1493 lines
46 KiB
C
1493 lines
46 KiB
C
#include <math.h>
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#include <stdlib.h>
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#include <string.h>
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#include <stdio.h>
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#include <complex.h>
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#ifdef complex
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#undef complex
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#endif
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#ifdef I
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#undef I
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#endif
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#if defined(_WIN64)
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typedef long long BLASLONG;
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typedef unsigned long long BLASULONG;
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#else
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typedef long BLASLONG;
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typedef unsigned long BLASULONG;
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#endif
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#ifdef LAPACK_ILP64
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typedef BLASLONG blasint;
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#if defined(_WIN64)
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#define blasabs(x) llabs(x)
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#else
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#define blasabs(x) labs(x)
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#endif
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#else
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typedef int blasint;
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#define blasabs(x) abs(x)
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#endif
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typedef blasint integer;
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typedef unsigned int uinteger;
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typedef char *address;
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typedef short int shortint;
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typedef float real;
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typedef double doublereal;
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typedef struct { real r, i; } complex;
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typedef struct { doublereal r, i; } doublecomplex;
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#ifdef _MSC_VER
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static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
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static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
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static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
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static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
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#else
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static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
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static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
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static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
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static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
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#endif
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#define pCf(z) (*_pCf(z))
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#define pCd(z) (*_pCd(z))
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typedef 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 integer c__1 = 1;
|
|
static integer c__2 = 2;
|
|
|
|
/* > \brief \b SLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each un
|
|
reduced block Ti, finds base representations and eigenvalues. */
|
|
|
|
/* =========== DOCUMENTATION =========== */
|
|
|
|
/* Online html documentation available at */
|
|
/* http://www.netlib.org/lapack/explore-html/ */
|
|
|
|
/* > \htmlonly */
|
|
/* > Download SLARRE + dependencies */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slarre.
|
|
f"> */
|
|
/* > [TGZ]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slarre.
|
|
f"> */
|
|
/* > [ZIP]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarre.
|
|
f"> */
|
|
/* > [TXT]</a> */
|
|
/* > \endhtmlonly */
|
|
|
|
/* Definition: */
|
|
/* =========== */
|
|
|
|
/* SUBROUTINE SLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2, */
|
|
/* RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, */
|
|
/* W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, */
|
|
/* WORK, IWORK, INFO ) */
|
|
|
|
/* CHARACTER RANGE */
|
|
/* INTEGER IL, INFO, IU, M, N, NSPLIT */
|
|
/* REAL PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU */
|
|
/* INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), */
|
|
/* $ INDEXW( * ) */
|
|
/* REAL D( * ), E( * ), E2( * ), GERS( * ), */
|
|
/* $ W( * ),WERR( * ), WGAP( * ), WORK( * ) */
|
|
|
|
|
|
/* > \par Purpose: */
|
|
/* ============= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > To find the desired eigenvalues of a given real symmetric */
|
|
/* > tridiagonal matrix T, SLARRE sets any "small" off-diagonal */
|
|
/* > elements to zero, and for each unreduced block T_i, it finds */
|
|
/* > (a) a suitable shift at one end of the block's spectrum, */
|
|
/* > (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and */
|
|
/* > (c) eigenvalues of each L_i D_i L_i^T. */
|
|
/* > The representations and eigenvalues found are then used by */
|
|
/* > SSTEMR to compute the eigenvectors of T. */
|
|
/* > The accuracy varies depending on whether bisection is used to */
|
|
/* > find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to */
|
|
/* > conpute all and then discard any unwanted one. */
|
|
/* > As an added benefit, SLARRE also outputs the n */
|
|
/* > Gerschgorin intervals for the matrices L_i D_i L_i^T. */
|
|
/* > \endverbatim */
|
|
|
|
/* Arguments: */
|
|
/* ========== */
|
|
|
|
/* > \param[in] RANGE */
|
|
/* > \verbatim */
|
|
/* > RANGE is CHARACTER*1 */
|
|
/* > = 'A': ("All") all eigenvalues will be found. */
|
|
/* > = 'V': ("Value") all eigenvalues in the half-open interval */
|
|
/* > (VL, VU] will be found. */
|
|
/* > = 'I': ("Index") the IL-th through IU-th eigenvalues (of the */
|
|
/* > entire matrix) will be found. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] N */
|
|
/* > \verbatim */
|
|
/* > N is INTEGER */
|
|
/* > The order of the matrix. N > 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] VL */
|
|
/* > \verbatim */
|
|
/* > VL is REAL */
|
|
/* > If RANGE='V', the lower bound for the eigenvalues. */
|
|
/* > Eigenvalues less than or equal to VL, or greater than VU, */
|
|
/* > will not be returned. VL < VU. */
|
|
/* > If RANGE='I' or ='A', SLARRE computes bounds on the desired */
|
|
/* > part of the spectrum. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] VU */
|
|
/* > \verbatim */
|
|
/* > VU is REAL */
|
|
/* > If RANGE='V', the upper bound for the eigenvalues. */
|
|
/* > Eigenvalues less than or equal to VL, or greater than VU, */
|
|
/* > will not be returned. VL < VU. */
|
|
/* > If RANGE='I' or ='A', SLARRE computes bounds on the desired */
|
|
/* > part of the spectrum. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] IL */
|
|
/* > \verbatim */
|
|
/* > IL is INTEGER */
|
|
/* > If RANGE='I', the index of the */
|
|
/* > smallest eigenvalue to be returned. */
|
|
/* > 1 <= IL <= IU <= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] IU */
|
|
/* > \verbatim */
|
|
/* > IU is INTEGER */
|
|
/* > If RANGE='I', the index of the */
|
|
/* > largest eigenvalue to be returned. */
|
|
/* > 1 <= IL <= IU <= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] D */
|
|
/* > \verbatim */
|
|
/* > D is REAL array, dimension (N) */
|
|
/* > On entry, the N diagonal elements of the tridiagonal */
|
|
/* > matrix T. */
|
|
/* > On exit, the N diagonal elements of the diagonal */
|
|
/* > matrices D_i. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] E */
|
|
/* > \verbatim */
|
|
/* > E is REAL array, dimension (N) */
|
|
/* > On entry, the first (N-1) entries contain the subdiagonal */
|
|
/* > elements of the tridiagonal matrix T; E(N) need not be set. */
|
|
/* > On exit, E contains the subdiagonal elements of the unit */
|
|
/* > bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), */
|
|
/* > 1 <= I <= NSPLIT, contain the base points sigma_i on output. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] E2 */
|
|
/* > \verbatim */
|
|
/* > E2 is REAL array, dimension (N) */
|
|
/* > On entry, the first (N-1) entries contain the SQUARES of the */
|
|
/* > subdiagonal elements of the tridiagonal matrix T; */
|
|
/* > E2(N) need not be set. */
|
|
/* > On exit, the entries E2( ISPLIT( I ) ), */
|
|
/* > 1 <= I <= NSPLIT, have been set to zero */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] RTOL1 */
|
|
/* > \verbatim */
|
|
/* > RTOL1 is REAL */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] RTOL2 */
|
|
/* > \verbatim */
|
|
/* > RTOL2 is REAL */
|
|
/* > Parameters for bisection. */
|
|
/* > An interval [LEFT,RIGHT] has converged if */
|
|
/* > RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] SPLTOL */
|
|
/* > \verbatim */
|
|
/* > SPLTOL is REAL */
|
|
/* > The threshold for splitting. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] NSPLIT */
|
|
/* > \verbatim */
|
|
/* > NSPLIT is INTEGER */
|
|
/* > The number of blocks T splits into. 1 <= NSPLIT <= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] ISPLIT */
|
|
/* > \verbatim */
|
|
/* > ISPLIT is INTEGER array, dimension (N) */
|
|
/* > The splitting points, at which T breaks up into blocks. */
|
|
/* > The first block consists of rows/columns 1 to ISPLIT(1), */
|
|
/* > the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), */
|
|
/* > etc., and the NSPLIT-th consists of rows/columns */
|
|
/* > ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] M */
|
|
/* > \verbatim */
|
|
/* > M is INTEGER */
|
|
/* > The total number of eigenvalues (of all L_i D_i L_i^T) */
|
|
/* > found. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] W */
|
|
/* > \verbatim */
|
|
/* > W is REAL array, dimension (N) */
|
|
/* > The first M elements contain the eigenvalues. The */
|
|
/* > eigenvalues of each of the blocks, L_i D_i L_i^T, are */
|
|
/* > sorted in ascending order ( SLARRE may use the */
|
|
/* > remaining N-M elements as workspace). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WERR */
|
|
/* > \verbatim */
|
|
/* > WERR is REAL array, dimension (N) */
|
|
/* > The error bound on the corresponding eigenvalue in W. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WGAP */
|
|
/* > \verbatim */
|
|
/* > WGAP is REAL array, dimension (N) */
|
|
/* > The separation from the right neighbor eigenvalue in W. */
|
|
/* > The gap is only with respect to the eigenvalues of the same block */
|
|
/* > as each block has its own representation tree. */
|
|
/* > Exception: at the right end of a block we store the left gap */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] IBLOCK */
|
|
/* > \verbatim */
|
|
/* > IBLOCK is INTEGER array, dimension (N) */
|
|
/* > The indices of the blocks (submatrices) associated with the */
|
|
/* > corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue */
|
|
/* > W(i) belongs to the first block from the top, =2 if W(i) */
|
|
/* > belongs to the second block, etc. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] INDEXW */
|
|
/* > \verbatim */
|
|
/* > INDEXW is INTEGER array, dimension (N) */
|
|
/* > The indices of the eigenvalues within each block (submatrix); */
|
|
/* > for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the */
|
|
/* > i-th eigenvalue W(i) is the 10-th eigenvalue in block 2 */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] GERS */
|
|
/* > \verbatim */
|
|
/* > GERS is REAL array, dimension (2*N) */
|
|
/* > The N Gerschgorin intervals (the i-th Gerschgorin interval */
|
|
/* > is (GERS(2*i-1), GERS(2*i)). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] PIVMIN */
|
|
/* > \verbatim */
|
|
/* > PIVMIN is REAL */
|
|
/* > The minimum pivot in the Sturm sequence for T. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WORK */
|
|
/* > \verbatim */
|
|
/* > WORK is REAL array, dimension (6*N) */
|
|
/* > Workspace. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] IWORK */
|
|
/* > \verbatim */
|
|
/* > IWORK is INTEGER array, dimension (5*N) */
|
|
/* > Workspace. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] INFO */
|
|
/* > \verbatim */
|
|
/* > INFO is INTEGER */
|
|
/* > = 0: successful exit */
|
|
/* > > 0: A problem occurred in SLARRE. */
|
|
/* > < 0: One of the called subroutines signaled an internal problem. */
|
|
/* > Needs inspection of the corresponding parameter IINFO */
|
|
/* > for further information. */
|
|
/* > */
|
|
/* > =-1: Problem in SLARRD. */
|
|
/* > = 2: No base representation could be found in MAXTRY iterations. */
|
|
/* > Increasing MAXTRY and recompilation might be a remedy. */
|
|
/* > =-3: Problem in SLARRB when computing the refined root */
|
|
/* > representation for SLASQ2. */
|
|
/* > =-4: Problem in SLARRB when preforming bisection on the */
|
|
/* > desired part of the spectrum. */
|
|
/* > =-5: Problem in SLASQ2. */
|
|
/* > =-6: Problem in SLASQ2. */
|
|
/* > \endverbatim */
|
|
|
|
/* Authors: */
|
|
/* ======== */
|
|
|
|
/* > \author Univ. of Tennessee */
|
|
/* > \author Univ. of California Berkeley */
|
|
/* > \author Univ. of Colorado Denver */
|
|
/* > \author NAG Ltd. */
|
|
|
|
/* > \date June 2016 */
|
|
|
|
/* > \ingroup OTHERauxiliary */
|
|
|
|
/* > \par Further Details: */
|
|
/* ===================== */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > The base representations are required to suffer very little */
|
|
/* > element growth and consequently define all their eigenvalues to */
|
|
/* > high relative accuracy. */
|
|
/* > \endverbatim */
|
|
|
|
/* > \par Contributors: */
|
|
/* ================== */
|
|
/* > */
|
|
/* > Beresford Parlett, University of California, Berkeley, USA \n */
|
|
/* > Jim Demmel, University of California, Berkeley, USA \n */
|
|
/* > Inderjit Dhillon, University of Texas, Austin, USA \n */
|
|
/* > Osni Marques, LBNL/NERSC, USA \n */
|
|
/* > Christof Voemel, University of California, Berkeley, USA \n */
|
|
/* > */
|
|
/* ===================================================================== */
|
|
/* Subroutine */ void slarre_(char *range, integer *n, real *vl, real *vu,
|
|
integer *il, integer *iu, real *d__, real *e, real *e2, real *rtol1,
|
|
real *rtol2, real *spltol, integer *nsplit, integer *isplit, integer *
|
|
m, real *w, real *werr, real *wgap, integer *iblock, integer *indexw,
|
|
real *gers, real *pivmin, real *work, integer *iwork, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer i__1, i__2;
|
|
real r__1, r__2, r__3;
|
|
|
|
/* Local variables */
|
|
real eabs;
|
|
integer iend, jblk;
|
|
real eold;
|
|
integer indl;
|
|
real dmax__, emax;
|
|
integer wend, idum, indu;
|
|
real rtol;
|
|
integer i__, j, iseed[4];
|
|
real avgap, sigma;
|
|
extern logical lsame_(char *, char *);
|
|
integer iinfo;
|
|
logical norep;
|
|
extern /* Subroutine */ void scopy_(integer *, real *, integer *, real *,
|
|
integer *);
|
|
real s1, s2;
|
|
extern /* Subroutine */ void slasq2_(integer *, real *, integer *);
|
|
integer mb;
|
|
real gl;
|
|
integer in, mm;
|
|
real gu;
|
|
integer ibegin;
|
|
logical forceb;
|
|
integer irange;
|
|
real sgndef;
|
|
extern real slamch_(char *);
|
|
integer wbegin;
|
|
real safmin, spdiam;
|
|
extern /* Subroutine */ void slarra_(integer *, real *, real *, real *,
|
|
real *, real *, integer *, integer *, integer *);
|
|
logical usedqd;
|
|
real clwdth, isleft;
|
|
extern /* Subroutine */ void slarrb_(integer *, real *, real *, integer *,
|
|
integer *, real *, real *, integer *, real *, real *, real *,
|
|
real *, integer *, real *, real *, integer *, integer *), slarrc_(
|
|
char *, integer *, real *, real *, real *, real *, real *,
|
|
integer *, integer *, integer *, integer *), slarrd_(char
|
|
*, char *, integer *, real *, real *, integer *, integer *, real *
|
|
, real *, real *, real *, real *, real *, integer *, integer *,
|
|
integer *, real *, real *, real *, real *, integer *, integer *,
|
|
real *, integer *, integer *), slarrk_(integer *,
|
|
integer *, real *, real *, real *, real *, real *, real *, real *,
|
|
real *, integer *);
|
|
real isrght, bsrtol, dpivot;
|
|
extern /* Subroutine */ void slarnv_(integer *, integer *, integer *, real
|
|
*);
|
|
integer cnt;
|
|
real eps, tau, tmp, rtl;
|
|
integer cnt1, cnt2;
|
|
real tmp1;
|
|
|
|
|
|
/* -- LAPACK auxiliary 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..-- */
|
|
/* June 2016 */
|
|
|
|
|
|
/* ===================================================================== */
|
|
|
|
|
|
/* Parameter adjustments */
|
|
--iwork;
|
|
--work;
|
|
--gers;
|
|
--indexw;
|
|
--iblock;
|
|
--wgap;
|
|
--werr;
|
|
--w;
|
|
--isplit;
|
|
--e2;
|
|
--e;
|
|
--d__;
|
|
|
|
/* Function Body */
|
|
*info = 0;
|
|
|
|
/* Quick return if possible */
|
|
|
|
if (*n <= 0) {
|
|
return;
|
|
}
|
|
|
|
/* Decode RANGE */
|
|
|
|
if (lsame_(range, "A")) {
|
|
irange = 1;
|
|
} else if (lsame_(range, "V")) {
|
|
irange = 3;
|
|
} else if (lsame_(range, "I")) {
|
|
irange = 2;
|
|
}
|
|
*m = 0;
|
|
/* Get machine constants */
|
|
safmin = slamch_("S");
|
|
eps = slamch_("P");
|
|
/* Set parameters */
|
|
rtl = eps * 100.f;
|
|
/* If one were ever to ask for less initial precision in BSRTOL, */
|
|
/* one should keep in mind that for the subset case, the extremal */
|
|
/* eigenvalues must be at least as accurate as the current setting */
|
|
/* (eigenvalues in the middle need not as much accuracy) */
|
|
bsrtol = sqrt(eps) * 5e-4f;
|
|
/* Treat case of 1x1 matrix for quick return */
|
|
if (*n == 1) {
|
|
if (irange == 1 || irange == 3 && d__[1] > *vl && d__[1] <= *vu ||
|
|
irange == 2 && *il == 1 && *iu == 1) {
|
|
*m = 1;
|
|
w[1] = d__[1];
|
|
/* The computation error of the eigenvalue is zero */
|
|
werr[1] = 0.f;
|
|
wgap[1] = 0.f;
|
|
iblock[1] = 1;
|
|
indexw[1] = 1;
|
|
gers[1] = d__[1];
|
|
gers[2] = d__[1];
|
|
}
|
|
/* store the shift for the initial RRR, which is zero in this case */
|
|
e[1] = 0.f;
|
|
return;
|
|
}
|
|
/* General case: tridiagonal matrix of order > 1 */
|
|
|
|
/* Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter. */
|
|
/* Compute maximum off-diagonal entry and pivmin. */
|
|
gl = d__[1];
|
|
gu = d__[1];
|
|
eold = 0.f;
|
|
emax = 0.f;
|
|
e[*n] = 0.f;
|
|
i__1 = *n;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
werr[i__] = 0.f;
|
|
wgap[i__] = 0.f;
|
|
eabs = (r__1 = e[i__], abs(r__1));
|
|
if (eabs >= emax) {
|
|
emax = eabs;
|
|
}
|
|
tmp1 = eabs + eold;
|
|
gers[(i__ << 1) - 1] = d__[i__] - tmp1;
|
|
/* Computing MIN */
|
|
r__1 = gl, r__2 = gers[(i__ << 1) - 1];
|
|
gl = f2cmin(r__1,r__2);
|
|
gers[i__ * 2] = d__[i__] + tmp1;
|
|
/* Computing MAX */
|
|
r__1 = gu, r__2 = gers[i__ * 2];
|
|
gu = f2cmax(r__1,r__2);
|
|
eold = eabs;
|
|
/* L5: */
|
|
}
|
|
/* The minimum pivot allowed in the Sturm sequence for T */
|
|
/* Computing MAX */
|
|
/* Computing 2nd power */
|
|
r__3 = emax;
|
|
r__1 = 1.f, r__2 = r__3 * r__3;
|
|
*pivmin = safmin * f2cmax(r__1,r__2);
|
|
/* Compute spectral diameter. The Gerschgorin bounds give an */
|
|
/* estimate that is wrong by at most a factor of SQRT(2) */
|
|
spdiam = gu - gl;
|
|
/* Compute splitting points */
|
|
slarra_(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], &
|
|
iinfo);
|
|
/* Can force use of bisection instead of faster DQDS. */
|
|
/* Option left in the code for future multisection work. */
|
|
forceb = FALSE_;
|
|
/* Initialize USEDQD, DQDS should be used for ALLRNG unless someone */
|
|
/* explicitly wants bisection. */
|
|
usedqd = irange == 1 && ! forceb;
|
|
if (irange == 1 && ! forceb) {
|
|
/* Set interval [VL,VU] that contains all eigenvalues */
|
|
*vl = gl;
|
|
*vu = gu;
|
|
} else {
|
|
/* We call SLARRD to find crude approximations to the eigenvalues */
|
|
/* in the desired range. In case IRANGE = INDRNG, we also obtain the */
|
|
/* interval (VL,VU] that contains all the wanted eigenvalues. */
|
|
/* An interval [LEFT,RIGHT] has converged if */
|
|
/* RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT)) */
|
|
/* SLARRD needs a WORK of size 4*N, IWORK of size 3*N */
|
|
slarrd_(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[
|
|
1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1],
|
|
vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return;
|
|
}
|
|
/* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
|
|
i__1 = *n;
|
|
for (i__ = mm + 1; i__ <= i__1; ++i__) {
|
|
w[i__] = 0.f;
|
|
werr[i__] = 0.f;
|
|
iblock[i__] = 0;
|
|
indexw[i__] = 0;
|
|
/* L14: */
|
|
}
|
|
}
|
|
/* ** */
|
|
/* Loop over unreduced blocks */
|
|
ibegin = 1;
|
|
wbegin = 1;
|
|
i__1 = *nsplit;
|
|
for (jblk = 1; jblk <= i__1; ++jblk) {
|
|
iend = isplit[jblk];
|
|
in = iend - ibegin + 1;
|
|
/* 1 X 1 block */
|
|
if (in == 1) {
|
|
if (irange == 1 || irange == 3 && d__[ibegin] > *vl && d__[ibegin]
|
|
<= *vu || irange == 2 && iblock[wbegin] == jblk) {
|
|
++(*m);
|
|
w[*m] = d__[ibegin];
|
|
werr[*m] = 0.f;
|
|
/* The gap for a single block doesn't matter for the later */
|
|
/* algorithm and is assigned an arbitrary large value */
|
|
wgap[*m] = 0.f;
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = 1;
|
|
++wbegin;
|
|
}
|
|
/* E( IEND ) holds the shift for the initial RRR */
|
|
e[iend] = 0.f;
|
|
ibegin = iend + 1;
|
|
goto L170;
|
|
}
|
|
|
|
/* Blocks of size larger than 1x1 */
|
|
|
|
/* E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
|
|
e[iend] = 0.f;
|
|
|
|
/* Find local outer bounds GL,GU for the block */
|
|
gl = d__[ibegin];
|
|
gu = d__[ibegin];
|
|
i__2 = iend;
|
|
for (i__ = ibegin; i__ <= i__2; ++i__) {
|
|
/* Computing MIN */
|
|
r__1 = gers[(i__ << 1) - 1];
|
|
gl = f2cmin(r__1,gl);
|
|
/* Computing MAX */
|
|
r__1 = gers[i__ * 2];
|
|
gu = f2cmax(r__1,gu);
|
|
/* L15: */
|
|
}
|
|
spdiam = gu - gl;
|
|
if (! (irange == 1 && ! forceb)) {
|
|
/* Count the number of eigenvalues in the current block. */
|
|
mb = 0;
|
|
i__2 = mm;
|
|
for (i__ = wbegin; i__ <= i__2; ++i__) {
|
|
if (iblock[i__] == jblk) {
|
|
++mb;
|
|
} else {
|
|
goto L21;
|
|
}
|
|
/* L20: */
|
|
}
|
|
L21:
|
|
if (mb == 0) {
|
|
/* No eigenvalue in the current block lies in the desired range */
|
|
/* E( IEND ) holds the shift for the initial RRR */
|
|
e[iend] = 0.f;
|
|
ibegin = iend + 1;
|
|
goto L170;
|
|
} else {
|
|
/* Decide whether dqds or bisection is more efficient */
|
|
usedqd = (real) mb > in * .5f && ! forceb;
|
|
wend = wbegin + mb - 1;
|
|
/* Calculate gaps for the current block */
|
|
/* In later stages, when representations for individual */
|
|
/* eigenvalues are different, we use SIGMA = E( IEND ). */
|
|
sigma = 0.f;
|
|
i__2 = wend - 1;
|
|
for (i__ = wbegin; i__ <= i__2; ++i__) {
|
|
/* Computing MAX */
|
|
r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
|
|
werr[i__]);
|
|
wgap[i__] = f2cmax(r__1,r__2);
|
|
/* L30: */
|
|
}
|
|
/* Computing MAX */
|
|
r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
|
|
wgap[wend] = f2cmax(r__1,r__2);
|
|
/* Find local index of the first and last desired evalue. */
|
|
indl = indexw[wbegin];
|
|
indu = indexw[wend];
|
|
}
|
|
}
|
|
if (irange == 1 && ! forceb || usedqd) {
|
|
/* Case of DQDS */
|
|
/* Find approximations to the extremal eigenvalues of the block */
|
|
slarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
|
|
rtl, &tmp, &tmp1, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return;
|
|
}
|
|
/* Computing MAX */
|
|
r__2 = gl, r__3 = tmp - tmp1 - eps * 100.f * (r__1 = tmp - tmp1,
|
|
abs(r__1));
|
|
isleft = f2cmax(r__2,r__3);
|
|
slarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
|
|
rtl, &tmp, &tmp1, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return;
|
|
}
|
|
/* Computing MIN */
|
|
r__2 = gu, r__3 = tmp + tmp1 + eps * 100.f * (r__1 = tmp + tmp1,
|
|
abs(r__1));
|
|
isrght = f2cmin(r__2,r__3);
|
|
/* Improve the estimate of the spectral diameter */
|
|
spdiam = isrght - isleft;
|
|
} else {
|
|
/* Case of bisection */
|
|
/* Find approximations to the wanted extremal eigenvalues */
|
|
/* Computing MAX */
|
|
r__2 = gl, r__3 = w[wbegin] - werr[wbegin] - eps * 100.f * (r__1 =
|
|
w[wbegin] - werr[wbegin], abs(r__1));
|
|
isleft = f2cmax(r__2,r__3);
|
|
/* Computing MIN */
|
|
r__2 = gu, r__3 = w[wend] + werr[wend] + eps * 100.f * (r__1 = w[
|
|
wend] + werr[wend], abs(r__1));
|
|
isrght = f2cmin(r__2,r__3);
|
|
}
|
|
/* Decide whether the base representation for the current block */
|
|
/* L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I */
|
|
/* should be on the left or the right end of the current block. */
|
|
/* The strategy is to shift to the end which is "more populated" */
|
|
/* Furthermore, decide whether to use DQDS for the computation of */
|
|
/* the eigenvalue approximations at the end of SLARRE or bisection. */
|
|
/* dqds is chosen if all eigenvalues are desired or the number of */
|
|
/* eigenvalues to be computed is large compared to the blocksize. */
|
|
if (irange == 1 && ! forceb) {
|
|
/* If all the eigenvalues have to be computed, we use dqd */
|
|
usedqd = TRUE_;
|
|
/* INDL is the local index of the first eigenvalue to compute */
|
|
indl = 1;
|
|
indu = in;
|
|
/* MB = number of eigenvalues to compute */
|
|
mb = in;
|
|
wend = wbegin + mb - 1;
|
|
/* Define 1/4 and 3/4 points of the spectrum */
|
|
s1 = isleft + spdiam * .25f;
|
|
s2 = isrght - spdiam * .25f;
|
|
} else {
|
|
/* SLARRD has computed IBLOCK and INDEXW for each eigenvalue */
|
|
/* approximation. */
|
|
/* choose sigma */
|
|
if (usedqd) {
|
|
s1 = isleft + spdiam * .25f;
|
|
s2 = isrght - spdiam * .25f;
|
|
} else {
|
|
tmp = f2cmin(isrght,*vu) - f2cmax(isleft,*vl);
|
|
s1 = f2cmax(isleft,*vl) + tmp * .25f;
|
|
s2 = f2cmin(isrght,*vu) - tmp * .25f;
|
|
}
|
|
}
|
|
/* Compute the negcount at the 1/4 and 3/4 points */
|
|
if (mb > 1) {
|
|
slarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
|
|
cnt, &cnt1, &cnt2, &iinfo);
|
|
}
|
|
if (mb == 1) {
|
|
sigma = gl;
|
|
sgndef = 1.f;
|
|
} else if (cnt1 - indl >= indu - cnt2) {
|
|
if (irange == 1 && ! forceb) {
|
|
sigma = f2cmax(isleft,gl);
|
|
} else if (usedqd) {
|
|
/* use Gerschgorin bound as shift to get pos def matrix */
|
|
/* for dqds */
|
|
sigma = isleft;
|
|
} else {
|
|
/* use approximation of the first desired eigenvalue of the */
|
|
/* block as shift */
|
|
sigma = f2cmax(isleft,*vl);
|
|
}
|
|
sgndef = 1.f;
|
|
} else {
|
|
if (irange == 1 && ! forceb) {
|
|
sigma = f2cmin(isrght,gu);
|
|
} else if (usedqd) {
|
|
/* use Gerschgorin bound as shift to get neg def matrix */
|
|
/* for dqds */
|
|
sigma = isrght;
|
|
} else {
|
|
/* use approximation of the first desired eigenvalue of the */
|
|
/* block as shift */
|
|
sigma = f2cmin(isrght,*vu);
|
|
}
|
|
sgndef = -1.f;
|
|
}
|
|
/* An initial SIGMA has been chosen that will be used for computing */
|
|
/* T - SIGMA I = L D L^T */
|
|
/* Define the increment TAU of the shift in case the initial shift */
|
|
/* needs to be refined to obtain a factorization with not too much */
|
|
/* element growth. */
|
|
if (usedqd) {
|
|
/* The initial SIGMA was to the outer end of the spectrum */
|
|
/* the matrix is definite and we need not retreat. */
|
|
tau = spdiam * eps * *n + *pivmin * 2.f;
|
|
/* Computing MAX */
|
|
r__1 = tau, r__2 = eps * 2.f * abs(sigma);
|
|
tau = f2cmax(r__1,r__2);
|
|
} else {
|
|
if (mb > 1) {
|
|
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
|
|
avgap = (r__1 = clwdth / (real) (wend - wbegin), abs(r__1));
|
|
if (sgndef == 1.f) {
|
|
/* Computing MAX */
|
|
r__1 = wgap[wbegin];
|
|
tau = f2cmax(r__1,avgap) * .5f;
|
|
/* Computing MAX */
|
|
r__1 = tau, r__2 = werr[wbegin];
|
|
tau = f2cmax(r__1,r__2);
|
|
} else {
|
|
/* Computing MAX */
|
|
r__1 = wgap[wend - 1];
|
|
tau = f2cmax(r__1,avgap) * .5f;
|
|
/* Computing MAX */
|
|
r__1 = tau, r__2 = werr[wend];
|
|
tau = f2cmax(r__1,r__2);
|
|
}
|
|
} else {
|
|
tau = werr[wbegin];
|
|
}
|
|
}
|
|
|
|
for (idum = 1; idum <= 6; ++idum) {
|
|
/* Compute L D L^T factorization of tridiagonal matrix T - sigma I. */
|
|
/* Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of */
|
|
/* pivots in WORK(2*IN+1:3*IN) */
|
|
dpivot = d__[ibegin] - sigma;
|
|
work[1] = dpivot;
|
|
dmax__ = abs(work[1]);
|
|
j = ibegin;
|
|
i__2 = in - 1;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
work[(in << 1) + i__] = 1.f / work[i__];
|
|
tmp = e[j] * work[(in << 1) + i__];
|
|
work[in + i__] = tmp;
|
|
dpivot = d__[j + 1] - sigma - tmp * e[j];
|
|
work[i__ + 1] = dpivot;
|
|
/* Computing MAX */
|
|
r__1 = dmax__, r__2 = abs(dpivot);
|
|
dmax__ = f2cmax(r__1,r__2);
|
|
++j;
|
|
/* L70: */
|
|
}
|
|
/* check for element growth */
|
|
if (dmax__ > spdiam * 64.f) {
|
|
norep = TRUE_;
|
|
} else {
|
|
norep = FALSE_;
|
|
}
|
|
if (usedqd && ! norep) {
|
|
/* Ensure the definiteness of the representation */
|
|
/* All entries of D (of L D L^T) must have the same sign */
|
|
i__2 = in;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
tmp = sgndef * work[i__];
|
|
if (tmp < 0.f) {
|
|
norep = TRUE_;
|
|
}
|
|
/* L71: */
|
|
}
|
|
}
|
|
if (norep) {
|
|
/* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin */
|
|
/* shift which makes the matrix definite. So we should end up */
|
|
/* here really only in the case of IRANGE = VALRNG or INDRNG. */
|
|
if (idum == 5) {
|
|
if (sgndef == 1.f) {
|
|
/* The fudged Gerschgorin shift should succeed */
|
|
sigma = gl - spdiam * 2.f * eps * *n - *pivmin * 4.f;
|
|
} else {
|
|
sigma = gu + spdiam * 2.f * eps * *n + *pivmin * 4.f;
|
|
}
|
|
} else {
|
|
sigma -= sgndef * tau;
|
|
tau *= 2.f;
|
|
}
|
|
} else {
|
|
/* an initial RRR is found */
|
|
goto L83;
|
|
}
|
|
/* L80: */
|
|
}
|
|
/* if the program reaches this point, no base representation could be */
|
|
/* found in MAXTRY iterations. */
|
|
*info = 2;
|
|
return;
|
|
L83:
|
|
/* At this point, we have found an initial base representation */
|
|
/* T - SIGMA I = L D L^T with not too much element growth. */
|
|
/* Store the shift. */
|
|
e[iend] = sigma;
|
|
/* Store D and L. */
|
|
scopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
|
|
i__2 = in - 1;
|
|
scopy_(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
|
|
if (mb > 1) {
|
|
|
|
/* Perturb each entry of the base representation by a small */
|
|
/* (but random) relative amount to overcome difficulties with */
|
|
/* glued matrices. */
|
|
|
|
for (i__ = 1; i__ <= 4; ++i__) {
|
|
iseed[i__ - 1] = 1;
|
|
/* L122: */
|
|
}
|
|
i__2 = (in << 1) - 1;
|
|
slarnv_(&c__2, iseed, &i__2, &work[1]);
|
|
i__2 = in - 1;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
d__[ibegin + i__ - 1] *= eps * 4.f * work[i__] + 1.f;
|
|
e[ibegin + i__ - 1] *= eps * 4.f * work[in + i__] + 1.f;
|
|
/* L125: */
|
|
}
|
|
d__[iend] *= eps * 4.f * work[in] + 1.f;
|
|
|
|
}
|
|
|
|
/* Don't update the Gerschgorin intervals because keeping track */
|
|
/* of the updates would be too much work in SLARRV. */
|
|
/* We update W instead and use it to locate the proper Gerschgorin */
|
|
/* intervals. */
|
|
/* Compute the required eigenvalues of L D L' by bisection or dqds */
|
|
if (! usedqd) {
|
|
/* If SLARRD has been used, shift the eigenvalue approximations */
|
|
/* according to their representation. This is necessary for */
|
|
/* a uniform SLARRV since dqds computes eigenvalues of the */
|
|
/* shifted representation. In SLARRV, W will always hold the */
|
|
/* UNshifted eigenvalue approximation. */
|
|
i__2 = wend;
|
|
for (j = wbegin; j <= i__2; ++j) {
|
|
w[j] -= sigma;
|
|
werr[j] += (r__1 = w[j], abs(r__1)) * eps;
|
|
/* L134: */
|
|
}
|
|
/* call SLARRB to reduce eigenvalue error of the approximations */
|
|
/* from SLARRD */
|
|
i__2 = iend - 1;
|
|
for (i__ = ibegin; i__ <= i__2; ++i__) {
|
|
/* Computing 2nd power */
|
|
r__1 = e[i__];
|
|
work[i__] = d__[i__] * (r__1 * r__1);
|
|
/* L135: */
|
|
}
|
|
/* use bisection to find EV from INDL to INDU */
|
|
i__2 = indl - 1;
|
|
slarrb_(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1,
|
|
rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], &
|
|
work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, &
|
|
iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -4;
|
|
return;
|
|
}
|
|
/* SLARRB computes all gaps correctly except for the last one */
|
|
/* Record distance to VU/GU */
|
|
/* Computing MAX */
|
|
r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
|
|
wgap[wend] = f2cmax(r__1,r__2);
|
|
i__2 = indu;
|
|
for (i__ = indl; i__ <= i__2; ++i__) {
|
|
++(*m);
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = i__;
|
|
/* L138: */
|
|
}
|
|
} else {
|
|
/* Call dqds to get all eigs (and then possibly delete unwanted */
|
|
/* eigenvalues). */
|
|
/* Note that dqds finds the eigenvalues of the L D L^T representation */
|
|
/* of T to high relative accuracy. High relative accuracy */
|
|
/* might be lost when the shift of the RRR is subtracted to obtain */
|
|
/* the eigenvalues of T. However, T is not guaranteed to define its */
|
|
/* eigenvalues to high relative accuracy anyway. */
|
|
/* Set RTOL to the order of the tolerance used in SLASQ2 */
|
|
/* This is an ESTIMATED error, the worst case bound is 4*N*EPS */
|
|
/* which is usually too large and requires unnecessary work to be */
|
|
/* done by bisection when computing the eigenvectors */
|
|
rtol = log((real) in) * 4.f * eps;
|
|
j = ibegin;
|
|
i__2 = in - 1;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
work[(i__ << 1) - 1] = (r__1 = d__[j], abs(r__1));
|
|
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
|
|
++j;
|
|
/* L140: */
|
|
}
|
|
work[(in << 1) - 1] = (r__1 = d__[iend], abs(r__1));
|
|
work[in * 2] = 0.f;
|
|
slasq2_(&in, &work[1], &iinfo);
|
|
if (iinfo != 0) {
|
|
/* If IINFO = -5 then an index is part of a tight cluster */
|
|
/* and should be changed. The index is in IWORK(1) and the */
|
|
/* gap is in WORK(N+1) */
|
|
*info = -5;
|
|
return;
|
|
} else {
|
|
/* Test that all eigenvalues are positive as expected */
|
|
i__2 = in;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
if (work[i__] < 0.f) {
|
|
*info = -6;
|
|
return;
|
|
}
|
|
/* L149: */
|
|
}
|
|
}
|
|
if (sgndef > 0.f) {
|
|
i__2 = indu;
|
|
for (i__ = indl; i__ <= i__2; ++i__) {
|
|
++(*m);
|
|
w[*m] = work[in - i__ + 1];
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = i__;
|
|
/* L150: */
|
|
}
|
|
} else {
|
|
i__2 = indu;
|
|
for (i__ = indl; i__ <= i__2; ++i__) {
|
|
++(*m);
|
|
w[*m] = -work[i__];
|
|
iblock[*m] = jblk;
|
|
indexw[*m] = i__;
|
|
/* L160: */
|
|
}
|
|
}
|
|
i__2 = *m;
|
|
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
|
|
/* the value of RTOL below should be the tolerance in SLASQ2 */
|
|
werr[i__] = rtol * (r__1 = w[i__], abs(r__1));
|
|
/* L165: */
|
|
}
|
|
i__2 = *m - 1;
|
|
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
|
|
/* compute the right gap between the intervals */
|
|
/* Computing MAX */
|
|
r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
|
|
werr[i__]);
|
|
wgap[i__] = f2cmax(r__1,r__2);
|
|
/* L166: */
|
|
}
|
|
/* Computing MAX */
|
|
r__1 = 0.f, r__2 = *vu - sigma - (w[*m] + werr[*m]);
|
|
wgap[*m] = f2cmax(r__1,r__2);
|
|
}
|
|
/* proceed with next block */
|
|
ibegin = iend + 1;
|
|
wbegin = wend + 1;
|
|
L170:
|
|
;
|
|
}
|
|
|
|
return;
|
|
|
|
/* end of SLARRE */
|
|
|
|
} /* slarre_ */
|
|
|