1521 lines
41 KiB
C
1521 lines
41 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;
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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 {
|
|
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 doublereal c_b15 = -.125;
|
|
static integer c__1 = 1;
|
|
static real c_b49 = 1.f;
|
|
static real c_b72 = -1.f;
|
|
|
|
/* > \brief \b SBDSQR */
|
|
|
|
/* =========== DOCUMENTATION =========== */
|
|
|
|
/* Online html documentation available at */
|
|
/* http://www.netlib.org/lapack/explore-html/ */
|
|
|
|
/* > \htmlonly */
|
|
/* > Download SBDSQR + dependencies */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sbdsqr.
|
|
f"> */
|
|
/* > [TGZ]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sbdsqr.
|
|
f"> */
|
|
/* > [ZIP]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sbdsqr.
|
|
f"> */
|
|
/* > [TXT]</a> */
|
|
/* > \endhtmlonly */
|
|
|
|
/* Definition: */
|
|
/* =========== */
|
|
|
|
/* SUBROUTINE SBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, */
|
|
/* LDU, C, LDC, WORK, INFO ) */
|
|
|
|
/* CHARACTER UPLO */
|
|
/* INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU */
|
|
/* REAL C( LDC, * ), D( * ), E( * ), U( LDU, * ), */
|
|
/* $ VT( LDVT, * ), WORK( * ) */
|
|
|
|
|
|
/* > \par Purpose: */
|
|
/* ============= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > SBDSQR computes the singular values and, optionally, the right and/or */
|
|
/* > left singular vectors from the singular value decomposition (SVD) of */
|
|
/* > a real N-by-N (upper or lower) bidiagonal matrix B using the implicit */
|
|
/* > zero-shift QR algorithm. The SVD of B has the form */
|
|
/* > */
|
|
/* > B = Q * S * P**T */
|
|
/* > */
|
|
/* > where S is the diagonal matrix of singular values, Q is an orthogonal */
|
|
/* > matrix of left singular vectors, and P is an orthogonal matrix of */
|
|
/* > right singular vectors. If left singular vectors are requested, this */
|
|
/* > subroutine actually returns U*Q instead of Q, and, if right singular */
|
|
/* > vectors are requested, this subroutine returns P**T*VT instead of */
|
|
/* > P**T, for given real input matrices U and VT. When U and VT are the */
|
|
/* > orthogonal matrices that reduce a general matrix A to bidiagonal */
|
|
/* > form: A = U*B*VT, as computed by SGEBRD, then */
|
|
/* > */
|
|
/* > A = (U*Q) * S * (P**T*VT) */
|
|
/* > */
|
|
/* > is the SVD of A. Optionally, the subroutine may also compute Q**T*C */
|
|
/* > for a given real input matrix C. */
|
|
/* > */
|
|
/* > See "Computing Small Singular Values of Bidiagonal Matrices With */
|
|
/* > Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, */
|
|
/* > LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, */
|
|
/* > no. 5, pp. 873-912, Sept 1990) and */
|
|
/* > "Accurate singular values and differential qd algorithms," by */
|
|
/* > B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics */
|
|
/* > Department, University of California at Berkeley, July 1992 */
|
|
/* > for a detailed description of the algorithm. */
|
|
/* > \endverbatim */
|
|
|
|
/* Arguments: */
|
|
/* ========== */
|
|
|
|
/* > \param[in] UPLO */
|
|
/* > \verbatim */
|
|
/* > UPLO is CHARACTER*1 */
|
|
/* > = 'U': B is upper bidiagonal; */
|
|
/* > = 'L': B is lower bidiagonal. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] N */
|
|
/* > \verbatim */
|
|
/* > N is INTEGER */
|
|
/* > The order of the matrix B. N >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] NCVT */
|
|
/* > \verbatim */
|
|
/* > NCVT is INTEGER */
|
|
/* > The number of columns of the matrix VT. NCVT >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] NRU */
|
|
/* > \verbatim */
|
|
/* > NRU is INTEGER */
|
|
/* > The number of rows of the matrix U. NRU >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] NCC */
|
|
/* > \verbatim */
|
|
/* > NCC is INTEGER */
|
|
/* > The number of columns of the matrix C. NCC >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] D */
|
|
/* > \verbatim */
|
|
/* > D is REAL array, dimension (N) */
|
|
/* > On entry, the n diagonal elements of the bidiagonal matrix B. */
|
|
/* > On exit, if INFO=0, the singular values of B in decreasing */
|
|
/* > order. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] E */
|
|
/* > \verbatim */
|
|
/* > E is REAL array, dimension (N-1) */
|
|
/* > On entry, the N-1 offdiagonal elements of the bidiagonal */
|
|
/* > matrix B. */
|
|
/* > On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E */
|
|
/* > will contain the diagonal and superdiagonal elements of a */
|
|
/* > bidiagonal matrix orthogonally equivalent to the one given */
|
|
/* > as input. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] VT */
|
|
/* > \verbatim */
|
|
/* > VT is REAL array, dimension (LDVT, NCVT) */
|
|
/* > On entry, an N-by-NCVT matrix VT. */
|
|
/* > On exit, VT is overwritten by P**T * VT. */
|
|
/* > Not referenced if NCVT = 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDVT */
|
|
/* > \verbatim */
|
|
/* > LDVT is INTEGER */
|
|
/* > The leading dimension of the array VT. */
|
|
/* > LDVT >= f2cmax(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] U */
|
|
/* > \verbatim */
|
|
/* > U is REAL array, dimension (LDU, N) */
|
|
/* > On entry, an NRU-by-N matrix U. */
|
|
/* > On exit, U is overwritten by U * Q. */
|
|
/* > Not referenced if NRU = 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDU */
|
|
/* > \verbatim */
|
|
/* > LDU is INTEGER */
|
|
/* > The leading dimension of the array U. LDU >= f2cmax(1,NRU). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] C */
|
|
/* > \verbatim */
|
|
/* > C is REAL array, dimension (LDC, NCC) */
|
|
/* > On entry, an N-by-NCC matrix C. */
|
|
/* > On exit, C is overwritten by Q**T * C. */
|
|
/* > Not referenced if NCC = 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDC */
|
|
/* > \verbatim */
|
|
/* > LDC is INTEGER */
|
|
/* > The leading dimension of the array C. */
|
|
/* > LDC >= f2cmax(1,N) if NCC > 0; LDC >=1 if NCC = 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WORK */
|
|
/* > \verbatim */
|
|
/* > WORK is REAL array, dimension (4*N) */
|
|
/* > \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 NCVT = NRU = NCC = 0, */
|
|
/* > = 1, a split was marked by a positive value in E */
|
|
/* > = 2, current block of Z not diagonalized after 30*N */
|
|
/* > iterations (in inner while loop) */
|
|
/* > = 3, termination criterion of outer while loop not met */
|
|
/* > (program created more than N unreduced blocks) */
|
|
/* > else NCVT = NRU = NCC = 0, */
|
|
/* > the algorithm did not converge; D and E contain the */
|
|
/* > elements of a bidiagonal matrix which is orthogonally */
|
|
/* > similar to the input matrix B; if INFO = i, i */
|
|
/* > elements of E have not converged to zero. */
|
|
/* > \endverbatim */
|
|
|
|
/* > \par Internal Parameters: */
|
|
/* ========================= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > TOLMUL REAL, default = f2cmax(10,f2cmin(100,EPS**(-1/8))) */
|
|
/* > TOLMUL controls the convergence criterion of the QR loop. */
|
|
/* > If it is positive, TOLMUL*EPS is the desired relative */
|
|
/* > precision in the computed singular values. */
|
|
/* > If it is negative, abs(TOLMUL*EPS*sigma_max) is the */
|
|
/* > desired absolute accuracy in the computed singular */
|
|
/* > values (corresponds to relative accuracy */
|
|
/* > abs(TOLMUL*EPS) in the largest singular value. */
|
|
/* > abs(TOLMUL) should be between 1 and 1/EPS, and preferably */
|
|
/* > between 10 (for fast convergence) and .1/EPS */
|
|
/* > (for there to be some accuracy in the results). */
|
|
/* > Default is to lose at either one eighth or 2 of the */
|
|
/* > available decimal digits in each computed singular value */
|
|
/* > (whichever is smaller). */
|
|
/* > */
|
|
/* > MAXITR INTEGER, default = 6 */
|
|
/* > MAXITR controls the maximum number of passes of the */
|
|
/* > algorithm through its inner loop. The algorithms stops */
|
|
/* > (and so fails to converge) if the number of passes */
|
|
/* > through the inner loop exceeds MAXITR*N**2. */
|
|
/* > \endverbatim */
|
|
|
|
/* > \par Note: */
|
|
/* =========== */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > Bug report from Cezary Dendek. */
|
|
/* > On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is */
|
|
/* > removed since it can overflow pretty easily (for N larger or equal */
|
|
/* > than 18,919). We instead use MAXITDIVN = MAXITR*N. */
|
|
/* > \endverbatim */
|
|
|
|
/* Authors: */
|
|
/* ======== */
|
|
|
|
/* > \author Univ. of Tennessee */
|
|
/* > \author Univ. of California Berkeley */
|
|
/* > \author Univ. of Colorado Denver */
|
|
/* > \author NAG Ltd. */
|
|
|
|
/* > \date June 2017 */
|
|
|
|
/* > \ingroup auxOTHERcomputational */
|
|
|
|
/* ===================================================================== */
|
|
/* Subroutine */ void sbdsqr_(char *uplo, integer *n, integer *ncvt, integer *
|
|
nru, integer *ncc, real *d__, real *e, real *vt, integer *ldvt, real *
|
|
u, integer *ldu, real *c__, integer *ldc, real *work, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1,
|
|
i__2;
|
|
real r__1, r__2, r__3, r__4;
|
|
doublereal d__1;
|
|
|
|
/* Local variables */
|
|
real abse;
|
|
integer idir;
|
|
real abss;
|
|
integer oldm;
|
|
real cosl;
|
|
integer isub, iter;
|
|
real unfl, sinl, cosr, smin, smax, sinr;
|
|
extern /* Subroutine */ void srot_(integer *, real *, integer *, real *,
|
|
integer *, real *, real *);
|
|
integer iterdivn;
|
|
extern /* Subroutine */ void slas2_(real *, real *, real *, real *, real *)
|
|
;
|
|
real f, g, h__;
|
|
integer i__, j, m;
|
|
real r__;
|
|
extern logical lsame_(char *, char *);
|
|
real oldcs;
|
|
extern /* Subroutine */ void sscal_(integer *, real *, real *, integer *);
|
|
integer oldll;
|
|
real shift, sigmn, oldsn, sminl;
|
|
extern /* Subroutine */ void slasr_(char *, char *, char *, integer *,
|
|
integer *, real *, real *, real *, integer *);
|
|
real sigmx;
|
|
logical lower;
|
|
extern /* Subroutine */ void sswap_(integer *, real *, integer *, real *,
|
|
integer *);
|
|
integer maxitdivn;
|
|
extern /* Subroutine */ void slasq1_(integer *, real *, real *, real *,
|
|
integer *), slasv2_(real *, real *, real *, real *, real *, real *
|
|
, real *, real *, real *);
|
|
real cs;
|
|
integer ll;
|
|
real sn, mu;
|
|
extern real slamch_(char *);
|
|
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
|
|
real sminoa;
|
|
extern /* Subroutine */ void slartg_(real *, real *, real *, real *, real *
|
|
);
|
|
real thresh;
|
|
logical rotate;
|
|
integer nm1;
|
|
real tolmul;
|
|
integer nm12, nm13, lll;
|
|
real eps, sll, tol;
|
|
|
|
|
|
/* -- 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 2017 */
|
|
|
|
|
|
/* ===================================================================== */
|
|
|
|
|
|
/* Test the input parameters. */
|
|
|
|
/* Parameter adjustments */
|
|
--d__;
|
|
--e;
|
|
vt_dim1 = *ldvt;
|
|
vt_offset = 1 + vt_dim1 * 1;
|
|
vt -= vt_offset;
|
|
u_dim1 = *ldu;
|
|
u_offset = 1 + u_dim1 * 1;
|
|
u -= u_offset;
|
|
c_dim1 = *ldc;
|
|
c_offset = 1 + c_dim1 * 1;
|
|
c__ -= c_offset;
|
|
--work;
|
|
|
|
/* Function Body */
|
|
*info = 0;
|
|
lower = lsame_(uplo, "L");
|
|
if (! lsame_(uplo, "U") && ! lower) {
|
|
*info = -1;
|
|
} else if (*n < 0) {
|
|
*info = -2;
|
|
} else if (*ncvt < 0) {
|
|
*info = -3;
|
|
} else if (*nru < 0) {
|
|
*info = -4;
|
|
} else if (*ncc < 0) {
|
|
*info = -5;
|
|
} else if (*ncvt == 0 && *ldvt < 1 || *ncvt > 0 && *ldvt < f2cmax(1,*n)) {
|
|
*info = -9;
|
|
} else if (*ldu < f2cmax(1,*nru)) {
|
|
*info = -11;
|
|
} else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < f2cmax(1,*n)) {
|
|
*info = -13;
|
|
}
|
|
if (*info != 0) {
|
|
i__1 = -(*info);
|
|
xerbla_("SBDSQR", &i__1, (ftnlen)6);
|
|
return;
|
|
}
|
|
if (*n == 0) {
|
|
return;
|
|
}
|
|
if (*n == 1) {
|
|
goto L160;
|
|
}
|
|
|
|
/* ROTATE is true if any singular vectors desired, false otherwise */
|
|
|
|
rotate = *ncvt > 0 || *nru > 0 || *ncc > 0;
|
|
|
|
/* If no singular vectors desired, use qd algorithm */
|
|
|
|
if (! rotate) {
|
|
slasq1_(n, &d__[1], &e[1], &work[1], info);
|
|
|
|
/* If INFO equals 2, dqds didn't finish, try to finish */
|
|
|
|
if (*info != 2) {
|
|
return;
|
|
}
|
|
*info = 0;
|
|
}
|
|
|
|
nm1 = *n - 1;
|
|
nm12 = nm1 + nm1;
|
|
nm13 = nm12 + nm1;
|
|
idir = 0;
|
|
|
|
/* Get machine constants */
|
|
|
|
eps = slamch_("Epsilon");
|
|
unfl = slamch_("Safe minimum");
|
|
|
|
/* If matrix lower bidiagonal, rotate to be upper bidiagonal */
|
|
/* by applying Givens rotations on the left */
|
|
|
|
if (lower) {
|
|
i__1 = *n - 1;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
|
|
d__[i__] = r__;
|
|
e[i__] = sn * d__[i__ + 1];
|
|
d__[i__ + 1] = cs * d__[i__ + 1];
|
|
work[i__] = cs;
|
|
work[nm1 + i__] = sn;
|
|
/* L10: */
|
|
}
|
|
|
|
/* Update singular vectors if desired */
|
|
|
|
if (*nru > 0) {
|
|
slasr_("R", "V", "F", nru, n, &work[1], &work[*n], &u[u_offset],
|
|
ldu);
|
|
}
|
|
if (*ncc > 0) {
|
|
slasr_("L", "V", "F", n, ncc, &work[1], &work[*n], &c__[c_offset],
|
|
ldc);
|
|
}
|
|
}
|
|
|
|
/* Compute singular values to relative accuracy TOL */
|
|
/* (By setting TOL to be negative, algorithm will compute */
|
|
/* singular values to absolute accuracy ABS(TOL)*norm(input matrix)) */
|
|
|
|
/* Computing MAX */
|
|
/* Computing MIN */
|
|
d__1 = (doublereal) eps;
|
|
r__3 = 100.f, r__4 = pow_dd(&d__1, &c_b15);
|
|
r__1 = 10.f, r__2 = f2cmin(r__3,r__4);
|
|
tolmul = f2cmax(r__1,r__2);
|
|
tol = tolmul * eps;
|
|
|
|
/* Compute approximate maximum, minimum singular values */
|
|
|
|
smax = 0.f;
|
|
i__1 = *n;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
/* Computing MAX */
|
|
r__2 = smax, r__3 = (r__1 = d__[i__], abs(r__1));
|
|
smax = f2cmax(r__2,r__3);
|
|
/* L20: */
|
|
}
|
|
i__1 = *n - 1;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
/* Computing MAX */
|
|
r__2 = smax, r__3 = (r__1 = e[i__], abs(r__1));
|
|
smax = f2cmax(r__2,r__3);
|
|
/* L30: */
|
|
}
|
|
sminl = 0.f;
|
|
if (tol >= 0.f) {
|
|
|
|
/* Relative accuracy desired */
|
|
|
|
sminoa = abs(d__[1]);
|
|
if (sminoa == 0.f) {
|
|
goto L50;
|
|
}
|
|
mu = sminoa;
|
|
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))));
|
|
sminoa = f2cmin(sminoa,mu);
|
|
if (sminoa == 0.f) {
|
|
goto L50;
|
|
}
|
|
/* L40: */
|
|
}
|
|
L50:
|
|
sminoa /= sqrt((real) (*n));
|
|
/* Computing MAX */
|
|
r__1 = tol * sminoa, r__2 = *n * (*n * unfl) * 6;
|
|
thresh = f2cmax(r__1,r__2);
|
|
} else {
|
|
|
|
/* Absolute accuracy desired */
|
|
|
|
/* Computing MAX */
|
|
r__1 = abs(tol) * smax, r__2 = *n * (*n * unfl) * 6;
|
|
thresh = f2cmax(r__1,r__2);
|
|
}
|
|
|
|
/* Prepare for main iteration loop for the singular values */
|
|
/* (MAXIT is the maximum number of passes through the inner */
|
|
/* loop permitted before nonconvergence signalled.) */
|
|
|
|
maxitdivn = *n * 6;
|
|
iterdivn = 0;
|
|
iter = -1;
|
|
oldll = -1;
|
|
oldm = -1;
|
|
|
|
/* M points to last element of unconverged part of matrix */
|
|
|
|
m = *n;
|
|
|
|
/* Begin main iteration loop */
|
|
|
|
L60:
|
|
|
|
/* Check for convergence or exceeding iteration count */
|
|
|
|
if (m <= 1) {
|
|
goto L160;
|
|
}
|
|
|
|
if (iter >= *n) {
|
|
iter -= *n;
|
|
++iterdivn;
|
|
if (iterdivn >= maxitdivn) {
|
|
goto L200;
|
|
}
|
|
}
|
|
|
|
/* Find diagonal block of matrix to work on */
|
|
|
|
if (tol < 0.f && (r__1 = d__[m], abs(r__1)) <= thresh) {
|
|
d__[m] = 0.f;
|
|
}
|
|
smax = (r__1 = d__[m], abs(r__1));
|
|
smin = smax;
|
|
i__1 = m - 1;
|
|
for (lll = 1; lll <= i__1; ++lll) {
|
|
ll = m - lll;
|
|
abss = (r__1 = d__[ll], abs(r__1));
|
|
abse = (r__1 = e[ll], abs(r__1));
|
|
if (tol < 0.f && abss <= thresh) {
|
|
d__[ll] = 0.f;
|
|
}
|
|
if (abse <= thresh) {
|
|
goto L80;
|
|
}
|
|
smin = f2cmin(smin,abss);
|
|
/* Computing MAX */
|
|
r__1 = f2cmax(smax,abss);
|
|
smax = f2cmax(r__1,abse);
|
|
/* L70: */
|
|
}
|
|
ll = 0;
|
|
goto L90;
|
|
L80:
|
|
e[ll] = 0.f;
|
|
|
|
/* Matrix splits since E(LL) = 0 */
|
|
|
|
if (ll == m - 1) {
|
|
|
|
/* Convergence of bottom singular value, return to top of loop */
|
|
|
|
--m;
|
|
goto L60;
|
|
}
|
|
L90:
|
|
++ll;
|
|
|
|
/* E(LL) through E(M-1) are nonzero, E(LL-1) is zero */
|
|
|
|
if (ll == m - 1) {
|
|
|
|
/* 2 by 2 block, handle separately */
|
|
|
|
slasv2_(&d__[m - 1], &e[m - 1], &d__[m], &sigmn, &sigmx, &sinr, &cosr,
|
|
&sinl, &cosl);
|
|
d__[m - 1] = sigmx;
|
|
e[m - 1] = 0.f;
|
|
d__[m] = sigmn;
|
|
|
|
/* Compute singular vectors, if desired */
|
|
|
|
if (*ncvt > 0) {
|
|
srot_(ncvt, &vt[m - 1 + vt_dim1], ldvt, &vt[m + vt_dim1], ldvt, &
|
|
cosr, &sinr);
|
|
}
|
|
if (*nru > 0) {
|
|
srot_(nru, &u[(m - 1) * u_dim1 + 1], &c__1, &u[m * u_dim1 + 1], &
|
|
c__1, &cosl, &sinl);
|
|
}
|
|
if (*ncc > 0) {
|
|
srot_(ncc, &c__[m - 1 + c_dim1], ldc, &c__[m + c_dim1], ldc, &
|
|
cosl, &sinl);
|
|
}
|
|
m += -2;
|
|
goto L60;
|
|
}
|
|
|
|
/* If working on new submatrix, choose shift direction */
|
|
/* (from larger end diagonal element towards smaller) */
|
|
|
|
if (ll > oldm || m < oldll) {
|
|
if ((r__1 = d__[ll], abs(r__1)) >= (r__2 = d__[m], abs(r__2))) {
|
|
|
|
/* Chase bulge from top (big end) to bottom (small end) */
|
|
|
|
idir = 1;
|
|
} else {
|
|
|
|
/* Chase bulge from bottom (big end) to top (small end) */
|
|
|
|
idir = 2;
|
|
}
|
|
}
|
|
|
|
/* Apply convergence tests */
|
|
|
|
if (idir == 1) {
|
|
|
|
/* Run convergence test in forward direction */
|
|
/* First apply standard test to bottom of matrix */
|
|
|
|
if ((r__2 = e[m - 1], abs(r__2)) <= abs(tol) * (r__1 = d__[m], abs(
|
|
r__1)) || tol < 0.f && (r__3 = e[m - 1], abs(r__3)) <= thresh)
|
|
{
|
|
e[m - 1] = 0.f;
|
|
goto L60;
|
|
}
|
|
|
|
if (tol >= 0.f) {
|
|
|
|
/* If relative accuracy desired, */
|
|
/* apply convergence criterion forward */
|
|
|
|
mu = (r__1 = d__[ll], abs(r__1));
|
|
sminl = mu;
|
|
i__1 = m - 1;
|
|
for (lll = ll; lll <= i__1; ++lll) {
|
|
if ((r__1 = e[lll], abs(r__1)) <= tol * mu) {
|
|
e[lll] = 0.f;
|
|
goto L60;
|
|
}
|
|
mu = (r__2 = d__[lll + 1], abs(r__2)) * (mu / (mu + (r__1 = e[
|
|
lll], abs(r__1))));
|
|
sminl = f2cmin(sminl,mu);
|
|
/* L100: */
|
|
}
|
|
}
|
|
|
|
} else {
|
|
|
|
/* Run convergence test in backward direction */
|
|
/* First apply standard test to top of matrix */
|
|
|
|
if ((r__2 = e[ll], abs(r__2)) <= abs(tol) * (r__1 = d__[ll], abs(r__1)
|
|
) || tol < 0.f && (r__3 = e[ll], abs(r__3)) <= thresh) {
|
|
e[ll] = 0.f;
|
|
goto L60;
|
|
}
|
|
|
|
if (tol >= 0.f) {
|
|
|
|
/* If relative accuracy desired, */
|
|
/* apply convergence criterion backward */
|
|
|
|
mu = (r__1 = d__[m], abs(r__1));
|
|
sminl = mu;
|
|
i__1 = ll;
|
|
for (lll = m - 1; lll >= i__1; --lll) {
|
|
if ((r__1 = e[lll], abs(r__1)) <= tol * mu) {
|
|
e[lll] = 0.f;
|
|
goto L60;
|
|
}
|
|
mu = (r__2 = d__[lll], abs(r__2)) * (mu / (mu + (r__1 = e[lll]
|
|
, abs(r__1))));
|
|
sminl = f2cmin(sminl,mu);
|
|
/* L110: */
|
|
}
|
|
}
|
|
}
|
|
oldll = ll;
|
|
oldm = m;
|
|
|
|
/* Compute shift. First, test if shifting would ruin relative */
|
|
/* accuracy, and if so set the shift to zero. */
|
|
|
|
/* Computing MAX */
|
|
r__1 = eps, r__2 = tol * .01f;
|
|
if (tol >= 0.f && *n * tol * (sminl / smax) <= f2cmax(r__1,r__2)) {
|
|
|
|
/* Use a zero shift to avoid loss of relative accuracy */
|
|
|
|
shift = 0.f;
|
|
} else {
|
|
|
|
/* Compute the shift from 2-by-2 block at end of matrix */
|
|
|
|
if (idir == 1) {
|
|
sll = (r__1 = d__[ll], abs(r__1));
|
|
slas2_(&d__[m - 1], &e[m - 1], &d__[m], &shift, &r__);
|
|
} else {
|
|
sll = (r__1 = d__[m], abs(r__1));
|
|
slas2_(&d__[ll], &e[ll], &d__[ll + 1], &shift, &r__);
|
|
}
|
|
|
|
/* Test if shift negligible, and if so set to zero */
|
|
|
|
if (sll > 0.f) {
|
|
/* Computing 2nd power */
|
|
r__1 = shift / sll;
|
|
if (r__1 * r__1 < eps) {
|
|
shift = 0.f;
|
|
}
|
|
}
|
|
}
|
|
|
|
/* Increment iteration count */
|
|
|
|
iter = iter + m - ll;
|
|
|
|
/* If SHIFT = 0, do simplified QR iteration */
|
|
|
|
if (shift == 0.f) {
|
|
if (idir == 1) {
|
|
|
|
/* Chase bulge from top to bottom */
|
|
/* Save cosines and sines for later singular vector updates */
|
|
|
|
cs = 1.f;
|
|
oldcs = 1.f;
|
|
i__1 = m - 1;
|
|
for (i__ = ll; i__ <= i__1; ++i__) {
|
|
r__1 = d__[i__] * cs;
|
|
slartg_(&r__1, &e[i__], &cs, &sn, &r__);
|
|
if (i__ > ll) {
|
|
e[i__ - 1] = oldsn * r__;
|
|
}
|
|
r__1 = oldcs * r__;
|
|
r__2 = d__[i__ + 1] * sn;
|
|
slartg_(&r__1, &r__2, &oldcs, &oldsn, &d__[i__]);
|
|
work[i__ - ll + 1] = cs;
|
|
work[i__ - ll + 1 + nm1] = sn;
|
|
work[i__ - ll + 1 + nm12] = oldcs;
|
|
work[i__ - ll + 1 + nm13] = oldsn;
|
|
/* L120: */
|
|
}
|
|
h__ = d__[m] * cs;
|
|
d__[m] = h__ * oldcs;
|
|
e[m - 1] = h__ * oldsn;
|
|
|
|
/* Update singular vectors */
|
|
|
|
if (*ncvt > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "F", &i__1, ncvt, &work[1], &work[*n], &vt[
|
|
ll + vt_dim1], ldvt);
|
|
}
|
|
if (*nru > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("R", "V", "F", nru, &i__1, &work[nm12 + 1], &work[nm13
|
|
+ 1], &u[ll * u_dim1 + 1], ldu);
|
|
}
|
|
if (*ncc > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "F", &i__1, ncc, &work[nm12 + 1], &work[nm13
|
|
+ 1], &c__[ll + c_dim1], ldc);
|
|
}
|
|
|
|
/* Test convergence */
|
|
|
|
if ((r__1 = e[m - 1], abs(r__1)) <= thresh) {
|
|
e[m - 1] = 0.f;
|
|
}
|
|
|
|
} else {
|
|
|
|
/* Chase bulge from bottom to top */
|
|
/* Save cosines and sines for later singular vector updates */
|
|
|
|
cs = 1.f;
|
|
oldcs = 1.f;
|
|
i__1 = ll + 1;
|
|
for (i__ = m; i__ >= i__1; --i__) {
|
|
r__1 = d__[i__] * cs;
|
|
slartg_(&r__1, &e[i__ - 1], &cs, &sn, &r__);
|
|
if (i__ < m) {
|
|
e[i__] = oldsn * r__;
|
|
}
|
|
r__1 = oldcs * r__;
|
|
r__2 = d__[i__ - 1] * sn;
|
|
slartg_(&r__1, &r__2, &oldcs, &oldsn, &d__[i__]);
|
|
work[i__ - ll] = cs;
|
|
work[i__ - ll + nm1] = -sn;
|
|
work[i__ - ll + nm12] = oldcs;
|
|
work[i__ - ll + nm13] = -oldsn;
|
|
/* L130: */
|
|
}
|
|
h__ = d__[ll] * cs;
|
|
d__[ll] = h__ * oldcs;
|
|
e[ll] = h__ * oldsn;
|
|
|
|
/* Update singular vectors */
|
|
|
|
if (*ncvt > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "B", &i__1, ncvt, &work[nm12 + 1], &work[
|
|
nm13 + 1], &vt[ll + vt_dim1], ldvt);
|
|
}
|
|
if (*nru > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("R", "V", "B", nru, &i__1, &work[1], &work[*n], &u[ll *
|
|
u_dim1 + 1], ldu);
|
|
}
|
|
if (*ncc > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "B", &i__1, ncc, &work[1], &work[*n], &c__[
|
|
ll + c_dim1], ldc);
|
|
}
|
|
|
|
/* Test convergence */
|
|
|
|
if ((r__1 = e[ll], abs(r__1)) <= thresh) {
|
|
e[ll] = 0.f;
|
|
}
|
|
}
|
|
} else {
|
|
|
|
/* Use nonzero shift */
|
|
|
|
if (idir == 1) {
|
|
|
|
/* Chase bulge from top to bottom */
|
|
/* Save cosines and sines for later singular vector updates */
|
|
|
|
f = ((r__1 = d__[ll], abs(r__1)) - shift) * (r_sign(&c_b49, &d__[
|
|
ll]) + shift / d__[ll]);
|
|
g = e[ll];
|
|
i__1 = m - 1;
|
|
for (i__ = ll; i__ <= i__1; ++i__) {
|
|
slartg_(&f, &g, &cosr, &sinr, &r__);
|
|
if (i__ > ll) {
|
|
e[i__ - 1] = r__;
|
|
}
|
|
f = cosr * d__[i__] + sinr * e[i__];
|
|
e[i__] = cosr * e[i__] - sinr * d__[i__];
|
|
g = sinr * d__[i__ + 1];
|
|
d__[i__ + 1] = cosr * d__[i__ + 1];
|
|
slartg_(&f, &g, &cosl, &sinl, &r__);
|
|
d__[i__] = r__;
|
|
f = cosl * e[i__] + sinl * d__[i__ + 1];
|
|
d__[i__ + 1] = cosl * d__[i__ + 1] - sinl * e[i__];
|
|
if (i__ < m - 1) {
|
|
g = sinl * e[i__ + 1];
|
|
e[i__ + 1] = cosl * e[i__ + 1];
|
|
}
|
|
work[i__ - ll + 1] = cosr;
|
|
work[i__ - ll + 1 + nm1] = sinr;
|
|
work[i__ - ll + 1 + nm12] = cosl;
|
|
work[i__ - ll + 1 + nm13] = sinl;
|
|
/* L140: */
|
|
}
|
|
e[m - 1] = f;
|
|
|
|
/* Update singular vectors */
|
|
|
|
if (*ncvt > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "F", &i__1, ncvt, &work[1], &work[*n], &vt[
|
|
ll + vt_dim1], ldvt);
|
|
}
|
|
if (*nru > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("R", "V", "F", nru, &i__1, &work[nm12 + 1], &work[nm13
|
|
+ 1], &u[ll * u_dim1 + 1], ldu);
|
|
}
|
|
if (*ncc > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "F", &i__1, ncc, &work[nm12 + 1], &work[nm13
|
|
+ 1], &c__[ll + c_dim1], ldc);
|
|
}
|
|
|
|
/* Test convergence */
|
|
|
|
if ((r__1 = e[m - 1], abs(r__1)) <= thresh) {
|
|
e[m - 1] = 0.f;
|
|
}
|
|
|
|
} else {
|
|
|
|
/* Chase bulge from bottom to top */
|
|
/* Save cosines and sines for later singular vector updates */
|
|
|
|
f = ((r__1 = d__[m], abs(r__1)) - shift) * (r_sign(&c_b49, &d__[m]
|
|
) + shift / d__[m]);
|
|
g = e[m - 1];
|
|
i__1 = ll + 1;
|
|
for (i__ = m; i__ >= i__1; --i__) {
|
|
slartg_(&f, &g, &cosr, &sinr, &r__);
|
|
if (i__ < m) {
|
|
e[i__] = r__;
|
|
}
|
|
f = cosr * d__[i__] + sinr * e[i__ - 1];
|
|
e[i__ - 1] = cosr * e[i__ - 1] - sinr * d__[i__];
|
|
g = sinr * d__[i__ - 1];
|
|
d__[i__ - 1] = cosr * d__[i__ - 1];
|
|
slartg_(&f, &g, &cosl, &sinl, &r__);
|
|
d__[i__] = r__;
|
|
f = cosl * e[i__ - 1] + sinl * d__[i__ - 1];
|
|
d__[i__ - 1] = cosl * d__[i__ - 1] - sinl * e[i__ - 1];
|
|
if (i__ > ll + 1) {
|
|
g = sinl * e[i__ - 2];
|
|
e[i__ - 2] = cosl * e[i__ - 2];
|
|
}
|
|
work[i__ - ll] = cosr;
|
|
work[i__ - ll + nm1] = -sinr;
|
|
work[i__ - ll + nm12] = cosl;
|
|
work[i__ - ll + nm13] = -sinl;
|
|
/* L150: */
|
|
}
|
|
e[ll] = f;
|
|
|
|
/* Test convergence */
|
|
|
|
if ((r__1 = e[ll], abs(r__1)) <= thresh) {
|
|
e[ll] = 0.f;
|
|
}
|
|
|
|
/* Update singular vectors if desired */
|
|
|
|
if (*ncvt > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "B", &i__1, ncvt, &work[nm12 + 1], &work[
|
|
nm13 + 1], &vt[ll + vt_dim1], ldvt);
|
|
}
|
|
if (*nru > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("R", "V", "B", nru, &i__1, &work[1], &work[*n], &u[ll *
|
|
u_dim1 + 1], ldu);
|
|
}
|
|
if (*ncc > 0) {
|
|
i__1 = m - ll + 1;
|
|
slasr_("L", "V", "B", &i__1, ncc, &work[1], &work[*n], &c__[
|
|
ll + c_dim1], ldc);
|
|
}
|
|
}
|
|
}
|
|
|
|
/* QR iteration finished, go back and check convergence */
|
|
|
|
goto L60;
|
|
|
|
/* All singular values converged, so make them positive */
|
|
|
|
L160:
|
|
i__1 = *n;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
if (d__[i__] < 0.f) {
|
|
d__[i__] = -d__[i__];
|
|
|
|
/* Change sign of singular vectors, if desired */
|
|
|
|
if (*ncvt > 0) {
|
|
sscal_(ncvt, &c_b72, &vt[i__ + vt_dim1], ldvt);
|
|
}
|
|
}
|
|
/* L170: */
|
|
}
|
|
|
|
/* Sort the singular values into decreasing order (insertion sort on */
|
|
/* singular values, but only one transposition per singular vector) */
|
|
|
|
i__1 = *n - 1;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
|
|
/* Scan for smallest D(I) */
|
|
|
|
isub = 1;
|
|
smin = d__[1];
|
|
i__2 = *n + 1 - i__;
|
|
for (j = 2; j <= i__2; ++j) {
|
|
if (d__[j] <= smin) {
|
|
isub = j;
|
|
smin = d__[j];
|
|
}
|
|
/* L180: */
|
|
}
|
|
if (isub != *n + 1 - i__) {
|
|
|
|
/* Swap singular values and vectors */
|
|
|
|
d__[isub] = d__[*n + 1 - i__];
|
|
d__[*n + 1 - i__] = smin;
|
|
if (*ncvt > 0) {
|
|
sswap_(ncvt, &vt[isub + vt_dim1], ldvt, &vt[*n + 1 - i__ +
|
|
vt_dim1], ldvt);
|
|
}
|
|
if (*nru > 0) {
|
|
sswap_(nru, &u[isub * u_dim1 + 1], &c__1, &u[(*n + 1 - i__) *
|
|
u_dim1 + 1], &c__1);
|
|
}
|
|
if (*ncc > 0) {
|
|
sswap_(ncc, &c__[isub + c_dim1], ldc, &c__[*n + 1 - i__ +
|
|
c_dim1], ldc);
|
|
}
|
|
}
|
|
/* L190: */
|
|
}
|
|
goto L220;
|
|
|
|
/* Maximum number of iterations exceeded, failure to converge */
|
|
|
|
L200:
|
|
*info = 0;
|
|
i__1 = *n - 1;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
if (e[i__] != 0.f) {
|
|
++(*info);
|
|
}
|
|
/* L210: */
|
|
}
|
|
L220:
|
|
return;
|
|
|
|
/* End of SBDSQR */
|
|
|
|
} /* sbdsqr_ */
|
|
|