1473 lines
42 KiB
C
1473 lines
42 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 int logical;
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typedef short int shortlogical;
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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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#define F2C_proc_par_types 1
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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;
|
||
else break;
|
||
}
|
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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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{
|
||
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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}
|
||
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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}
|
||
} 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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||
}
|
||
} else {
|
||
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]);
|
||
}
|
||
}
|
||
pCf(z) = zdotc;
|
||
}
|
||
#endif
|
||
static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
|
||
integer n = *n_, incx = *incx_, incy = *incy_, i;
|
||
#ifdef _MSC_VER
|
||
_Dcomplex zdotc = {0.0, 0.0};
|
||
if (incx == 1 && incy == 1) {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
|
||
zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
|
||
}
|
||
} else {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
|
||
zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
|
||
}
|
||
}
|
||
pCd(z) = zdotc;
|
||
}
|
||
#else
|
||
_Complex double zdotc = 0.0;
|
||
if (incx == 1 && incy == 1) {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
|
||
}
|
||
} else {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
|
||
}
|
||
}
|
||
pCd(z) = zdotc;
|
||
}
|
||
#endif
|
||
static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
|
||
integer n = *n_, incx = *incx_, incy = *incy_, i;
|
||
#ifdef _MSC_VER
|
||
_Fcomplex zdotc = {0.0, 0.0};
|
||
if (incx == 1 && incy == 1) {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
|
||
zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
|
||
}
|
||
} else {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
|
||
zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
|
||
}
|
||
}
|
||
pCf(z) = zdotc;
|
||
}
|
||
#else
|
||
_Complex float zdotc = 0.0;
|
||
if (incx == 1 && incy == 1) {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc += Cf(&x[i]) * Cf(&y[i]);
|
||
}
|
||
} else {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
|
||
}
|
||
}
|
||
pCf(z) = zdotc;
|
||
}
|
||
#endif
|
||
static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
|
||
integer n = *n_, incx = *incx_, incy = *incy_, i;
|
||
#ifdef _MSC_VER
|
||
_Dcomplex zdotc = {0.0, 0.0};
|
||
if (incx == 1 && incy == 1) {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
|
||
zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
|
||
}
|
||
} else {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
|
||
zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
|
||
}
|
||
}
|
||
pCd(z) = zdotc;
|
||
}
|
||
#else
|
||
_Complex double zdotc = 0.0;
|
||
if (incx == 1 && incy == 1) {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc += Cd(&x[i]) * Cd(&y[i]);
|
||
}
|
||
} else {
|
||
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
||
zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
|
||
}
|
||
}
|
||
pCd(z) = zdotc;
|
||
}
|
||
#endif
|
||
/* -- translated by f2c (version 20000121).
|
||
You must link the resulting object file with the libraries:
|
||
-lf2c -lm (in that order)
|
||
*/
|
||
|
||
|
||
|
||
|
||
/* Table of constant values */
|
||
|
||
static real c_b10 = 1.f;
|
||
static doublereal c_b14 = -.125;
|
||
static integer c__1 = 1;
|
||
static real c_b19 = 0.f;
|
||
static integer c__2 = 2;
|
||
|
||
/* > \brief \b SBDSVDX */
|
||
|
||
/* =========== DOCUMENTATION =========== */
|
||
|
||
/* Online html documentation available at */
|
||
/* http://www.netlib.org/lapack/explore-html/ */
|
||
|
||
/* > \htmlonly */
|
||
/* > Download SBDSVDX + dependencies */
|
||
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sbdsvdx
|
||
.f"> */
|
||
/* > [TGZ]</a> */
|
||
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sbdsvdx
|
||
.f"> */
|
||
/* > [ZIP]</a> */
|
||
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sbdsvdx
|
||
.f"> */
|
||
/* > [TXT]</a> */
|
||
/* > \endhtmlonly */
|
||
|
||
/* Definition: */
|
||
/* =========== */
|
||
|
||
/* SUBROUTINE SBDSVDX( UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, */
|
||
/* $ NS, S, Z, LDZ, WORK, IWORK, INFO ) */
|
||
|
||
/* CHARACTER JOBZ, RANGE, UPLO */
|
||
/* INTEGER IL, INFO, IU, LDZ, N, NS */
|
||
/* REAL VL, VU */
|
||
/* INTEGER IWORK( * ) */
|
||
/* REAL D( * ), E( * ), S( * ), WORK( * ), */
|
||
/* Z( LDZ, * ) */
|
||
|
||
/* > \par Purpose: */
|
||
/* ============= */
|
||
/* > */
|
||
/* > \verbatim */
|
||
/* > */
|
||
/* > SBDSVDX computes the singular value decomposition (SVD) of a real */
|
||
/* > N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, */
|
||
/* > where S is a diagonal matrix with non-negative diagonal elements */
|
||
/* > (the singular values of B), and U and VT are orthogonal matrices */
|
||
/* > of left and right singular vectors, respectively. */
|
||
/* > */
|
||
/* > Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] */
|
||
/* > and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the */
|
||
/* > singular value decompositon of B through the eigenvalues and */
|
||
/* > eigenvectors of the N*2-by-N*2 tridiagonal matrix */
|
||
/* > */
|
||
/* > | 0 d_1 | */
|
||
/* > | d_1 0 e_1 | */
|
||
/* > TGK = | e_1 0 d_2 | */
|
||
/* > | d_2 . . | */
|
||
/* > | . . . | */
|
||
/* > */
|
||
/* > If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then */
|
||
/* > (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / */
|
||
/* > sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and */
|
||
/* > P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ]. */
|
||
/* > */
|
||
/* > Given a TGK matrix, one can either a) compute -s,-v and change signs */
|
||
/* > so that the singular values (and corresponding vectors) are already in */
|
||
/* > descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder */
|
||
/* > the values (and corresponding vectors). SBDSVDX implements a) by */
|
||
/* > calling SSTEVX (bisection plus inverse iteration, to be replaced */
|
||
/* > with a version of the Multiple Relative Robust Representation */
|
||
/* > algorithm. (See P. Willems and B. Lang, A framework for the MR^3 */
|
||
/* > algorithm: theory and implementation, SIAM J. Sci. Comput., */
|
||
/* > 35:740-766, 2013.) */
|
||
/* > \endverbatim */
|
||
|
||
/* Arguments: */
|
||
/* ========== */
|
||
|
||
/* > \param[in] UPLO */
|
||
/* > \verbatim */
|
||
/* > UPLO is CHARACTER*1 */
|
||
/* > = 'U': B is upper bidiagonal; */
|
||
/* > = 'L': B is lower bidiagonal. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] JOBZ */
|
||
/* > \verbatim */
|
||
/* > JOBZ is CHARACTER*1 */
|
||
/* > = 'N': Compute singular values only; */
|
||
/* > = 'V': Compute singular values and singular vectors. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] RANGE */
|
||
/* > \verbatim */
|
||
/* > RANGE is CHARACTER*1 */
|
||
/* > = 'A': all singular values will be found. */
|
||
/* > = 'V': all singular values in the half-open interval [VL,VU) */
|
||
/* > will be found. */
|
||
/* > = 'I': the IL-th through IU-th singular values will be found. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] N */
|
||
/* > \verbatim */
|
||
/* > N is INTEGER */
|
||
/* > The order of the bidiagonal matrix. N >= 0. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] D */
|
||
/* > \verbatim */
|
||
/* > D is REAL array, dimension (N) */
|
||
/* > The n diagonal elements of the bidiagonal matrix B. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] E */
|
||
/* > \verbatim */
|
||
/* > E is REAL array, dimension (f2cmax(1,N-1)) */
|
||
/* > The (n-1) superdiagonal elements of the bidiagonal matrix */
|
||
/* > B in elements 1 to N-1. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] VL */
|
||
/* > \verbatim */
|
||
/* > VL is REAL */
|
||
/* > If RANGE='V', the lower bound of the interval to */
|
||
/* > be searched for singular values. VU > VL. */
|
||
/* > Not referenced if RANGE = 'A' or 'I'. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] VU */
|
||
/* > \verbatim */
|
||
/* > VU is REAL */
|
||
/* > If RANGE='V', the upper bound of the interval to */
|
||
/* > be searched for singular values. VU > VL. */
|
||
/* > Not referenced if RANGE = 'A' or 'I'. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] IL */
|
||
/* > \verbatim */
|
||
/* > IL is INTEGER */
|
||
/* > If RANGE='I', the index of the */
|
||
/* > smallest singular value to be returned. */
|
||
/* > 1 <= IL <= IU <= f2cmin(M,N), if f2cmin(M,N) > 0. */
|
||
/* > Not referenced if RANGE = 'A' or 'V'. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] IU */
|
||
/* > \verbatim */
|
||
/* > IU is INTEGER */
|
||
/* > If RANGE='I', the index of the */
|
||
/* > largest singular value to be returned. */
|
||
/* > 1 <= IL <= IU <= f2cmin(M,N), if f2cmin(M,N) > 0. */
|
||
/* > Not referenced if RANGE = 'A' or 'V'. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[out] NS */
|
||
/* > \verbatim */
|
||
/* > NS is INTEGER */
|
||
/* > The total number of singular values found. 0 <= NS <= N. */
|
||
/* > If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[out] S */
|
||
/* > \verbatim */
|
||
/* > S is REAL array, dimension (N) */
|
||
/* > The first NS elements contain the selected singular values in */
|
||
/* > ascending order. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[out] Z */
|
||
/* > \verbatim */
|
||
/* > Z is REAL array, dimension (2*N,K) */
|
||
/* > If JOBZ = 'V', then if INFO = 0 the first NS columns of Z */
|
||
/* > contain the singular vectors of the matrix B corresponding to */
|
||
/* > the selected singular values, with U in rows 1 to N and V */
|
||
/* > in rows N+1 to N*2, i.e. */
|
||
/* > Z = [ U ] */
|
||
/* > [ V ] */
|
||
/* > If JOBZ = 'N', then Z is not referenced. */
|
||
/* > Note: The user must ensure that at least K = NS+1 columns are */
|
||
/* > supplied in the array Z; if RANGE = 'V', the exact value of */
|
||
/* > NS is not known in advance and an upper bound must be used. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[in] LDZ */
|
||
/* > \verbatim */
|
||
/* > LDZ is INTEGER */
|
||
/* > The leading dimension of the array Z. LDZ >= 1, and if */
|
||
/* > JOBZ = 'V', LDZ >= f2cmax(2,N*2). */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[out] WORK */
|
||
/* > \verbatim */
|
||
/* > WORK is REAL array, dimension (14*N) */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[out] IWORK */
|
||
/* > \verbatim */
|
||
/* > IWORK is INTEGER array, dimension (12*N) */
|
||
/* > If JOBZ = 'V', then if INFO = 0, the first NS elements of */
|
||
/* > IWORK are zero. If INFO > 0, then IWORK contains the indices */
|
||
/* > of the eigenvectors that failed to converge in DSTEVX. */
|
||
/* > \endverbatim */
|
||
/* > */
|
||
/* > \param[out] INFO */
|
||
/* > \verbatim */
|
||
/* > INFO is INTEGER */
|
||
/* > = 0: successful exit */
|
||
/* > < 0: if INFO = -i, the i-th argument had an illegal value */
|
||
/* > > 0: if INFO = i, then i eigenvectors failed to converge */
|
||
/* > in SSTEVX. The indices of the eigenvectors */
|
||
/* > (as returned by SSTEVX) are stored in the */
|
||
/* > array IWORK. */
|
||
/* > if INFO = N*2 + 1, an internal error occurred. */
|
||
/* > \endverbatim */
|
||
|
||
/* Authors: */
|
||
/* ======== */
|
||
|
||
/* > \author Univ. of Tennessee */
|
||
/* > \author Univ. of California Berkeley */
|
||
/* > \author Univ. of Colorado Denver */
|
||
/* > \author NAG Ltd. */
|
||
|
||
/* > \date June 2016 */
|
||
|
||
/* > \ingroup realOTHEReigen */
|
||
|
||
/* ===================================================================== */
|
||
/* Subroutine */ void sbdsvdx_(char *uplo, char *jobz, char *range, integer *n,
|
||
real *d__, real *e, real *vl, real *vu, integer *il, integer *iu,
|
||
integer *ns, real *s, real *z__, integer *ldz, real *work, integer *
|
||
iwork, integer *info)
|
||
{
|
||
/* System generated locals */
|
||
integer z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
|
||
real r__1, r__2, r__3, r__4;
|
||
doublereal d__1;
|
||
|
||
/* Local variables */
|
||
real emin;
|
||
integer ntgk;
|
||
real smin, smax;
|
||
extern real sdot_(integer *, real *, integer *, real *, integer *);
|
||
real nrmu, nrmv;
|
||
logical sveq0;
|
||
extern real snrm2_(integer *, real *, integer *);
|
||
integer i__, idbeg, j, k;
|
||
real sqrt2;
|
||
integer idend, isbeg;
|
||
extern logical lsame_(char *, char *);
|
||
integer idtgk, ietgk;
|
||
extern /* Subroutine */ void sscal_(integer *, real *, real *, integer *);
|
||
integer iltgk, itemp, icolz;
|
||
logical allsv;
|
||
integer idptr;
|
||
logical indsv;
|
||
integer ieptr, iutgk;
|
||
real vltgk;
|
||
logical lower;
|
||
real zjtji;
|
||
logical split, valsv;
|
||
integer isplt;
|
||
real ortol, vutgk;
|
||
extern /* Subroutine */ void scopy_(integer *, real *, integer *, real *,
|
||
integer *), sswap_(integer *, real *, integer *, real *, integer *
|
||
);
|
||
logical wantz;
|
||
char rngvx[1];
|
||
integer irowu, irowv;
|
||
extern /* Subroutine */ void saxpy_(integer *, real *, real *, integer *,
|
||
real *, integer *);
|
||
integer irowz, iifail;
|
||
real mu;
|
||
extern real slamch_(char *);
|
||
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
|
||
extern integer isamax_(integer *, real *, integer *);
|
||
real abstol;
|
||
extern /* Subroutine */ void slaset_(char *, integer *, integer *, real *,
|
||
real *, real *, integer *);
|
||
real thresh;
|
||
integer iiwork;
|
||
extern /* Subroutine */ void mecago_(), sstevx_(char *, char *,
|
||
integer *, real *, real *, real *, real *, integer *, integer *,
|
||
real *, integer *, real *, real *, integer *, real *, integer *,
|
||
integer *, integer *);
|
||
real eps;
|
||
integer nsl;
|
||
real tol, ulp;
|
||
integer nru, nrv;
|
||
|
||
|
||
/* -- LAPACK driver routine (version 3.8.0) -- */
|
||
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
|
||
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
|
||
/* November 2017 */
|
||
|
||
|
||
/* ===================================================================== */
|
||
|
||
|
||
/* Test the input parameters. */
|
||
|
||
/* Parameter adjustments */
|
||
--d__;
|
||
--e;
|
||
--s;
|
||
z_dim1 = *ldz;
|
||
z_offset = 1 + z_dim1 * 1;
|
||
z__ -= z_offset;
|
||
--work;
|
||
--iwork;
|
||
|
||
/* Function Body */
|
||
allsv = lsame_(range, "A");
|
||
valsv = lsame_(range, "V");
|
||
indsv = lsame_(range, "I");
|
||
wantz = lsame_(jobz, "V");
|
||
lower = lsame_(uplo, "L");
|
||
|
||
*info = 0;
|
||
if (! lsame_(uplo, "U") && ! lower) {
|
||
*info = -1;
|
||
} else if (! (wantz || lsame_(jobz, "N"))) {
|
||
*info = -2;
|
||
} else if (! (allsv || valsv || indsv)) {
|
||
*info = -3;
|
||
} else if (*n < 0) {
|
||
*info = -4;
|
||
} else if (*n > 0) {
|
||
if (valsv) {
|
||
if (*vl < 0.f) {
|
||
*info = -7;
|
||
} else if (*vu <= *vl) {
|
||
*info = -8;
|
||
}
|
||
} else if (indsv) {
|
||
if (*il < 1 || *il > f2cmax(1,*n)) {
|
||
*info = -9;
|
||
} else if (*iu < f2cmin(*n,*il) || *iu > *n) {
|
||
*info = -10;
|
||
}
|
||
}
|
||
}
|
||
if (*info == 0) {
|
||
if (*ldz < 1 || wantz && *ldz < *n << 1) {
|
||
*info = -14;
|
||
}
|
||
}
|
||
|
||
if (*info != 0) {
|
||
i__1 = -(*info);
|
||
xerbla_("SBDSVDX", &i__1, (ftnlen)7);
|
||
return;
|
||
}
|
||
|
||
/* Quick return if possible (N.LE.1) */
|
||
|
||
*ns = 0;
|
||
if (*n == 0) {
|
||
return;
|
||
}
|
||
|
||
if (*n == 1) {
|
||
if (allsv || indsv) {
|
||
*ns = 1;
|
||
s[1] = abs(d__[1]);
|
||
} else {
|
||
if (*vl < abs(d__[1]) && *vu >= abs(d__[1])) {
|
||
*ns = 1;
|
||
s[1] = abs(d__[1]);
|
||
}
|
||
}
|
||
if (wantz) {
|
||
z__[z_dim1 + 1] = r_sign(&c_b10, &d__[1]);
|
||
z__[z_dim1 + 2] = 1.f;
|
||
}
|
||
return;
|
||
}
|
||
|
||
abstol = slamch_("Safe Minimum") * 2;
|
||
ulp = slamch_("Precision");
|
||
eps = slamch_("Epsilon");
|
||
sqrt2 = sqrt(2.f);
|
||
ortol = sqrt(ulp);
|
||
|
||
/* Criterion for splitting is taken from SBDSQR when singular */
|
||
/* values are computed to relative accuracy TOL. (See J. Demmel and */
|
||
/* W. Kahan, Accurate singular values of bidiagonal matrices, SIAM */
|
||
/* J. Sci. and Stat. Comput., 11:873–912, 1990.) */
|
||
|
||
/* Computing MAX */
|
||
/* Computing MIN */
|
||
d__1 = (doublereal) eps;
|
||
r__3 = 100.f, r__4 = pow_dd(&d__1, &c_b14);
|
||
r__1 = 10.f, r__2 = f2cmin(r__3,r__4);
|
||
tol = f2cmax(r__1,r__2) * eps;
|
||
|
||
/* Compute approximate maximum, minimum singular values. */
|
||
|
||
i__ = isamax_(n, &d__[1], &c__1);
|
||
smax = (r__1 = d__[i__], abs(r__1));
|
||
i__1 = *n - 1;
|
||
i__ = isamax_(&i__1, &e[1], &c__1);
|
||
/* Computing MAX */
|
||
r__2 = smax, r__3 = (r__1 = e[i__], abs(r__1));
|
||
smax = f2cmax(r__2,r__3);
|
||
|
||
/* Compute threshold for neglecting D's and E's. */
|
||
|
||
smin = abs(d__[1]);
|
||
if (smin != 0.f) {
|
||
mu = smin;
|
||
i__1 = *n;
|
||
for (i__ = 2; i__ <= i__1; ++i__) {
|
||
mu = (r__2 = d__[i__], abs(r__2)) * (mu / (mu + (r__1 = e[i__ - 1]
|
||
, abs(r__1))));
|
||
smin = f2cmin(smin,mu);
|
||
if (smin == 0.f) {
|
||
myexit_();
|
||
}
|
||
}
|
||
}
|
||
smin /= sqrt((real) (*n));
|
||
thresh = tol * smin;
|
||
|
||
/* Check for zeros in D and E (splits), i.e. submatrices. */
|
||
|
||
i__1 = *n - 1;
|
||
for (i__ = 1; i__ <= i__1; ++i__) {
|
||
if ((r__1 = d__[i__], abs(r__1)) <= thresh) {
|
||
d__[i__] = 0.f;
|
||
}
|
||
if ((r__1 = e[i__], abs(r__1)) <= thresh) {
|
||
e[i__] = 0.f;
|
||
}
|
||
}
|
||
if ((r__1 = d__[*n], abs(r__1)) <= thresh) {
|
||
d__[*n] = 0.f;
|
||
}
|
||
|
||
/* Pointers for arrays used by SSTEVX. */
|
||
|
||
idtgk = 1;
|
||
ietgk = idtgk + (*n << 1);
|
||
itemp = ietgk + (*n << 1);
|
||
iifail = 1;
|
||
iiwork = iifail + (*n << 1);
|
||
|
||
/* Set RNGVX, which corresponds to RANGE for SSTEVX in TGK mode. */
|
||
/* VL,VU or IL,IU are redefined to conform to implementation a) */
|
||
/* described in the leading comments. */
|
||
|
||
iltgk = 0;
|
||
iutgk = 0;
|
||
vltgk = 0.f;
|
||
vutgk = 0.f;
|
||
|
||
if (allsv) {
|
||
|
||
/* All singular values will be found. We aim at -s (see */
|
||
/* leading comments) with RNGVX = 'I'. IL and IU are set */
|
||
/* later (as ILTGK and IUTGK) according to the dimension */
|
||
/* of the active submatrix. */
|
||
|
||
*(unsigned char *)rngvx = 'I';
|
||
if (wantz) {
|
||
i__1 = *n << 1;
|
||
i__2 = *n + 1;
|
||
slaset_("F", &i__1, &i__2, &c_b19, &c_b19, &z__[z_offset], ldz);
|
||
}
|
||
} else if (valsv) {
|
||
|
||
/* Find singular values in a half-open interval. We aim */
|
||
/* at -s (see leading comments) and we swap VL and VU */
|
||
/* (as VUTGK and VLTGK), changing their signs. */
|
||
|
||
*(unsigned char *)rngvx = 'V';
|
||
vltgk = -(*vu);
|
||
vutgk = -(*vl);
|
||
i__1 = idtgk + (*n << 1) - 1;
|
||
for (i__ = idtgk; i__ <= i__1; ++i__) {
|
||
work[i__] = 0.f;
|
||
}
|
||
/* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */
|
||
scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2);
|
||
i__1 = *n - 1;
|
||
scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2);
|
||
i__1 = *n << 1;
|
||
sstevx_("N", "V", &i__1, &work[idtgk], &work[ietgk], &vltgk, &vutgk, &
|
||
iltgk, &iltgk, &abstol, ns, &s[1], &z__[z_offset], ldz, &work[
|
||
itemp], &iwork[iiwork], &iwork[iifail], info);
|
||
if (*ns == 0) {
|
||
return;
|
||
} else {
|
||
if (wantz) {
|
||
i__1 = *n << 1;
|
||
slaset_("F", &i__1, ns, &c_b19, &c_b19, &z__[z_offset], ldz);
|
||
}
|
||
}
|
||
} else if (indsv) {
|
||
|
||
/* Find the IL-th through the IU-th singular values. We aim */
|
||
/* at -s (see leading comments) and indices are mapped into */
|
||
/* values, therefore mimicking SSTEBZ, where */
|
||
|
||
/* GL = GL - FUDGE*TNORM*ULP*N - FUDGE*TWO*PIVMIN */
|
||
/* GU = GU + FUDGE*TNORM*ULP*N + FUDGE*PIVMIN */
|
||
|
||
iltgk = *il;
|
||
iutgk = *iu;
|
||
*(unsigned char *)rngvx = 'V';
|
||
i__1 = idtgk + (*n << 1) - 1;
|
||
for (i__ = idtgk; i__ <= i__1; ++i__) {
|
||
work[i__] = 0.f;
|
||
}
|
||
/* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */
|
||
scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2);
|
||
i__1 = *n - 1;
|
||
scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2);
|
||
i__1 = *n << 1;
|
||
sstevx_("N", "I", &i__1, &work[idtgk], &work[ietgk], &vltgk, &vltgk, &
|
||
iltgk, &iltgk, &abstol, ns, &s[1], &z__[z_offset], ldz, &work[
|
||
itemp], &iwork[iiwork], &iwork[iifail], info);
|
||
vltgk = s[1] - smax * 2.f * ulp * *n;
|
||
i__1 = idtgk + (*n << 1) - 1;
|
||
for (i__ = idtgk; i__ <= i__1; ++i__) {
|
||
work[i__] = 0.f;
|
||
}
|
||
/* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */
|
||
scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2);
|
||
i__1 = *n - 1;
|
||
scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2);
|
||
i__1 = *n << 1;
|
||
sstevx_("N", "I", &i__1, &work[idtgk], &work[ietgk], &vutgk, &vutgk, &
|
||
iutgk, &iutgk, &abstol, ns, &s[1], &z__[z_offset], ldz, &work[
|
||
itemp], &iwork[iiwork], &iwork[iifail], info);
|
||
vutgk = s[1] + smax * 2.f * ulp * *n;
|
||
vutgk = f2cmin(vutgk,0.f);
|
||
|
||
/* If VLTGK=VUTGK, SSTEVX returns an error message, */
|
||
/* so if needed we change VUTGK slightly. */
|
||
|
||
if (vltgk == vutgk) {
|
||
vltgk -= tol;
|
||
}
|
||
|
||
if (wantz) {
|
||
i__1 = *n << 1;
|
||
i__2 = *iu - *il + 1;
|
||
slaset_("F", &i__1, &i__2, &c_b19, &c_b19, &z__[z_offset], ldz);
|
||
}
|
||
}
|
||
|
||
/* Initialize variables and pointers for S, Z, and WORK. */
|
||
|
||
/* NRU, NRV: number of rows in U and V for the active submatrix */
|
||
/* IDBEG, ISBEG: offsets for the entries of D and S */
|
||
/* IROWZ, ICOLZ: offsets for the rows and columns of Z */
|
||
/* IROWU, IROWV: offsets for the rows of U and V */
|
||
|
||
*ns = 0;
|
||
nru = 0;
|
||
nrv = 0;
|
||
idbeg = 1;
|
||
isbeg = 1;
|
||
irowz = 1;
|
||
icolz = 1;
|
||
irowu = 2;
|
||
irowv = 1;
|
||
split = FALSE_;
|
||
sveq0 = FALSE_;
|
||
|
||
/* Form the tridiagonal TGK matrix. */
|
||
|
||
i__1 = *n;
|
||
for (i__ = 1; i__ <= i__1; ++i__) {
|
||
s[i__] = 0.f;
|
||
}
|
||
/* S( 1:N ) = ZERO */
|
||
work[ietgk + (*n << 1) - 1] = 0.f;
|
||
i__1 = idtgk + (*n << 1) - 1;
|
||
for (i__ = idtgk; i__ <= i__1; ++i__) {
|
||
work[i__] = 0.f;
|
||
}
|
||
/* WORK( IDTGK:IDTGK+2*N-1 ) = ZERO */
|
||
scopy_(n, &d__[1], &c__1, &work[ietgk], &c__2);
|
||
i__1 = *n - 1;
|
||
scopy_(&i__1, &e[1], &c__1, &work[ietgk + 1], &c__2);
|
||
|
||
|
||
/* Check for splits in two levels, outer level */
|
||
/* in E and inner level in D. */
|
||
|
||
i__1 = *n << 1;
|
||
for (ieptr = 2; ieptr <= i__1; ieptr += 2) {
|
||
if (work[ietgk + ieptr - 1] == 0.f) {
|
||
|
||
/* Split in E (this piece of B is square) or bottom */
|
||
/* of the (input bidiagonal) matrix. */
|
||
|
||
isplt = idbeg;
|
||
idend = ieptr - 1;
|
||
i__2 = idend;
|
||
for (idptr = idbeg; idptr <= i__2; idptr += 2) {
|
||
if (work[ietgk + idptr - 1] == 0.f) {
|
||
|
||
/* Split in D (rectangular submatrix). Set the number */
|
||
/* of rows in U and V (NRU and NRV) accordingly. */
|
||
|
||
if (idptr == idbeg) {
|
||
|
||
/* D=0 at the top. */
|
||
|
||
sveq0 = TRUE_;
|
||
if (idbeg == idend) {
|
||
nru = 1;
|
||
nrv = 1;
|
||
}
|
||
} else if (idptr == idend) {
|
||
|
||
/* D=0 at the bottom. */
|
||
|
||
sveq0 = TRUE_;
|
||
nru = (idend - isplt) / 2 + 1;
|
||
nrv = nru;
|
||
if (isplt != idbeg) {
|
||
++nru;
|
||
}
|
||
} else {
|
||
if (isplt == idbeg) {
|
||
|
||
/* Split: top rectangular submatrix. */
|
||
|
||
nru = (idptr - idbeg) / 2;
|
||
nrv = nru + 1;
|
||
} else {
|
||
|
||
/* Split: middle square submatrix. */
|
||
|
||
nru = (idptr - isplt) / 2 + 1;
|
||
nrv = nru;
|
||
}
|
||
}
|
||
} else if (idptr == idend) {
|
||
|
||
/* Last entry of D in the active submatrix. */
|
||
|
||
if (isplt == idbeg) {
|
||
|
||
/* No split (trivial case). */
|
||
|
||
nru = (idend - idbeg) / 2 + 1;
|
||
nrv = nru;
|
||
} else {
|
||
|
||
/* Split: bottom rectangular submatrix. */
|
||
|
||
nrv = (idend - isplt) / 2 + 1;
|
||
nru = nrv + 1;
|
||
}
|
||
}
|
||
|
||
ntgk = nru + nrv;
|
||
|
||
if (ntgk > 0) {
|
||
|
||
/* Compute eigenvalues/vectors of the active */
|
||
/* submatrix according to RANGE: */
|
||
/* if RANGE='A' (ALLSV) then RNGVX = 'I' */
|
||
/* if RANGE='V' (VALSV) then RNGVX = 'V' */
|
||
/* if RANGE='I' (INDSV) then RNGVX = 'V' */
|
||
|
||
iltgk = 1;
|
||
iutgk = ntgk / 2;
|
||
if (allsv || vutgk == 0.f) {
|
||
if (sveq0 || smin < eps || ntgk % 2 > 0) {
|
||
/* Special case: eigenvalue equal to zero or very */
|
||
/* small, additional eigenvector is needed. */
|
||
++iutgk;
|
||
}
|
||
}
|
||
|
||
/* Workspace needed by SSTEVX: */
|
||
/* WORK( ITEMP: ): 2*5*NTGK */
|
||
/* IWORK( 1: ): 2*6*NTGK */
|
||
|
||
sstevx_(jobz, rngvx, &ntgk, &work[idtgk + isplt - 1], &
|
||
work[ietgk + isplt - 1], &vltgk, &vutgk, &iltgk, &
|
||
iutgk, &abstol, &nsl, &s[isbeg], &z__[irowz +
|
||
icolz * z_dim1], ldz, &work[itemp], &iwork[iiwork]
|
||
, &iwork[iifail], info);
|
||
if (*info != 0) {
|
||
/* Exit with the error code from SSTEVX. */
|
||
return;
|
||
}
|
||
emin = (r__1 = s[isbeg], abs(r__1));
|
||
i__3 = isbeg + nsl - 1;
|
||
for (i__ = isbeg; i__ <= i__3; ++i__) {
|
||
if ((r__1 = s[i__], abs(r__1)) > emin) {
|
||
emin = s[i__];
|
||
}
|
||
}
|
||
/* EMIN = ABS( MAXVAL( S( ISBEG:ISBEG+NSL-1 ) ) ) */
|
||
|
||
if (nsl > 0 && wantz) {
|
||
|
||
/* Normalize u=Z([2,4,...],:) and v=Z([1,3,...],:), */
|
||
/* changing the sign of v as discussed in the leading */
|
||
/* comments. The norms of u and v may be (slightly) */
|
||
/* different from 1/sqrt(2) if the corresponding */
|
||
/* eigenvalues are very small or too close. We check */
|
||
/* those norms and, if needed, reorthogonalize the */
|
||
/* vectors. */
|
||
|
||
if (nsl > 1 && vutgk == 0.f && ntgk % 2 == 0 && emin
|
||
== 0.f && ! split) {
|
||
|
||
/* D=0 at the top or bottom of the active submatrix: */
|
||
/* one eigenvalue is equal to zero; concatenate the */
|
||
/* eigenvectors corresponding to the two smallest */
|
||
/* eigenvalues. */
|
||
|
||
i__3 = irowz + ntgk - 1;
|
||
for (i__ = irowz; i__ <= i__3; ++i__) {
|
||
z__[i__ + (icolz + nsl - 2) * z_dim1] += z__[
|
||
i__ + (icolz + nsl - 1) * z_dim1];
|
||
z__[i__ + (icolz + nsl - 1) * z_dim1] = 0.f;
|
||
}
|
||
/* Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-2 ) = */
|
||
/* $ Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-2 ) + */
|
||
/* $ Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-1 ) */
|
||
/* Z( IROWZ:IROWZ+NTGK-1,ICOLZ+NSL-1 ) = */
|
||
/* $ ZERO */
|
||
/* IF( IUTGK*2.GT.NTGK ) THEN */
|
||
/* Eigenvalue equal to zero or very small. */
|
||
/* NSL = NSL - 1 */
|
||
/* END IF */
|
||
}
|
||
|
||
/* Computing MIN */
|
||
i__4 = nsl - 1, i__5 = nru - 1;
|
||
i__3 = f2cmin(i__4,i__5);
|
||
for (i__ = 0; i__ <= i__3; ++i__) {
|
||
nrmu = snrm2_(&nru, &z__[irowu + (icolz + i__) *
|
||
z_dim1], &c__2);
|
||
if (nrmu == 0.f) {
|
||
*info = (*n << 1) + 1;
|
||
return;
|
||
}
|
||
r__1 = 1.f / nrmu;
|
||
sscal_(&nru, &r__1, &z__[irowu + (icolz + i__) *
|
||
z_dim1], &c__2);
|
||
if (nrmu != 1.f && (r__1 = nrmu - ortol, abs(r__1)
|
||
) * sqrt2 > 1.f) {
|
||
i__4 = i__ - 1;
|
||
for (j = 0; j <= i__4; ++j) {
|
||
zjtji = -sdot_(&nru, &z__[irowu + (icolz
|
||
+ j) * z_dim1], &c__2, &z__[irowu
|
||
+ (icolz + i__) * z_dim1], &c__2);
|
||
saxpy_(&nru, &zjtji, &z__[irowu + (icolz
|
||
+ j) * z_dim1], &c__2, &z__[irowu
|
||
+ (icolz + i__) * z_dim1], &c__2);
|
||
}
|
||
nrmu = snrm2_(&nru, &z__[irowu + (icolz + i__)
|
||
* z_dim1], &c__2);
|
||
r__1 = 1.f / nrmu;
|
||
sscal_(&nru, &r__1, &z__[irowu + (icolz + i__)
|
||
* z_dim1], &c__2);
|
||
}
|
||
}
|
||
/* Computing MIN */
|
||
i__4 = nsl - 1, i__5 = nrv - 1;
|
||
i__3 = f2cmin(i__4,i__5);
|
||
for (i__ = 0; i__ <= i__3; ++i__) {
|
||
nrmv = snrm2_(&nrv, &z__[irowv + (icolz + i__) *
|
||
z_dim1], &c__2);
|
||
if (nrmv == 0.f) {
|
||
*info = (*n << 1) + 1;
|
||
return;
|
||
}
|
||
r__1 = -1.f / nrmv;
|
||
sscal_(&nrv, &r__1, &z__[irowv + (icolz + i__) *
|
||
z_dim1], &c__2);
|
||
if (nrmv != 1.f && (r__1 = nrmv - ortol, abs(r__1)
|
||
) * sqrt2 > 1.f) {
|
||
i__4 = i__ - 1;
|
||
for (j = 0; j <= i__4; ++j) {
|
||
zjtji = -sdot_(&nrv, &z__[irowv + (icolz
|
||
+ j) * z_dim1], &c__2, &z__[irowv
|
||
+ (icolz + i__) * z_dim1], &c__2);
|
||
saxpy_(&nru, &zjtji, &z__[irowv + (icolz
|
||
+ j) * z_dim1], &c__2, &z__[irowv
|
||
+ (icolz + i__) * z_dim1], &c__2);
|
||
}
|
||
nrmv = snrm2_(&nrv, &z__[irowv + (icolz + i__)
|
||
* z_dim1], &c__2);
|
||
r__1 = 1.f / nrmv;
|
||
sscal_(&nrv, &r__1, &z__[irowv + (icolz + i__)
|
||
* z_dim1], &c__2);
|
||
}
|
||
}
|
||
if (vutgk == 0.f && idptr < idend && ntgk % 2 > 0) {
|
||
|
||
/* D=0 in the middle of the active submatrix (one */
|
||
/* eigenvalue is equal to zero): save the corresponding */
|
||
/* eigenvector for later use (when bottom of the */
|
||
/* active submatrix is reached). */
|
||
|
||
split = TRUE_;
|
||
i__3 = irowz + ntgk - 1;
|
||
for (i__ = irowz; i__ <= i__3; ++i__) {
|
||
z__[i__ + (*n + 1) * z_dim1] = z__[i__ + (*ns
|
||
+ nsl) * z_dim1];
|
||
z__[i__ + (*ns + nsl) * z_dim1] = 0.f;
|
||
}
|
||
/* Z( IROWZ:IROWZ+NTGK-1,N+1 ) = */
|
||
/* $ Z( IROWZ:IROWZ+NTGK-1,NS+NSL ) */
|
||
/* Z( IROWZ:IROWZ+NTGK-1,NS+NSL ) = */
|
||
/* $ ZERO */
|
||
}
|
||
}
|
||
|
||
/* ** WANTZ **! */
|
||
nsl = f2cmin(nsl,nru);
|
||
sveq0 = FALSE_;
|
||
|
||
/* Absolute values of the eigenvalues of TGK. */
|
||
|
||
i__3 = nsl - 1;
|
||
for (i__ = 0; i__ <= i__3; ++i__) {
|
||
s[isbeg + i__] = (r__1 = s[isbeg + i__], abs(r__1));
|
||
}
|
||
|
||
/* Update pointers for TGK, S and Z. */
|
||
|
||
isbeg += nsl;
|
||
irowz += ntgk;
|
||
icolz += nsl;
|
||
irowu = irowz;
|
||
irowv = irowz + 1;
|
||
isplt = idptr + 1;
|
||
*ns += nsl;
|
||
nru = 0;
|
||
nrv = 0;
|
||
}
|
||
/* ** NTGK.GT.0 **! */
|
||
if (irowz < *n << 1 && wantz) {
|
||
i__3 = irowz - 1;
|
||
for (i__ = 1; i__ <= i__3; ++i__) {
|
||
z__[i__ + icolz * z_dim1] = 0.f;
|
||
}
|
||
/* Z( 1:IROWZ-1, ICOLZ ) = ZERO */
|
||
}
|
||
}
|
||
/* ** IDPTR loop **! */
|
||
if (split && wantz) {
|
||
|
||
/* Bring back eigenvector corresponding */
|
||
/* to eigenvalue equal to zero. */
|
||
|
||
i__2 = idend - ntgk + 1;
|
||
for (i__ = idbeg; i__ <= i__2; ++i__) {
|
||
z__[i__ + (isbeg - 1) * z_dim1] += z__[i__ + (*n + 1) *
|
||
z_dim1];
|
||
z__[i__ + (*n + 1) * z_dim1] = 0.f;
|
||
}
|
||
/* Z( IDBEG:IDEND-NTGK+1,ISBEG-1 ) = */
|
||
/* $ Z( IDBEG:IDEND-NTGK+1,ISBEG-1 ) + */
|
||
/* $ Z( IDBEG:IDEND-NTGK+1,N+1 ) */
|
||
/* Z( IDBEG:IDEND-NTGK+1,N+1 ) = 0 */
|
||
}
|
||
--irowv;
|
||
++irowu;
|
||
idbeg = ieptr + 1;
|
||
sveq0 = FALSE_;
|
||
split = FALSE_;
|
||
}
|
||
/* ** Check for split in E **! */
|
||
}
|
||
|
||
/* Sort the singular values into decreasing order (insertion sort on */
|
||
/* singular values, but only one transposition per singular vector) */
|
||
|
||
/* ** IEPTR loop **! */
|
||
i__1 = *ns - 1;
|
||
for (i__ = 1; i__ <= i__1; ++i__) {
|
||
k = 1;
|
||
smin = s[1];
|
||
i__2 = *ns + 1 - i__;
|
||
for (j = 2; j <= i__2; ++j) {
|
||
if (s[j] <= smin) {
|
||
k = j;
|
||
smin = s[j];
|
||
}
|
||
}
|
||
if (k != *ns + 1 - i__) {
|
||
s[k] = s[*ns + 1 - i__];
|
||
s[*ns + 1 - i__] = smin;
|
||
if (wantz) {
|
||
i__2 = *n << 1;
|
||
sswap_(&i__2, &z__[k * z_dim1 + 1], &c__1, &z__[(*ns + 1 -
|
||
i__) * z_dim1 + 1], &c__1);
|
||
}
|
||
}
|
||
}
|
||
|
||
/* If RANGE=I, check for singular values/vectors to be discarded. */
|
||
|
||
if (indsv) {
|
||
k = *iu - *il + 1;
|
||
if (k < *ns) {
|
||
i__1 = *ns;
|
||
for (i__ = k + 1; i__ <= i__1; ++i__) {
|
||
s[i__] = 0.f;
|
||
}
|
||
/* S( K+1:NS ) = ZERO */
|
||
if (wantz) {
|
||
i__1 = *n << 1;
|
||
for (i__ = 1; i__ <= i__1; ++i__) {
|
||
i__2 = *ns;
|
||
for (j = k + 1; j <= i__2; ++j) {
|
||
z__[i__ + j * z_dim1] = 0.f;
|
||
}
|
||
}
|
||
/* Z( 1:N*2,K+1:NS ) = ZERO */
|
||
}
|
||
*ns = k;
|
||
}
|
||
}
|
||
|
||
/* Reorder Z: U = Z( 1:N,1:NS ), V = Z( N+1:N*2,1:NS ). */
|
||
/* If B is a lower diagonal, swap U and V. */
|
||
|
||
if (wantz) {
|
||
i__1 = *ns;
|
||
for (i__ = 1; i__ <= i__1; ++i__) {
|
||
i__2 = *n << 1;
|
||
scopy_(&i__2, &z__[i__ * z_dim1 + 1], &c__1, &work[1], &c__1);
|
||
if (lower) {
|
||
scopy_(n, &work[2], &c__2, &z__[*n + 1 + i__ * z_dim1], &c__1)
|
||
;
|
||
scopy_(n, &work[1], &c__2, &z__[i__ * z_dim1 + 1], &c__1);
|
||
} else {
|
||
scopy_(n, &work[2], &c__2, &z__[i__ * z_dim1 + 1], &c__1);
|
||
scopy_(n, &work[1], &c__2, &z__[*n + 1 + i__ * z_dim1], &c__1)
|
||
;
|
||
}
|
||
}
|
||
}
|
||
|
||
return;
|
||
|
||
/* End of SBDSVDX */
|
||
|
||
} /* sbdsvdx_ */
|
||
|