
|
|
/* rank deficiency and possibilities for deflation, or possible */
|
|
/* ill-conditioning. Depending on the user specified flag L2RANK, */
|
|
/* the procedure explores possibilities to reduce the numerical */
|
|
/* rank by inspecting the computed upper triangular factor. If */
|
|
/* L2RANK or L2ABER are up, then DGEJSV will compute the SVD of */
|
|
/* A + dA, where ||dA|| <= f(M,N)*EPSLN. */
|
|
|
|
nr = 1;
|
|
if (l2aber) {
|
|
/* Standard absolute error bound suffices. All sigma_i with */
|
|
/* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an */
|
|
/* aggressive enforcement of lower numerical rank by introducing a */
|
|
/* backward error of the order of N*EPSLN*||A||. */
|
|
temp1 = sqrt((doublereal) (*n)) * epsln;
|
|
i__1 = *n;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
if ((d__2 = a[p + p * a_dim1], abs(d__2)) >= temp1 * (d__1 = a[
|
|
a_dim1 + 1], abs(d__1))) {
|
|
++nr;
|
|
} else {
|
|
goto L3002;
|
|
}
|
|
/* L3001: */
|
|
}
|
|
L3002:
|
|
;
|
|
} else if (l2rank) {
|
|
/* Sudden drop on the diagonal of R1 is used as the criterion for */
|
|
/* close-to-rank-deficient. */
|
|
temp1 = sqrt(sfmin);
|
|
i__1 = *n;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
if ((d__2 = a[p + p * a_dim1], abs(d__2)) < epsln * (d__1 = a[p -
|
|
1 + (p - 1) * a_dim1], abs(d__1)) || (d__3 = a[p + p *
|
|
a_dim1], abs(d__3)) < small || l2kill && (d__4 = a[p + p *
|
|
a_dim1], abs(d__4)) < temp1) {
|
|
goto L3402;
|
|
}
|
|
++nr;
|
|
/* L3401: */
|
|
}
|
|
L3402:
|
|
|
|
;
|
|
} else {
|
|
/* The goal is high relative accuracy. However, if the matrix */
|
|
/* has high scaled condition number the relative accuracy is in */
|
|
/* general not feasible. Later on, a condition number estimator */
|
|
/* will be deployed to estimate the scaled condition number. */
|
|
/* Here we just remove the underflowed part of the triangular */
|
|
/* factor. This prevents the situation in which the code is */
|
|
/* working hard to get the accuracy not warranted by the data. */
|
|
temp1 = sqrt(sfmin);
|
|
i__1 = *n;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
if ((d__1 = a[p + p * a_dim1], abs(d__1)) < small || l2kill && (
|
|
d__2 = a[p + p * a_dim1], abs(d__2)) < temp1) {
|
|
goto L3302;
|
|
}
|
|
++nr;
|
|
/* L3301: */
|
|
}
|
|
L3302:
|
|
|
|
;
|
|
}
|
|
|
|
almort = FALSE_;
|
|
if (nr == *n) {
|
|
maxprj = 1.;
|
|
i__1 = *n;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
temp1 = (d__1 = a[p + p * a_dim1], abs(d__1)) / sva[iwork[p]];
|
|
maxprj = f2cmin(maxprj,temp1);
|
|
/* L3051: */
|
|
}
|
|
/* Computing 2nd power */
|
|
d__1 = maxprj;
|
|
if (d__1 * d__1 >= 1. - (doublereal) (*n) * epsln) {
|
|
almort = TRUE_;
|
|
}
|
|
}
|
|
|
|
|
|
sconda = -1.;
|
|
condr1 = -1.;
|
|
condr2 = -1.;
|
|
|
|
if (errest) {
|
|
if (*n == nr) {
|
|
if (rsvec) {
|
|
dlacpy_("U", n, n, &a[a_offset], lda, &v[v_offset], ldv);
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
temp1 = sva[iwork[p]];
|
|
d__1 = 1. / temp1;
|
|
dscal_(&p, &d__1, &v[p * v_dim1 + 1], &c__1);
|
|
/* L3053: */
|
|
}
|
|
dpocon_("U", n, &v[v_offset], ldv, &c_b35, &temp1, &work[*n +
|
|
1], &iwork[(*n << 1) + *m + 1], &ierr);
|
|
} else if (lsvec) {
|
|
dlacpy_("U", n, n, &a[a_offset], lda, &u[u_offset], ldu);
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
temp1 = sva[iwork[p]];
|
|
d__1 = 1. / temp1;
|
|
dscal_(&p, &d__1, &u[p * u_dim1 + 1], &c__1);
|
|
/* L3054: */
|
|
}
|
|
dpocon_("U", n, &u[u_offset], ldu, &c_b35, &temp1, &work[*n +
|
|
1], &iwork[(*n << 1) + *m + 1], &ierr);
|
|
} else {
|
|
dlacpy_("U", n, n, &a[a_offset], lda, &work[*n + 1], n);
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
temp1 = sva[iwork[p]];
|
|
d__1 = 1. / temp1;
|
|
dscal_(&p, &d__1, &work[*n + (p - 1) * *n + 1], &c__1);
|
|
/* L3052: */
|
|
}
|
|
dpocon_("U", n, &work[*n + 1], n, &c_b35, &temp1, &work[*n + *
|
|
n * *n + 1], &iwork[(*n << 1) + *m + 1], &ierr);
|
|
}
|
|
sconda = 1. / sqrt(temp1);
|
|
/* SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). */
|
|
/* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA */
|
|
} else {
|
|
sconda = -1.;
|
|
}
|
|
}
|
|
|
|
l2pert = l2pert && (d__1 = a[a_dim1 + 1] / a[nr + nr * a_dim1], abs(d__1))
|
|
> sqrt(big1);
|
|
/* If there is no violent scaling, artificial perturbation is not needed. */
|
|
|
|
/* Phase 3: */
|
|
|
|
if (! (rsvec || lsvec)) {
|
|
|
|
/* Singular Values only */
|
|
|
|
/* Computing MIN */
|
|
i__2 = *n - 1;
|
|
i__1 = f2cmin(i__2,nr);
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *n - p;
|
|
dcopy_(&i__2, &a[p + (p + 1) * a_dim1], lda, &a[p + 1 + p *
|
|
a_dim1], &c__1);
|
|
/* L1946: */
|
|
}
|
|
|
|
/* The following two DO-loops introduce small relative perturbation */
|
|
/* into the strict upper triangle of the lower triangular matrix. */
|
|
/* Small entries below the main diagonal are also changed. */
|
|
/* This modification is useful if the computing environment does not */
|
|
/* provide/allow FLUSH TO ZERO underflow, for it prevents many */
|
|
/* annoying denormalized numbers in case of strongly scaled matrices. */
|
|
/* The perturbation is structured so that it does not introduce any */
|
|
/* new perturbation of the singular values, and it does not destroy */
|
|
/* the job done by the preconditioner. */
|
|
/* The licence for this perturbation is in the variable L2PERT, which */
|
|
/* should be .FALSE. if FLUSH TO ZERO underflow is active. */
|
|
|
|
if (! almort) {
|
|
|
|
if (l2pert) {
|
|
/* XSC = DSQRT(SMALL) */
|
|
xsc = epsln / (doublereal) (*n);
|
|
i__1 = nr;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
temp1 = xsc * (d__1 = a[q + q * a_dim1], abs(d__1));
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
if (p > q && (d__1 = a[p + q * a_dim1], abs(d__1)) <=
|
|
temp1 || p < q) {
|
|
a[p + q * a_dim1] = d_sign(&temp1, &a[p + q *
|
|
a_dim1]);
|
|
}
|
|
/* L4949: */
|
|
}
|
|
/* L4947: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &a[(a_dim1 << 1) +
|
|
1], lda);
|
|
}
|
|
|
|
|
|
i__1 = *lwork - *n;
|
|
dgeqrf_(n, &nr, &a[a_offset], lda, &work[1], &work[*n + 1], &i__1,
|
|
&ierr);
|
|
|
|
i__1 = nr - 1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = nr - p;
|
|
dcopy_(&i__2, &a[p + (p + 1) * a_dim1], lda, &a[p + 1 + p *
|
|
a_dim1], &c__1);
|
|
/* L1948: */
|
|
}
|
|
|
|
}
|
|
|
|
/* Row-cyclic Jacobi SVD algorithm with column pivoting */
|
|
|
|
/* to drown denormals */
|
|
if (l2pert) {
|
|
/* XSC = DSQRT(SMALL) */
|
|
xsc = epsln / (doublereal) (*n);
|
|
i__1 = nr;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
temp1 = xsc * (d__1 = a[q + q * a_dim1], abs(d__1));
|
|
i__2 = nr;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
if (p > q && (d__1 = a[p + q * a_dim1], abs(d__1)) <=
|
|
temp1 || p < q) {
|
|
a[p + q * a_dim1] = d_sign(&temp1, &a[p + q * a_dim1])
|
|
;
|
|
}
|
|
/* L1949: */
|
|
}
|
|
/* L1947: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &a[(a_dim1 << 1) + 1],
|
|
lda);
|
|
}
|
|
|
|
/* triangular matrix (plus perturbation which is ignored in */
|
|
/* the part which destroys triangular form (confusing?!)) */
|
|
|
|
dgesvj_("L", "NoU", "NoV", &nr, &nr, &a[a_offset], lda, &sva[1], n, &
|
|
v[v_offset], ldv, &work[1], lwork, info);
|
|
|
|
scalem = work[1];
|
|
numrank = i_dnnt(&work[2]);
|
|
|
|
|
|
} else if (rsvec && ! lsvec) {
|
|
|
|
/* -> Singular Values and Right Singular Vectors <- */
|
|
|
|
if (almort) {
|
|
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *n - p + 1;
|
|
dcopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], &
|
|
c__1);
|
|
/* L1998: */
|
|
}
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) +
|
|
1], ldv);
|
|
|
|
dgesvj_("L", "U", "N", n, &nr, &v[v_offset], ldv, &sva[1], &nr, &
|
|
a[a_offset], lda, &work[1], lwork, info);
|
|
scalem = work[1];
|
|
numrank = i_dnnt(&work[2]);
|
|
} else {
|
|
|
|
/* accumulated product of Jacobi rotations, three are perfect ) */
|
|
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("Lower", &i__1, &i__2, &c_b34, &c_b34, &a[a_dim1 + 2],
|
|
lda);
|
|
i__1 = *lwork - *n;
|
|
dgelqf_(&nr, n, &a[a_offset], lda, &work[1], &work[*n + 1], &i__1,
|
|
&ierr);
|
|
dlacpy_("Lower", &nr, &nr, &a[a_offset], lda, &v[v_offset], ldv);
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) +
|
|
1], ldv);
|
|
i__1 = *lwork - (*n << 1);
|
|
dgeqrf_(&nr, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*n <<
|
|
1) + 1], &i__1, &ierr);
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = nr - p + 1;
|
|
dcopy_(&i__2, &v[p + p * v_dim1], ldv, &v[p + p * v_dim1], &
|
|
c__1);
|
|
/* L8998: */
|
|
}
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) +
|
|
1], ldv);
|
|
|
|
dgesvj_("Lower", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[1], &
|
|
nr, &u[u_offset], ldu, &work[*n + 1], lwork, info);
|
|
scalem = work[*n + 1];
|
|
numrank = i_dnnt(&work[*n + 2]);
|
|
if (nr < *n) {
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1],
|
|
ldv);
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1
|
|
+ 1], ldv);
|
|
i__1 = *n - nr;
|
|
i__2 = *n - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr +
|
|
1) * v_dim1], ldv);
|
|
}
|
|
|
|
i__1 = *lwork - *n;
|
|
dormlq_("Left", "Transpose", n, n, &nr, &a[a_offset], lda, &work[
|
|
1], &v[v_offset], ldv, &work[*n + 1], &i__1, &ierr);
|
|
|
|
}
|
|
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
dcopy_(n, &v[p + v_dim1], ldv, &a[iwork[p] + a_dim1], lda);
|
|
/* L8991: */
|
|
}
|
|
dlacpy_("All", n, n, &a[a_offset], lda, &v[v_offset], ldv);
|
|
|
|
if (transp) {
|
|
dlacpy_("All", n, n, &v[v_offset], ldv, &u[u_offset], ldu);
|
|
}
|
|
|
|
} else if (lsvec && ! rsvec) {
|
|
|
|
|
|
/* Jacobi rotations in the Jacobi iterations. */
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *n - p + 1;
|
|
dcopy_(&i__2, &a[p + p * a_dim1], lda, &u[p + p * u_dim1], &c__1);
|
|
/* L1965: */
|
|
}
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1],
|
|
ldu);
|
|
|
|
i__1 = *lwork - (*n << 1);
|
|
dgeqrf_(n, &nr, &u[u_offset], ldu, &work[*n + 1], &work[(*n << 1) + 1]
|
|
, &i__1, &ierr);
|
|
|
|
i__1 = nr - 1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = nr - p;
|
|
dcopy_(&i__2, &u[p + (p + 1) * u_dim1], ldu, &u[p + 1 + p *
|
|
u_dim1], &c__1);
|
|
/* L1967: */
|
|
}
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1],
|
|
ldu);
|
|
|
|
i__1 = *lwork - *n;
|
|
dgesvj_("Lower", "U", "N", &nr, &nr, &u[u_offset], ldu, &sva[1], &nr,
|
|
&a[a_offset], lda, &work[*n + 1], &i__1, info);
|
|
scalem = work[*n + 1];
|
|
numrank = i_dnnt(&work[*n + 2]);
|
|
|
|
if (nr < *m) {
|
|
i__1 = *m - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + u_dim1], ldu);
|
|
if (nr < n1) {
|
|
i__1 = n1 - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) * u_dim1
|
|
+ 1], ldu);
|
|
i__1 = *m - nr;
|
|
i__2 = n1 - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + (nr +
|
|
1) * u_dim1], ldu);
|
|
}
|
|
}
|
|
|
|
i__1 = *lwork - *n;
|
|
dormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[1], &u[
|
|
u_offset], ldu, &work[*n + 1], &i__1, &ierr);
|
|
|
|
if (rowpiv) {
|
|
i__1 = *m - 1;
|
|
dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1) +
|
|
1], &c_n1);
|
|
}
|
|
|
|
i__1 = n1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
xsc = 1. / dnrm2_(m, &u[p * u_dim1 + 1], &c__1);
|
|
dscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1);
|
|
/* L1974: */
|
|
}
|
|
|
|
if (transp) {
|
|
dlacpy_("All", n, n, &u[u_offset], ldu, &v[v_offset], ldv);
|
|
}
|
|
|
|
} else {
|
|
|
|
|
|
if (! jracc) {
|
|
|
|
if (! almort) {
|
|
|
|
/* Second Preconditioning Step (QRF [with pivoting]) */
|
|
/* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is */
|
|
/* equivalent to an LQF CALL. Since in many libraries the QRF */
|
|
/* seems to be better optimized than the LQF, we do explicit */
|
|
/* transpose and use the QRF. This is subject to changes in an */
|
|
/* optimized implementation of DGEJSV. */
|
|
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *n - p + 1;
|
|
dcopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1],
|
|
&c__1);
|
|
/* L1968: */
|
|
}
|
|
|
|
/* denormals in the second QR factorization, where they are */
|
|
/* as good as zeros. This is done to avoid painfully slow */
|
|
/* computation with denormals. The relative size of the perturbation */
|
|
/* is a parameter that can be changed by the implementer. */
|
|
/* This perturbation device will be obsolete on machines with */
|
|
/* properly implemented arithmetic. */
|
|
/* To switch it off, set L2PERT=.FALSE. To remove it from the */
|
|
/* code, remove the action under L2PERT=.TRUE., leave the ELSE part. */
|
|
/* The following two loops should be blocked and fused with the */
|
|
/* transposed copy above. */
|
|
|
|
if (l2pert) {
|
|
xsc = sqrt(small);
|
|
i__1 = nr;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
temp1 = xsc * (d__1 = v[q + q * v_dim1], abs(d__1));
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
if (p > q && (d__1 = v[p + q * v_dim1], abs(d__1))
|
|
<= temp1 || p < q) {
|
|
v[p + q * v_dim1] = d_sign(&temp1, &v[p + q *
|
|
v_dim1]);
|
|
}
|
|
if (p < q) {
|
|
v[p + q * v_dim1] = -v[p + q * v_dim1];
|
|
}
|
|
/* L2968: */
|
|
}
|
|
/* L2969: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 <<
|
|
1) + 1], ldv);
|
|
}
|
|
|
|
/* Estimate the row scaled condition number of R1 */
|
|
/* (If R1 is rectangular, N > NR, then the condition number */
|
|
/* of the leading NR x NR submatrix is estimated.) */
|
|
|
|
dlacpy_("L", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1]
|
|
, &nr);
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = nr - p + 1;
|
|
temp1 = dnrm2_(&i__2, &work[(*n << 1) + (p - 1) * nr + p],
|
|
&c__1);
|
|
i__2 = nr - p + 1;
|
|
d__1 = 1. / temp1;
|
|
dscal_(&i__2, &d__1, &work[(*n << 1) + (p - 1) * nr + p],
|
|
&c__1);
|
|
/* L3950: */
|
|
}
|
|
dpocon_("Lower", &nr, &work[(*n << 1) + 1], &nr, &c_b35, &
|
|
temp1, &work[(*n << 1) + nr * nr + 1], &iwork[*m + (*
|
|
n << 1) + 1], &ierr);
|
|
condr1 = 1. / sqrt(temp1);
|
|
/* R1 is OK for inverse <=> CONDR1 .LT. DBLE(N) */
|
|
/* more conservative <=> CONDR1 .LT. DSQRT(DBLE(N)) */
|
|
|
|
cond_ok__ = sqrt((doublereal) nr);
|
|
/* [TP] COND_OK is a tuning parameter. */
|
|
if (condr1 < cond_ok__) {
|
|
/* implementation, this QRF should be implemented as the QRF */
|
|
/* of a lower triangular matrix. */
|
|
/* R1^t = Q2 * R2 */
|
|
i__1 = *lwork - (*n << 1);
|
|
dgeqrf_(n, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*
|
|
n << 1) + 1], &i__1, &ierr);
|
|
|
|
if (l2pert) {
|
|
xsc = sqrt(small) / epsln;
|
|
i__1 = nr;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
i__2 = p - 1;
|
|
for (q = 1; q <= i__2; ++q) {
|
|
/* Computing MIN */
|
|
d__3 = (d__1 = v[p + p * v_dim1], abs(d__1)),
|
|
d__4 = (d__2 = v[q + q * v_dim1], abs(
|
|
d__2));
|
|
temp1 = xsc * f2cmin(d__3,d__4);
|
|
if ((d__1 = v[q + p * v_dim1], abs(d__1)) <=
|
|
temp1) {
|
|
v[q + p * v_dim1] = d_sign(&temp1, &v[q +
|
|
p * v_dim1]);
|
|
}
|
|
/* L3958: */
|
|
}
|
|
/* L3959: */
|
|
}
|
|
}
|
|
|
|
if (nr != *n) {
|
|
dlacpy_("A", n, &nr, &v[v_offset], ldv, &work[(*n <<
|
|
1) + 1], n);
|
|
}
|
|
|
|
i__1 = nr - 1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = nr - p;
|
|
dcopy_(&i__2, &v[p + (p + 1) * v_dim1], ldv, &v[p + 1
|
|
+ p * v_dim1], &c__1);
|
|
/* L1969: */
|
|
}
|
|
|
|
condr2 = condr1;
|
|
|
|
} else {
|
|
|
|
/* Note that windowed pivoting would be equally good */
|
|
/* numerically, and more run-time efficient. So, in */
|
|
/* an optimal implementation, the next call to DGEQP3 */
|
|
/* should be replaced with eg. CALL SGEQPX (ACM TOMS #782) */
|
|
/* with properly (carefully) chosen parameters. */
|
|
|
|
/* R1^t * P2 = Q2 * R2 */
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
iwork[*n + p] = 0;
|
|
/* L3003: */
|
|
}
|
|
i__1 = *lwork - (*n << 1);
|
|
dgeqp3_(n, &nr, &v[v_offset], ldv, &iwork[*n + 1], &work[*
|
|
n + 1], &work[(*n << 1) + 1], &i__1, &ierr);
|
|
/* * CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1), */
|
|
/* * $ LWORK-2*N, IERR ) */
|
|
if (l2pert) {
|
|
xsc = sqrt(small);
|
|
i__1 = nr;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
i__2 = p - 1;
|
|
for (q = 1; q <= i__2; ++q) {
|
|
/* Computing MIN */
|
|
d__3 = (d__1 = v[p + p * v_dim1], abs(d__1)),
|
|
d__4 = (d__2 = v[q + q * v_dim1], abs(
|
|
d__2));
|
|
temp1 = xsc * f2cmin(d__3,d__4);
|
|
if ((d__1 = v[q + p * v_dim1], abs(d__1)) <=
|
|
temp1) {
|
|
v[q + p * v_dim1] = d_sign(&temp1, &v[q +
|
|
p * v_dim1]);
|
|
}
|
|
/* L3968: */
|
|
}
|
|
/* L3969: */
|
|
}
|
|
}
|
|
|
|
dlacpy_("A", n, &nr, &v[v_offset], ldv, &work[(*n << 1) +
|
|
1], n);
|
|
|
|
if (l2pert) {
|
|
xsc = sqrt(small);
|
|
i__1 = nr;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
i__2 = p - 1;
|
|
for (q = 1; q <= i__2; ++q) {
|
|
/* Computing MIN */
|
|
d__3 = (d__1 = v[p + p * v_dim1], abs(d__1)),
|
|
d__4 = (d__2 = v[q + q * v_dim1], abs(
|
|
d__2));
|
|
temp1 = xsc * f2cmin(d__3,d__4);
|
|
v[p + q * v_dim1] = -d_sign(&temp1, &v[q + p *
|
|
v_dim1]);
|
|
/* L8971: */
|
|
}
|
|
/* L8970: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("L", &i__1, &i__2, &c_b34, &c_b34, &v[v_dim1
|
|
+ 2], ldv);
|
|
}
|
|
/* Now, compute R2 = L3 * Q3, the LQ factorization. */
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dgelqf_(&nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + *n
|
|
* nr + 1], &work[(*n << 1) + *n * nr + nr + 1], &
|
|
i__1, &ierr);
|
|
dlacpy_("L", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1)
|
|
+ *n * nr + nr + 1], &nr);
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
temp1 = dnrm2_(&p, &work[(*n << 1) + *n * nr + nr + p]
|
|
, &nr);
|
|
d__1 = 1. / temp1;
|
|
dscal_(&p, &d__1, &work[(*n << 1) + *n * nr + nr + p],
|
|
&nr);
|
|
/* L4950: */
|
|
}
|
|
dpocon_("L", &nr, &work[(*n << 1) + *n * nr + nr + 1], &
|
|
nr, &c_b35, &temp1, &work[(*n << 1) + *n * nr +
|
|
nr + nr * nr + 1], &iwork[*m + (*n << 1) + 1], &
|
|
ierr);
|
|
condr2 = 1. / sqrt(temp1);
|
|
|
|
if (condr2 >= cond_ok__) {
|
|
/* (this overwrites the copy of R2, as it will not be */
|
|
/* needed in this branch, but it does not overwritte the */
|
|
/* Huseholder vectors of Q2.). */
|
|
dlacpy_("U", &nr, &nr, &v[v_offset], ldv, &work[(*n <<
|
|
1) + 1], n);
|
|
/* WORK(2*N+N*NR+1:2*N+N*NR+N) */
|
|
}
|
|
|
|
}
|
|
|
|
if (l2pert) {
|
|
xsc = sqrt(small);
|
|
i__1 = nr;
|
|
for (q = 2; q <= i__1; ++q) {
|
|
temp1 = xsc * v[q + q * v_dim1];
|
|
i__2 = q - 1;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
/* V(p,q) = - DSIGN( TEMP1, V(q,p) ) */
|
|
v[p + q * v_dim1] = -d_sign(&temp1, &v[p + q *
|
|
v_dim1]);
|
|
/* L4969: */
|
|
}
|
|
/* L4968: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 <<
|
|
1) + 1], ldv);
|
|
}
|
|
|
|
/* Second preconditioning finished; continue with Jacobi SVD */
|
|
/* The input matrix is lower trinagular. */
|
|
|
|
/* Recover the right singular vectors as solution of a well */
|
|
/* conditioned triangular matrix equation. */
|
|
|
|
if (condr1 < cond_ok__) {
|
|
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dgesvj_("L", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[
|
|
1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n *
|
|
nr + nr + 1], &i__1, info);
|
|
scalem = work[(*n << 1) + *n * nr + nr + 1];
|
|
numrank = i_dnnt(&work[(*n << 1) + *n * nr + nr + 2]);
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
dcopy_(&nr, &v[p * v_dim1 + 1], &c__1, &u[p * u_dim1
|
|
+ 1], &c__1);
|
|
dscal_(&nr, &sva[p], &v[p * v_dim1 + 1], &c__1);
|
|
/* L3970: */
|
|
}
|
|
|
|
if (nr == *n) {
|
|
/* :)) .. best case, R1 is inverted. The solution of this matrix */
|
|
/* equation is Q2*V2 = the product of the Jacobi rotations */
|
|
/* used in DGESVJ, premultiplied with the orthogonal matrix */
|
|
/* from the second QR factorization. */
|
|
dtrsm_("L", "U", "N", "N", &nr, &nr, &c_b35, &a[
|
|
a_offset], lda, &v[v_offset], ldv);
|
|
} else {
|
|
/* is inverted to get the product of the Jacobi rotations */
|
|
/* used in DGESVJ. The Q-factor from the second QR */
|
|
/* factorization is then built in explicitly. */
|
|
dtrsm_("L", "U", "T", "N", &nr, &nr, &c_b35, &work[(*
|
|
n << 1) + 1], n, &v[v_offset], ldv);
|
|
if (nr < *n) {
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr +
|
|
1 + v_dim1], ldv);
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr +
|
|
1) * v_dim1 + 1], ldv);
|
|
i__1 = *n - nr;
|
|
i__2 = *n - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr
|
|
+ 1 + (nr + 1) * v_dim1], ldv);
|
|
}
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n,
|
|
&work[*n + 1], &v[v_offset], ldv, &work[(*n <<
|
|
1) + *n * nr + nr + 1], &i__1, &ierr);
|
|
}
|
|
|
|
} else if (condr2 < cond_ok__) {
|
|
|
|
/* :) .. the input matrix A is very likely a relative of */
|
|
/* the Kahan matrix :) */
|
|
/* The matrix R2 is inverted. The solution of the matrix equation */
|
|
/* is Q3^T*V3 = the product of the Jacobi rotations (appplied to */
|
|
/* the lower triangular L3 from the LQ factorization of */
|
|
/* R2=L3*Q3), pre-multiplied with the transposed Q3. */
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dgesvj_("L", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[
|
|
1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n *
|
|
nr + nr + 1], &i__1, info);
|
|
scalem = work[(*n << 1) + *n * nr + nr + 1];
|
|
numrank = i_dnnt(&work[(*n << 1) + *n * nr + nr + 2]);
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
dcopy_(&nr, &v[p * v_dim1 + 1], &c__1, &u[p * u_dim1
|
|
+ 1], &c__1);
|
|
dscal_(&nr, &sva[p], &u[p * u_dim1 + 1], &c__1);
|
|
/* L3870: */
|
|
}
|
|
dtrsm_("L", "U", "N", "N", &nr, &nr, &c_b35, &work[(*n <<
|
|
1) + 1], n, &u[u_offset], ldu);
|
|
i__1 = nr;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
i__2 = nr;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
work[(*n << 1) + *n * nr + nr + iwork[*n + p]] =
|
|
u[p + q * u_dim1];
|
|
/* L872: */
|
|
}
|
|
i__2 = nr;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
u[p + q * u_dim1] = work[(*n << 1) + *n * nr + nr
|
|
+ p];
|
|
/* L874: */
|
|
}
|
|
/* L873: */
|
|
}
|
|
if (nr < *n) {
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 +
|
|
v_dim1], ldv);
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) *
|
|
v_dim1 + 1], ldv);
|
|
i__1 = *n - nr;
|
|
i__2 = *n - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1
|
|
+ (nr + 1) * v_dim1], ldv);
|
|
}
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &
|
|
work[*n + 1], &v[v_offset], ldv, &work[(*n << 1)
|
|
+ *n * nr + nr + 1], &i__1, &ierr);
|
|
} else {
|
|
/* Last line of defense. */
|
|
/* #:( This is a rather pathological case: no scaled condition */
|
|
/* improvement after two pivoted QR factorizations. Other */
|
|
/* possibility is that the rank revealing QR factorization */
|
|
/* or the condition estimator has failed, or the COND_OK */
|
|
/* is set very close to ONE (which is unnecessary). Normally, */
|
|
/* this branch should never be executed, but in rare cases of */
|
|
/* failure of the RRQR or condition estimator, the last line of */
|
|
/* defense ensures that DGEJSV completes the task. */
|
|
/* Compute the full SVD of L3 using DGESVJ with explicit */
|
|
/* accumulation of Jacobi rotations. */
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dgesvj_("L", "U", "V", &nr, &nr, &v[v_offset], ldv, &sva[
|
|
1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n *
|
|
nr + nr + 1], &i__1, info);
|
|
scalem = work[(*n << 1) + *n * nr + nr + 1];
|
|
numrank = i_dnnt(&work[(*n << 1) + *n * nr + nr + 2]);
|
|
if (nr < *n) {
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 +
|
|
v_dim1], ldv);
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) *
|
|
v_dim1 + 1], ldv);
|
|
i__1 = *n - nr;
|
|
i__2 = *n - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1
|
|
+ (nr + 1) * v_dim1], ldv);
|
|
}
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &
|
|
work[*n + 1], &v[v_offset], ldv, &work[(*n << 1)
|
|
+ *n * nr + nr + 1], &i__1, &ierr);
|
|
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dormlq_("L", "T", &nr, &nr, &nr, &work[(*n << 1) + 1], n,
|
|
&work[(*n << 1) + *n * nr + 1], &u[u_offset], ldu,
|
|
&work[(*n << 1) + *n * nr + nr + 1], &i__1, &
|
|
ierr);
|
|
i__1 = nr;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
i__2 = nr;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
work[(*n << 1) + *n * nr + nr + iwork[*n + p]] =
|
|
u[p + q * u_dim1];
|
|
/* L772: */
|
|
}
|
|
i__2 = nr;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
u[p + q * u_dim1] = work[(*n << 1) + *n * nr + nr
|
|
+ p];
|
|
/* L774: */
|
|
}
|
|
/* L773: */
|
|
}
|
|
|
|
}
|
|
|
|
/* Permute the rows of V using the (column) permutation from the */
|
|
/* first QRF. Also, scale the columns to make them unit in */
|
|
/* Euclidean norm. This applies to all cases. */
|
|
|
|
temp1 = sqrt((doublereal) (*n)) * epsln;
|
|
i__1 = *n;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
work[(*n << 1) + *n * nr + nr + iwork[p]] = v[p + q *
|
|
v_dim1];
|
|
/* L972: */
|
|
}
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
v[p + q * v_dim1] = work[(*n << 1) + *n * nr + nr + p]
|
|
;
|
|
/* L973: */
|
|
}
|
|
xsc = 1. / dnrm2_(n, &v[q * v_dim1 + 1], &c__1);
|
|
if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
|
|
dscal_(n, &xsc, &v[q * v_dim1 + 1], &c__1);
|
|
}
|
|
/* L1972: */
|
|
}
|
|
/* At this moment, V contains the right singular vectors of A. */
|
|
/* Next, assemble the left singular vector matrix U (M x N). */
|
|
if (nr < *m) {
|
|
i__1 = *m - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 +
|
|
u_dim1], ldu);
|
|
if (nr < n1) {
|
|
i__1 = n1 - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) *
|
|
u_dim1 + 1], ldu);
|
|
i__1 = *m - nr;
|
|
i__2 = n1 - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1
|
|
+ (nr + 1) * u_dim1], ldu);
|
|
}
|
|
}
|
|
|
|
/* The Q matrix from the first QRF is built into the left singular */
|
|
/* matrix U. This applies to all cases. */
|
|
|
|
i__1 = *lwork - *n;
|
|
dormqr_("Left", "No_Tr", m, &n1, n, &a[a_offset], lda, &work[
|
|
1], &u[u_offset], ldu, &work[*n + 1], &i__1, &ierr);
|
|
/* The columns of U are normalized. The cost is O(M*N) flops. */
|
|
temp1 = sqrt((doublereal) (*m)) * epsln;
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
xsc = 1. / dnrm2_(m, &u[p * u_dim1 + 1], &c__1);
|
|
if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
|
|
dscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1);
|
|
}
|
|
/* L1973: */
|
|
}
|
|
|
|
/* If the initial QRF is computed with row pivoting, the left */
|
|
/* singular vectors must be adjusted. */
|
|
|
|
if (rowpiv) {
|
|
i__1 = *m - 1;
|
|
dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n
|
|
<< 1) + 1], &c_n1);
|
|
}
|
|
|
|
} else {
|
|
|
|
/* the second QRF is not needed */
|
|
|
|
dlacpy_("Upper", n, n, &a[a_offset], lda, &work[*n + 1], n);
|
|
if (l2pert) {
|
|
xsc = sqrt(small);
|
|
i__1 = *n;
|
|
for (p = 2; p <= i__1; ++p) {
|
|
temp1 = xsc * work[*n + (p - 1) * *n + p];
|
|
i__2 = p - 1;
|
|
for (q = 1; q <= i__2; ++q) {
|
|
work[*n + (q - 1) * *n + p] = -d_sign(&temp1, &
|
|
work[*n + (p - 1) * *n + q]);
|
|
/* L5971: */
|
|
}
|
|
/* L5970: */
|
|
}
|
|
} else {
|
|
i__1 = *n - 1;
|
|
i__2 = *n - 1;
|
|
dlaset_("Lower", &i__1, &i__2, &c_b34, &c_b34, &work[*n +
|
|
2], n);
|
|
}
|
|
|
|
i__1 = *lwork - *n - *n * *n;
|
|
dgesvj_("Upper", "U", "N", n, n, &work[*n + 1], n, &sva[1], n,
|
|
&u[u_offset], ldu, &work[*n + *n * *n + 1], &i__1,
|
|
info);
|
|
|
|
scalem = work[*n + *n * *n + 1];
|
|
numrank = i_dnnt(&work[*n + *n * *n + 2]);
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
dcopy_(n, &work[*n + (p - 1) * *n + 1], &c__1, &u[p *
|
|
u_dim1 + 1], &c__1);
|
|
dscal_(n, &sva[p], &work[*n + (p - 1) * *n + 1], &c__1);
|
|
/* L6970: */
|
|
}
|
|
|
|
dtrsm_("Left", "Upper", "NoTrans", "No UD", n, n, &c_b35, &a[
|
|
a_offset], lda, &work[*n + 1], n);
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
dcopy_(n, &work[*n + p], n, &v[iwork[p] + v_dim1], ldv);
|
|
/* L6972: */
|
|
}
|
|
temp1 = sqrt((doublereal) (*n)) * epsln;
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
xsc = 1. / dnrm2_(n, &v[p * v_dim1 + 1], &c__1);
|
|
if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
|
|
dscal_(n, &xsc, &v[p * v_dim1 + 1], &c__1);
|
|
}
|
|
/* L6971: */
|
|
}
|
|
|
|
/* Assemble the left singular vector matrix U (M x N). */
|
|
|
|
if (*n < *m) {
|
|
i__1 = *m - *n;
|
|
dlaset_("A", &i__1, n, &c_b34, &c_b34, &u[*n + 1 + u_dim1]
|
|
, ldu);
|
|
if (*n < n1) {
|
|
i__1 = n1 - *n;
|
|
dlaset_("A", n, &i__1, &c_b34, &c_b34, &u[(*n + 1) *
|
|
u_dim1 + 1], ldu);
|
|
i__1 = *m - *n;
|
|
i__2 = n1 - *n;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[*n + 1
|
|
+ (*n + 1) * u_dim1], ldu);
|
|
}
|
|
}
|
|
i__1 = *lwork - *n;
|
|
dormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[
|
|
1], &u[u_offset], ldu, &work[*n + 1], &i__1, &ierr);
|
|
temp1 = sqrt((doublereal) (*m)) * epsln;
|
|
i__1 = n1;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
xsc = 1. / dnrm2_(m, &u[p * u_dim1 + 1], &c__1);
|
|
if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
|
|
dscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1);
|
|
}
|
|
/* L6973: */
|
|
}
|
|
|
|
if (rowpiv) {
|
|
i__1 = *m - 1;
|
|
dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n
|
|
<< 1) + 1], &c_n1);
|
|
}
|
|
|
|
}
|
|
|
|
/* end of the >> almost orthogonal case << in the full SVD */
|
|
|
|
} else {
|
|
|
|
/* This branch deploys a preconditioned Jacobi SVD with explicitly */
|
|
/* accumulated rotations. It is included as optional, mainly for */
|
|
/* experimental purposes. It does perform well, and can also be used. */
|
|
/* In this implementation, this branch will be automatically activated */
|
|
/* if the condition number sigma_max(A) / sigma_min(A) is predicted */
|
|
/* to be greater than the overflow threshold. This is because the */
|
|
/* a posteriori computation of the singular vectors assumes robust */
|
|
/* implementation of BLAS and some LAPACK procedures, capable of working */
|
|
/* in presence of extreme values. Since that is not always the case, ... */
|
|
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = *n - p + 1;
|
|
dcopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], &
|
|
c__1);
|
|
/* L7968: */
|
|
}
|
|
|
|
if (l2pert) {
|
|
xsc = sqrt(small / epsln);
|
|
i__1 = nr;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
temp1 = xsc * (d__1 = v[q + q * v_dim1], abs(d__1));
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
if (p > q && (d__1 = v[p + q * v_dim1], abs(d__1)) <=
|
|
temp1 || p < q) {
|
|
v[p + q * v_dim1] = d_sign(&temp1, &v[p + q *
|
|
v_dim1]);
|
|
}
|
|
if (p < q) {
|
|
v[p + q * v_dim1] = -v[p + q * v_dim1];
|
|
}
|
|
/* L5968: */
|
|
}
|
|
/* L5969: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) +
|
|
1], ldv);
|
|
}
|
|
i__1 = *lwork - (*n << 1);
|
|
dgeqrf_(n, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*n << 1)
|
|
+ 1], &i__1, &ierr);
|
|
dlacpy_("L", n, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1], n);
|
|
|
|
i__1 = nr;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
i__2 = nr - p + 1;
|
|
dcopy_(&i__2, &v[p + p * v_dim1], ldv, &u[p + p * u_dim1], &
|
|
c__1);
|
|
/* L7969: */
|
|
}
|
|
if (l2pert) {
|
|
xsc = sqrt(small / epsln);
|
|
i__1 = nr;
|
|
for (q = 2; q <= i__1; ++q) {
|
|
i__2 = q - 1;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
/* Computing MIN */
|
|
d__3 = (d__1 = u[p + p * u_dim1], abs(d__1)), d__4 = (
|
|
d__2 = u[q + q * u_dim1], abs(d__2));
|
|
temp1 = xsc * f2cmin(d__3,d__4);
|
|
u[p + q * u_dim1] = -d_sign(&temp1, &u[q + p * u_dim1]
|
|
);
|
|
/* L9971: */
|
|
}
|
|
/* L9970: */
|
|
}
|
|
} else {
|
|
i__1 = nr - 1;
|
|
i__2 = nr - 1;
|
|
dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) +
|
|
1], ldu);
|
|
}
|
|
i__1 = *lwork - (*n << 1) - *n * nr;
|
|
dgesvj_("G", "U", "V", &nr, &nr, &u[u_offset], ldu, &sva[1], n, &
|
|
v[v_offset], ldv, &work[(*n << 1) + *n * nr + 1], &i__1,
|
|
info);
|
|
scalem = work[(*n << 1) + *n * nr + 1];
|
|
numrank = i_dnnt(&work[(*n << 1) + *n * nr + 2]);
|
|
if (nr < *n) {
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1],
|
|
ldv);
|
|
i__1 = *n - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1
|
|
+ 1], ldv);
|
|
i__1 = *n - nr;
|
|
i__2 = *n - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr +
|
|
1) * v_dim1], ldv);
|
|
}
|
|
i__1 = *lwork - (*n << 1) - *n * nr - nr;
|
|
dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &work[*n +
|
|
1], &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1]
|
|
, &i__1, &ierr);
|
|
|
|
/* Permute the rows of V using the (column) permutation from the */
|
|
/* first QRF. Also, scale the columns to make them unit in */
|
|
/* Euclidean norm. This applies to all cases. */
|
|
|
|
temp1 = sqrt((doublereal) (*n)) * epsln;
|
|
i__1 = *n;
|
|
for (q = 1; q <= i__1; ++q) {
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
work[(*n << 1) + *n * nr + nr + iwork[p]] = v[p + q *
|
|
v_dim1];
|
|
/* L8972: */
|
|
}
|
|
i__2 = *n;
|
|
for (p = 1; p <= i__2; ++p) {
|
|
v[p + q * v_dim1] = work[(*n << 1) + *n * nr + nr + p];
|
|
/* L8973: */
|
|
}
|
|
xsc = 1. / dnrm2_(n, &v[q * v_dim1 + 1], &c__1);
|
|
if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
|
|
dscal_(n, &xsc, &v[q * v_dim1 + 1], &c__1);
|
|
}
|
|
/* L7972: */
|
|
}
|
|
|
|
/* At this moment, V contains the right singular vectors of A. */
|
|
/* Next, assemble the left singular vector matrix U (M x N). */
|
|
|
|
if (nr < *m) {
|
|
i__1 = *m - nr;
|
|
dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + u_dim1],
|
|
ldu);
|
|
if (nr < n1) {
|
|
i__1 = n1 - nr;
|
|
dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) *
|
|
u_dim1 + 1], ldu);
|
|
i__1 = *m - nr;
|
|
i__2 = n1 - nr;
|
|
dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + (
|
|
nr + 1) * u_dim1], ldu);
|
|
}
|
|
}
|
|
|
|
i__1 = *lwork - *n;
|
|
dormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[1], &
|
|
u[u_offset], ldu, &work[*n + 1], &i__1, &ierr);
|
|
|
|
if (rowpiv) {
|
|
i__1 = *m - 1;
|
|
dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1)
|
|
+ 1], &c_n1);
|
|
}
|
|
|
|
|
|
}
|
|
if (transp) {
|
|
i__1 = *n;
|
|
for (p = 1; p <= i__1; ++p) {
|
|
dswap_(n, &u[p * u_dim1 + 1], &c__1, &v[p * v_dim1 + 1], &
|
|
c__1);
|
|
/* L6974: */
|
|
}
|
|
}
|
|
|
|
}
|
|
/* end of the full SVD */
|
|
|
|
/* Undo scaling, if necessary (and possible) */
|
|
|
|
if (uscal2 <= big / sva[1] * uscal1) {
|
|
dlascl_("G", &c__0, &c__0, &uscal1, &uscal2, &nr, &c__1, &sva[1], n, &
|
|
ierr);
|
|
uscal1 = 1.;
|
|
uscal2 = 1.;
|
|
}
|
|
|
|
if (nr < *n) {
|
|
i__1 = *n;
|
|
for (p = nr + 1; p <= i__1; ++p) {
|
|
sva[p] = 0.;
|
|
/* L3004: */
|
|
}
|
|
}
|
|
|
|
work[1] = uscal2 * scalem;
|
|
work[2] = uscal1;
|
|
if (errest) {
|
|
work[3] = sconda;
|
|
}
|
|
if (lsvec && rsvec) {
|
|
work[4] = condr1;
|
|
work[5] = condr2;
|
|
}
|
|
if (l2tran) {
|
|
work[6] = entra;
|
|
work[7] = entrat;
|
|
}
|
|
|
|
iwork[1] = nr;
|
|
iwork[2] = numrank;
|
|
iwork[3] = warning;
|
|
|
|
return;
|
|
} /* dgejsv_ */
|
|
|