1619 lines
54 KiB
C
1619 lines
54 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_b5 = 0.;
|
|
static integer c__1 = 1;
|
|
static integer c__2 = 2;
|
|
|
|
/* > \brief \b DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenv
|
|
alues of L D LT. */
|
|
|
|
/* =========== DOCUMENTATION =========== */
|
|
|
|
/* Online html documentation available at */
|
|
/* http://www.netlib.org/lapack/explore-html/ */
|
|
|
|
/* > \htmlonly */
|
|
/* > Download DLARRV + dependencies */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrv.
|
|
f"> */
|
|
/* > [TGZ]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrv.
|
|
f"> */
|
|
/* > [ZIP]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrv.
|
|
f"> */
|
|
/* > [TXT]</a> */
|
|
/* > \endhtmlonly */
|
|
|
|
/* Definition: */
|
|
/* =========== */
|
|
|
|
/* SUBROUTINE DLARRV( N, VL, VU, D, L, PIVMIN, */
|
|
/* ISPLIT, M, DOL, DOU, MINRGP, */
|
|
/* RTOL1, RTOL2, W, WERR, WGAP, */
|
|
/* IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, */
|
|
/* WORK, IWORK, INFO ) */
|
|
|
|
/* INTEGER DOL, DOU, INFO, LDZ, M, N */
|
|
/* DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU */
|
|
/* INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), */
|
|
/* $ ISUPPZ( * ), IWORK( * ) */
|
|
/* DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ), */
|
|
/* $ WGAP( * ), WORK( * ) */
|
|
/* DOUBLE PRECISION Z( LDZ, * ) */
|
|
|
|
|
|
/* > \par Purpose: */
|
|
/* ============= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > DLARRV computes the eigenvectors of the tridiagonal matrix */
|
|
/* > T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. */
|
|
/* > The input eigenvalues should have been computed by DLARRE. */
|
|
/* > \endverbatim */
|
|
|
|
/* Arguments: */
|
|
/* ========== */
|
|
|
|
/* > \param[in] N */
|
|
/* > \verbatim */
|
|
/* > N is INTEGER */
|
|
/* > The order of the matrix. N >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] VL */
|
|
/* > \verbatim */
|
|
/* > VL is DOUBLE PRECISION */
|
|
/* > Lower bound of the interval that contains the desired */
|
|
/* > eigenvalues. VL < VU. Needed to compute gaps on the left or right */
|
|
/* > end of the extremal eigenvalues in the desired RANGE. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] VU */
|
|
/* > \verbatim */
|
|
/* > VU is DOUBLE PRECISION */
|
|
/* > Upper bound of the interval that contains the desired */
|
|
/* > eigenvalues. VL < VU. */
|
|
/* > Note: VU is currently not used by this implementation of DLARRV, VU is */
|
|
/* > passed to DLARRV because it could be used compute gaps on the right end */
|
|
/* > of the extremal eigenvalues. However, with not much initial accuracy in */
|
|
/* > LAMBDA and VU, the formula can lead to an overestimation of the right gap */
|
|
/* > and thus to inadequately early RQI 'convergence'. This is currently */
|
|
/* > prevented this by forcing a small right gap. And so it turns out that VU */
|
|
/* > is currently not used by this implementation of DLARRV. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] D */
|
|
/* > \verbatim */
|
|
/* > D is DOUBLE PRECISION array, dimension (N) */
|
|
/* > On entry, the N diagonal elements of the diagonal matrix D. */
|
|
/* > On exit, D may be overwritten. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] L */
|
|
/* > \verbatim */
|
|
/* > L is DOUBLE PRECISION array, dimension (N) */
|
|
/* > On entry, the (N-1) subdiagonal elements of the unit */
|
|
/* > bidiagonal matrix L are in elements 1 to N-1 of L */
|
|
/* > (if the matrix is not split.) At the end of each block */
|
|
/* > is stored the corresponding shift as given by DLARRE. */
|
|
/* > On exit, L is overwritten. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] PIVMIN */
|
|
/* > \verbatim */
|
|
/* > PIVMIN is DOUBLE PRECISION */
|
|
/* > The minimum pivot allowed in the Sturm sequence. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] 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. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] M */
|
|
/* > \verbatim */
|
|
/* > M is INTEGER */
|
|
/* > The total number of input eigenvalues. 0 <= M <= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] DOL */
|
|
/* > \verbatim */
|
|
/* > DOL is INTEGER */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] DOU */
|
|
/* > \verbatim */
|
|
/* > DOU is INTEGER */
|
|
/* > If the user wants to compute only selected eigenvectors from all */
|
|
/* > the eigenvalues supplied, he can specify an index range DOL:DOU. */
|
|
/* > Or else the setting DOL=1, DOU=M should be applied. */
|
|
/* > Note that DOL and DOU refer to the order in which the eigenvalues */
|
|
/* > are stored in W. */
|
|
/* > If the user wants to compute only selected eigenpairs, then */
|
|
/* > the columns DOL-1 to DOU+1 of the eigenvector space Z contain the */
|
|
/* > computed eigenvectors. All other columns of Z are set to zero. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] MINRGP */
|
|
/* > \verbatim */
|
|
/* > MINRGP is DOUBLE PRECISION */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] RTOL1 */
|
|
/* > \verbatim */
|
|
/* > RTOL1 is DOUBLE PRECISION */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] RTOL2 */
|
|
/* > \verbatim */
|
|
/* > RTOL2 is DOUBLE PRECISION */
|
|
/* > Parameters for bisection. */
|
|
/* > An interval [LEFT,RIGHT] has converged if */
|
|
/* > RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] W */
|
|
/* > \verbatim */
|
|
/* > W is DOUBLE PRECISION array, dimension (N) */
|
|
/* > The first M elements of W contain the APPROXIMATE eigenvalues for */
|
|
/* > which eigenvectors are to be computed. The eigenvalues */
|
|
/* > should be grouped by split-off block and ordered from */
|
|
/* > smallest to largest within the block ( The output array */
|
|
/* > W from DLARRE is expected here ). Furthermore, they are with */
|
|
/* > respect to the shift of the corresponding root representation */
|
|
/* > for their block. On exit, W holds the eigenvalues of the */
|
|
/* > UNshifted matrix. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] WERR */
|
|
/* > \verbatim */
|
|
/* > WERR is DOUBLE PRECISION array, dimension (N) */
|
|
/* > The first M elements contain the semiwidth of the uncertainty */
|
|
/* > interval of the corresponding eigenvalue in W */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] WGAP */
|
|
/* > \verbatim */
|
|
/* > WGAP is DOUBLE PRECISION array, dimension (N) */
|
|
/* > The separation from the right neighbor eigenvalue in W. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] 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[in] 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 the second block. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] GERS */
|
|
/* > \verbatim */
|
|
/* > GERS is DOUBLE PRECISION array, dimension (2*N) */
|
|
/* > The N Gerschgorin intervals (the i-th Gerschgorin interval */
|
|
/* > is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should */
|
|
/* > be computed from the original UNshifted matrix. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] Z */
|
|
/* > \verbatim */
|
|
/* > Z is DOUBLE PRECISION array, dimension (LDZ, f2cmax(1,M) ) */
|
|
/* > If INFO = 0, the first M columns of Z contain the */
|
|
/* > orthonormal eigenvectors of the matrix T */
|
|
/* > corresponding to the input eigenvalues, with the i-th */
|
|
/* > column of Z holding the eigenvector associated with W(i). */
|
|
/* > Note: the user must ensure that at least f2cmax(1,M) columns are */
|
|
/* > supplied in the array Z. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDZ */
|
|
/* > \verbatim */
|
|
/* > LDZ is INTEGER */
|
|
/* > The leading dimension of the array Z. LDZ >= 1, and if */
|
|
/* > JOBZ = 'V', LDZ >= f2cmax(1,N). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] ISUPPZ */
|
|
/* > \verbatim */
|
|
/* > ISUPPZ is INTEGER array, dimension ( 2*f2cmax(1,M) ) */
|
|
/* > The support of the eigenvectors in Z, i.e., the indices */
|
|
/* > indicating the nonzero elements in Z. The I-th eigenvector */
|
|
/* > is nonzero only in elements ISUPPZ( 2*I-1 ) through */
|
|
/* > ISUPPZ( 2*I ). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WORK */
|
|
/* > \verbatim */
|
|
/* > WORK is DOUBLE PRECISION array, dimension (12*N) */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] IWORK */
|
|
/* > \verbatim */
|
|
/* > IWORK is INTEGER array, dimension (7*N) */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] INFO */
|
|
/* > \verbatim */
|
|
/* > INFO is INTEGER */
|
|
/* > = 0: successful exit */
|
|
/* > */
|
|
/* > > 0: A problem occurred in DLARRV. */
|
|
/* > < 0: One of the called subroutines signaled an internal problem. */
|
|
/* > Needs inspection of the corresponding parameter IINFO */
|
|
/* > for further information. */
|
|
/* > */
|
|
/* > =-1: Problem in DLARRB when refining a child's eigenvalues. */
|
|
/* > =-2: Problem in DLARRF when computing the RRR of a child. */
|
|
/* > When a child is inside a tight cluster, it can be difficult */
|
|
/* > to find an RRR. A partial remedy from the user's point of */
|
|
/* > view is to make the parameter MINRGP smaller and recompile. */
|
|
/* > However, as the orthogonality of the computed vectors is */
|
|
/* > proportional to 1/MINRGP, the user should be aware that */
|
|
/* > he might be trading in precision when he decreases MINRGP. */
|
|
/* > =-3: Problem in DLARRB when refining a single eigenvalue */
|
|
/* > after the Rayleigh correction was rejected. */
|
|
/* > = 5: The Rayleigh Quotient Iteration failed to converge to */
|
|
/* > full accuracy in MAXITR steps. */
|
|
/* > \endverbatim */
|
|
|
|
/* Authors: */
|
|
/* ======== */
|
|
|
|
/* > \author Univ. of Tennessee */
|
|
/* > \author Univ. of California Berkeley */
|
|
/* > \author Univ. of Colorado Denver */
|
|
/* > \author NAG Ltd. */
|
|
|
|
/* > \date June 2016 */
|
|
|
|
/* > \ingroup doubleOTHERauxiliary */
|
|
|
|
/* > \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 */
|
|
|
|
/* ===================================================================== */
|
|
/* Subroutine */ void dlarrv_(integer *n, doublereal *vl, doublereal *vu,
|
|
doublereal *d__, doublereal *l, doublereal *pivmin, integer *isplit,
|
|
integer *m, integer *dol, integer *dou, doublereal *minrgp,
|
|
doublereal *rtol1, doublereal *rtol2, doublereal *w, doublereal *werr,
|
|
doublereal *wgap, integer *iblock, integer *indexw, doublereal *gers,
|
|
doublereal *z__, integer *ldz, integer *isuppz, doublereal *work,
|
|
integer *iwork, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
|
|
doublereal d__1, d__2;
|
|
logical L__1;
|
|
|
|
/* Local variables */
|
|
integer iend, jblk;
|
|
doublereal lgap;
|
|
integer done;
|
|
doublereal rgap, left;
|
|
integer wend, iter;
|
|
doublereal bstw;
|
|
integer minwsize, itmp1, i__, j, k, p, q;
|
|
extern /* Subroutine */ void dscal_(integer *, doublereal *, doublereal *,
|
|
integer *);
|
|
integer indld;
|
|
doublereal fudge;
|
|
integer idone;
|
|
doublereal sigma;
|
|
integer iinfo, iindr;
|
|
doublereal resid;
|
|
logical eskip;
|
|
doublereal right;
|
|
extern /* Subroutine */ void dcopy_(integer *, doublereal *, integer *,
|
|
doublereal *, integer *);
|
|
integer nclus, zfrom;
|
|
doublereal rqtol;
|
|
integer iindc1, iindc2;
|
|
extern /* Subroutine */ void dlar1v_(integer *, integer *, integer *,
|
|
doublereal *, doublereal *, doublereal *, doublereal *,
|
|
doublereal *, doublereal *, doublereal *, doublereal *, logical *,
|
|
integer *, doublereal *, doublereal *, integer *, integer *,
|
|
doublereal *, doublereal *, doublereal *, doublereal *);
|
|
integer miniwsize;
|
|
logical stp2ii;
|
|
doublereal lambda;
|
|
integer ii;
|
|
doublereal gl;
|
|
integer im, in;
|
|
extern doublereal dlamch_(char *);
|
|
doublereal gu;
|
|
integer ibegin, indeig;
|
|
logical needbs;
|
|
integer indlld;
|
|
doublereal sgndef, mingma;
|
|
extern /* Subroutine */ void dlarrb_(integer *, doublereal *, doublereal *,
|
|
integer *, integer *, doublereal *, doublereal *, integer *,
|
|
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
|
|
doublereal *, doublereal *, integer *, integer *);
|
|
integer oldien, oldncl, wbegin, negcnt;
|
|
doublereal spdiam;
|
|
integer oldcls;
|
|
doublereal savgap;
|
|
integer ndepth;
|
|
doublereal ssigma;
|
|
extern /* Subroutine */ void dlarrf_(integer *, doublereal *, doublereal *,
|
|
doublereal *, integer *, integer *, doublereal *, doublereal *,
|
|
doublereal *, doublereal *, doublereal *, doublereal *,
|
|
doublereal *, doublereal *, doublereal *, doublereal *,
|
|
doublereal *, integer *);
|
|
logical usedbs;
|
|
integer iindwk, offset;
|
|
doublereal gaptol;
|
|
extern /* Subroutine */ void dlaset_(char *, integer *, integer *,
|
|
doublereal *, doublereal *, doublereal *, integer *);
|
|
integer newcls, oldfst, indwrk, windex, oldlst;
|
|
logical usedrq;
|
|
integer newfst, newftt, parity, windmn, windpl, isupmn, newlst, zusedl;
|
|
doublereal bstres;
|
|
integer newsiz, zusedu, zusedw;
|
|
doublereal nrminv, rqcorr;
|
|
logical tryrqc;
|
|
integer isupmx;
|
|
doublereal gap, eps, tau, tol, tmp;
|
|
integer zto;
|
|
doublereal ztz;
|
|
|
|
|
|
/* -- 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 */
|
|
--d__;
|
|
--l;
|
|
--isplit;
|
|
--w;
|
|
--werr;
|
|
--wgap;
|
|
--iblock;
|
|
--indexw;
|
|
--gers;
|
|
z_dim1 = *ldz;
|
|
z_offset = 1 + z_dim1 * 1;
|
|
z__ -= z_offset;
|
|
--isuppz;
|
|
--work;
|
|
--iwork;
|
|
|
|
/* Function Body */
|
|
*info = 0;
|
|
|
|
/* Quick return if possible */
|
|
|
|
if (*n <= 0 || *m <= 0) {
|
|
return;
|
|
}
|
|
|
|
/* The first N entries of WORK are reserved for the eigenvalues */
|
|
indld = *n + 1;
|
|
indlld = (*n << 1) + 1;
|
|
indwrk = *n * 3 + 1;
|
|
minwsize = *n * 12;
|
|
i__1 = minwsize;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
work[i__] = 0.;
|
|
/* L5: */
|
|
}
|
|
/* IWORK(IINDR+1:IINDR+N) hold the twist indices R for the */
|
|
/* factorization used to compute the FP vector */
|
|
iindr = 0;
|
|
/* IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current */
|
|
/* layer and the one above. */
|
|
iindc1 = *n;
|
|
iindc2 = *n << 1;
|
|
iindwk = *n * 3 + 1;
|
|
miniwsize = *n * 7;
|
|
i__1 = miniwsize;
|
|
for (i__ = 1; i__ <= i__1; ++i__) {
|
|
iwork[i__] = 0;
|
|
/* L10: */
|
|
}
|
|
zusedl = 1;
|
|
if (*dol > 1) {
|
|
/* Set lower bound for use of Z */
|
|
zusedl = *dol - 1;
|
|
}
|
|
zusedu = *m;
|
|
if (*dou < *m) {
|
|
/* Set lower bound for use of Z */
|
|
zusedu = *dou + 1;
|
|
}
|
|
/* The width of the part of Z that is used */
|
|
zusedw = zusedu - zusedl + 1;
|
|
dlaset_("Full", n, &zusedw, &c_b5, &c_b5, &z__[zusedl * z_dim1 + 1], ldz);
|
|
eps = dlamch_("Precision");
|
|
rqtol = eps * 2.;
|
|
|
|
/* Set expert flags for standard code. */
|
|
tryrqc = TRUE_;
|
|
if (*dol == 1 && *dou == *m) {
|
|
} else {
|
|
/* Only selected eigenpairs are computed. Since the other evalues */
|
|
/* are not refined by RQ iteration, bisection has to compute to full */
|
|
/* accuracy. */
|
|
*rtol1 = eps * 4.;
|
|
*rtol2 = eps * 4.;
|
|
}
|
|
/* The entries WBEGIN:WEND in W, WERR, WGAP correspond to the */
|
|
/* desired eigenvalues. The support of the nonzero eigenvector */
|
|
/* entries is contained in the interval IBEGIN:IEND. */
|
|
/* Remark that if k eigenpairs are desired, then the eigenvectors */
|
|
/* are stored in k contiguous columns of Z. */
|
|
/* DONE is the number of eigenvectors already computed */
|
|
done = 0;
|
|
ibegin = 1;
|
|
wbegin = 1;
|
|
i__1 = iblock[*m];
|
|
for (jblk = 1; jblk <= i__1; ++jblk) {
|
|
iend = isplit[jblk];
|
|
sigma = l[iend];
|
|
/* Find the eigenvectors of the submatrix indexed IBEGIN */
|
|
/* through IEND. */
|
|
wend = wbegin - 1;
|
|
L15:
|
|
if (wend < *m) {
|
|
if (iblock[wend + 1] == jblk) {
|
|
++wend;
|
|
goto L15;
|
|
}
|
|
}
|
|
if (wend < wbegin) {
|
|
ibegin = iend + 1;
|
|
goto L170;
|
|
} else if (wend < *dol || wbegin > *dou) {
|
|
ibegin = iend + 1;
|
|
wbegin = wend + 1;
|
|
goto L170;
|
|
}
|
|
/* Find local spectral diameter of the block */
|
|
gl = gers[(ibegin << 1) - 1];
|
|
gu = gers[ibegin * 2];
|
|
i__2 = iend;
|
|
for (i__ = ibegin + 1; i__ <= i__2; ++i__) {
|
|
/* Computing MIN */
|
|
d__1 = gers[(i__ << 1) - 1];
|
|
gl = f2cmin(d__1,gl);
|
|
/* Computing MAX */
|
|
d__1 = gers[i__ * 2];
|
|
gu = f2cmax(d__1,gu);
|
|
/* L20: */
|
|
}
|
|
spdiam = gu - gl;
|
|
/* OLDIEN is the last index of the previous block */
|
|
oldien = ibegin - 1;
|
|
/* Calculate the size of the current block */
|
|
in = iend - ibegin + 1;
|
|
/* The number of eigenvalues in the current block */
|
|
im = wend - wbegin + 1;
|
|
/* This is for a 1x1 block */
|
|
if (ibegin == iend) {
|
|
++done;
|
|
z__[ibegin + wbegin * z_dim1] = 1.;
|
|
isuppz[(wbegin << 1) - 1] = ibegin;
|
|
isuppz[wbegin * 2] = ibegin;
|
|
w[wbegin] += sigma;
|
|
work[wbegin] = w[wbegin];
|
|
ibegin = iend + 1;
|
|
++wbegin;
|
|
goto L170;
|
|
}
|
|
/* The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND) */
|
|
/* Note that these can be approximations, in this case, the corresp. */
|
|
/* entries of WERR give the size of the uncertainty interval. */
|
|
/* The eigenvalue approximations will be refined when necessary as */
|
|
/* high relative accuracy is required for the computation of the */
|
|
/* corresponding eigenvectors. */
|
|
dcopy_(&im, &w[wbegin], &c__1, &work[wbegin], &c__1);
|
|
/* We store in W the eigenvalue approximations w.r.t. the original */
|
|
/* matrix T. */
|
|
i__2 = im;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
w[wbegin + i__ - 1] += sigma;
|
|
/* L30: */
|
|
}
|
|
/* NDEPTH is the current depth of the representation tree */
|
|
ndepth = 0;
|
|
/* PARITY is either 1 or 0 */
|
|
parity = 1;
|
|
/* NCLUS is the number of clusters for the next level of the */
|
|
/* representation tree, we start with NCLUS = 1 for the root */
|
|
nclus = 1;
|
|
iwork[iindc1 + 1] = 1;
|
|
iwork[iindc1 + 2] = im;
|
|
/* IDONE is the number of eigenvectors already computed in the current */
|
|
/* block */
|
|
idone = 0;
|
|
/* loop while( IDONE.LT.IM ) */
|
|
/* generate the representation tree for the current block and */
|
|
/* compute the eigenvectors */
|
|
L40:
|
|
if (idone < im) {
|
|
/* This is a crude protection against infinitely deep trees */
|
|
if (ndepth > *m) {
|
|
*info = -2;
|
|
return;
|
|
}
|
|
/* breadth first processing of the current level of the representation */
|
|
/* tree: OLDNCL = number of clusters on current level */
|
|
oldncl = nclus;
|
|
/* reset NCLUS to count the number of child clusters */
|
|
nclus = 0;
|
|
|
|
parity = 1 - parity;
|
|
if (parity == 0) {
|
|
oldcls = iindc1;
|
|
newcls = iindc2;
|
|
} else {
|
|
oldcls = iindc2;
|
|
newcls = iindc1;
|
|
}
|
|
/* Process the clusters on the current level */
|
|
i__2 = oldncl;
|
|
for (i__ = 1; i__ <= i__2; ++i__) {
|
|
j = oldcls + (i__ << 1);
|
|
/* OLDFST, OLDLST = first, last index of current cluster. */
|
|
/* cluster indices start with 1 and are relative */
|
|
/* to WBEGIN when accessing W, WGAP, WERR, Z */
|
|
oldfst = iwork[j - 1];
|
|
oldlst = iwork[j];
|
|
if (ndepth > 0) {
|
|
/* Retrieve relatively robust representation (RRR) of cluster */
|
|
/* that has been computed at the previous level */
|
|
/* The RRR is stored in Z and overwritten once the eigenvectors */
|
|
/* have been computed or when the cluster is refined */
|
|
if (*dol == 1 && *dou == *m) {
|
|
/* Get representation from location of the leftmost evalue */
|
|
/* of the cluster */
|
|
j = wbegin + oldfst - 1;
|
|
} else {
|
|
if (wbegin + oldfst - 1 < *dol) {
|
|
/* Get representation from the left end of Z array */
|
|
j = *dol - 1;
|
|
} else if (wbegin + oldfst - 1 > *dou) {
|
|
/* Get representation from the right end of Z array */
|
|
j = *dou;
|
|
} else {
|
|
j = wbegin + oldfst - 1;
|
|
}
|
|
}
|
|
dcopy_(&in, &z__[ibegin + j * z_dim1], &c__1, &d__[ibegin]
|
|
, &c__1);
|
|
i__3 = in - 1;
|
|
dcopy_(&i__3, &z__[ibegin + (j + 1) * z_dim1], &c__1, &l[
|
|
ibegin], &c__1);
|
|
sigma = z__[iend + (j + 1) * z_dim1];
|
|
/* Set the corresponding entries in Z to zero */
|
|
dlaset_("Full", &in, &c__2, &c_b5, &c_b5, &z__[ibegin + j
|
|
* z_dim1], ldz);
|
|
}
|
|
/* Compute DL and DLL of current RRR */
|
|
i__3 = iend - 1;
|
|
for (j = ibegin; j <= i__3; ++j) {
|
|
tmp = d__[j] * l[j];
|
|
work[indld - 1 + j] = tmp;
|
|
work[indlld - 1 + j] = tmp * l[j];
|
|
/* L50: */
|
|
}
|
|
if (ndepth > 0) {
|
|
/* P and Q are index of the first and last eigenvalue to compute */
|
|
/* within the current block */
|
|
p = indexw[wbegin - 1 + oldfst];
|
|
q = indexw[wbegin - 1 + oldlst];
|
|
/* Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET */
|
|
/* through the Q-OFFSET elements of these arrays are to be used. */
|
|
/* OFFSET = P-OLDFST */
|
|
offset = indexw[wbegin] - 1;
|
|
/* perform limited bisection (if necessary) to get approximate */
|
|
/* eigenvalues to the precision needed. */
|
|
dlarrb_(&in, &d__[ibegin], &work[indlld + ibegin - 1], &p,
|
|
&q, rtol1, rtol2, &offset, &work[wbegin], &wgap[
|
|
wbegin], &werr[wbegin], &work[indwrk], &iwork[
|
|
iindwk], pivmin, &spdiam, &in, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -1;
|
|
return;
|
|
}
|
|
/* We also recompute the extremal gaps. W holds all eigenvalues */
|
|
/* of the unshifted matrix and must be used for computation */
|
|
/* of WGAP, the entries of WORK might stem from RRRs with */
|
|
/* different shifts. The gaps from WBEGIN-1+OLDFST to */
|
|
/* WBEGIN-1+OLDLST are correctly computed in DLARRB. */
|
|
/* However, we only allow the gaps to become greater since */
|
|
/* this is what should happen when we decrease WERR */
|
|
if (oldfst > 1) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[wbegin + oldfst - 2], d__2 = w[wbegin +
|
|
oldfst - 1] - werr[wbegin + oldfst - 1] - w[
|
|
wbegin + oldfst - 2] - werr[wbegin + oldfst -
|
|
2];
|
|
wgap[wbegin + oldfst - 2] = f2cmax(d__1,d__2);
|
|
}
|
|
if (wbegin + oldlst - 1 < wend) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[wbegin + oldlst - 1], d__2 = w[wbegin +
|
|
oldlst] - werr[wbegin + oldlst] - w[wbegin +
|
|
oldlst - 1] - werr[wbegin + oldlst - 1];
|
|
wgap[wbegin + oldlst - 1] = f2cmax(d__1,d__2);
|
|
}
|
|
/* Each time the eigenvalues in WORK get refined, we store */
|
|
/* the newly found approximation with all shifts applied in W */
|
|
i__3 = oldlst;
|
|
for (j = oldfst; j <= i__3; ++j) {
|
|
w[wbegin + j - 1] = work[wbegin + j - 1] + sigma;
|
|
/* L53: */
|
|
}
|
|
}
|
|
/* Process the current node. */
|
|
newfst = oldfst;
|
|
i__3 = oldlst;
|
|
for (j = oldfst; j <= i__3; ++j) {
|
|
if (j == oldlst) {
|
|
/* we are at the right end of the cluster, this is also the */
|
|
/* boundary of the child cluster */
|
|
newlst = j;
|
|
} else if (wgap[wbegin + j - 1] >= *minrgp * (d__1 = work[
|
|
wbegin + j - 1], abs(d__1))) {
|
|
/* the right relative gap is big enough, the child cluster */
|
|
/* (NEWFST,..,NEWLST) is well separated from the following */
|
|
newlst = j;
|
|
} else {
|
|
/* inside a child cluster, the relative gap is not */
|
|
/* big enough. */
|
|
goto L140;
|
|
}
|
|
/* Compute size of child cluster found */
|
|
newsiz = newlst - newfst + 1;
|
|
/* NEWFTT is the place in Z where the new RRR or the computed */
|
|
/* eigenvector is to be stored */
|
|
if (*dol == 1 && *dou == *m) {
|
|
/* Store representation at location of the leftmost evalue */
|
|
/* of the cluster */
|
|
newftt = wbegin + newfst - 1;
|
|
} else {
|
|
if (wbegin + newfst - 1 < *dol) {
|
|
/* Store representation at the left end of Z array */
|
|
newftt = *dol - 1;
|
|
} else if (wbegin + newfst - 1 > *dou) {
|
|
/* Store representation at the right end of Z array */
|
|
newftt = *dou;
|
|
} else {
|
|
newftt = wbegin + newfst - 1;
|
|
}
|
|
}
|
|
if (newsiz > 1) {
|
|
|
|
/* Current child is not a singleton but a cluster. */
|
|
/* Compute and store new representation of child. */
|
|
|
|
|
|
/* Compute left and right cluster gap. */
|
|
|
|
/* LGAP and RGAP are not computed from WORK because */
|
|
/* the eigenvalue approximations may stem from RRRs */
|
|
/* different shifts. However, W hold all eigenvalues */
|
|
/* of the unshifted matrix. Still, the entries in WGAP */
|
|
/* have to be computed from WORK since the entries */
|
|
/* in W might be of the same order so that gaps are not */
|
|
/* exhibited correctly for very close eigenvalues. */
|
|
if (newfst == 1) {
|
|
/* Computing MAX */
|
|
d__1 = 0., d__2 = w[wbegin] - werr[wbegin] - *vl;
|
|
lgap = f2cmax(d__1,d__2);
|
|
} else {
|
|
lgap = wgap[wbegin + newfst - 2];
|
|
}
|
|
rgap = wgap[wbegin + newlst - 1];
|
|
|
|
/* Compute left- and rightmost eigenvalue of child */
|
|
/* to high precision in order to shift as close */
|
|
/* as possible and obtain as large relative gaps */
|
|
/* as possible */
|
|
|
|
for (k = 1; k <= 2; ++k) {
|
|
if (k == 1) {
|
|
p = indexw[wbegin - 1 + newfst];
|
|
} else {
|
|
p = indexw[wbegin - 1 + newlst];
|
|
}
|
|
offset = indexw[wbegin] - 1;
|
|
dlarrb_(&in, &d__[ibegin], &work[indlld + ibegin
|
|
- 1], &p, &p, &rqtol, &rqtol, &offset, &
|
|
work[wbegin], &wgap[wbegin], &werr[wbegin]
|
|
, &work[indwrk], &iwork[iindwk], pivmin, &
|
|
spdiam, &in, &iinfo);
|
|
/* L55: */
|
|
}
|
|
|
|
if (wbegin + newlst - 1 < *dol || wbegin + newfst - 1
|
|
> *dou) {
|
|
/* if the cluster contains no desired eigenvalues */
|
|
/* skip the computation of that branch of the rep. tree */
|
|
|
|
/* We could skip before the refinement of the extremal */
|
|
/* eigenvalues of the child, but then the representation */
|
|
/* tree could be different from the one when nothing is */
|
|
/* skipped. For this reason we skip at this place. */
|
|
idone = idone + newlst - newfst + 1;
|
|
goto L139;
|
|
}
|
|
|
|
/* Compute RRR of child cluster. */
|
|
/* Note that the new RRR is stored in Z */
|
|
|
|
/* DLARRF needs LWORK = 2*N */
|
|
dlarrf_(&in, &d__[ibegin], &l[ibegin], &work[indld +
|
|
ibegin - 1], &newfst, &newlst, &work[wbegin],
|
|
&wgap[wbegin], &werr[wbegin], &spdiam, &lgap,
|
|
&rgap, pivmin, &tau, &z__[ibegin + newftt *
|
|
z_dim1], &z__[ibegin + (newftt + 1) * z_dim1],
|
|
&work[indwrk], &iinfo);
|
|
if (iinfo == 0) {
|
|
/* a new RRR for the cluster was found by DLARRF */
|
|
/* update shift and store it */
|
|
ssigma = sigma + tau;
|
|
z__[iend + (newftt + 1) * z_dim1] = ssigma;
|
|
/* WORK() are the midpoints and WERR() the semi-width */
|
|
/* Note that the entries in W are unchanged. */
|
|
i__4 = newlst;
|
|
for (k = newfst; k <= i__4; ++k) {
|
|
fudge = eps * 3. * (d__1 = work[wbegin + k -
|
|
1], abs(d__1));
|
|
work[wbegin + k - 1] -= tau;
|
|
fudge += eps * 4. * (d__1 = work[wbegin + k -
|
|
1], abs(d__1));
|
|
/* Fudge errors */
|
|
werr[wbegin + k - 1] += fudge;
|
|
/* Gaps are not fudged. Provided that WERR is small */
|
|
/* when eigenvalues are close, a zero gap indicates */
|
|
/* that a new representation is needed for resolving */
|
|
/* the cluster. A fudge could lead to a wrong decision */
|
|
/* of judging eigenvalues 'separated' which in */
|
|
/* reality are not. This could have a negative impact */
|
|
/* on the orthogonality of the computed eigenvectors. */
|
|
/* L116: */
|
|
}
|
|
++nclus;
|
|
k = newcls + (nclus << 1);
|
|
iwork[k - 1] = newfst;
|
|
iwork[k] = newlst;
|
|
} else {
|
|
*info = -2;
|
|
return;
|
|
}
|
|
} else {
|
|
|
|
/* Compute eigenvector of singleton */
|
|
|
|
iter = 0;
|
|
|
|
tol = log((doublereal) in) * 4. * eps;
|
|
|
|
k = newfst;
|
|
windex = wbegin + k - 1;
|
|
/* Computing MAX */
|
|
i__4 = windex - 1;
|
|
windmn = f2cmax(i__4,1);
|
|
/* Computing MIN */
|
|
i__4 = windex + 1;
|
|
windpl = f2cmin(i__4,*m);
|
|
lambda = work[windex];
|
|
++done;
|
|
/* Check if eigenvector computation is to be skipped */
|
|
if (windex < *dol || windex > *dou) {
|
|
eskip = TRUE_;
|
|
goto L125;
|
|
} else {
|
|
eskip = FALSE_;
|
|
}
|
|
left = work[windex] - werr[windex];
|
|
right = work[windex] + werr[windex];
|
|
indeig = indexw[windex];
|
|
/* Note that since we compute the eigenpairs for a child, */
|
|
/* all eigenvalue approximations are w.r.t the same shift. */
|
|
/* In this case, the entries in WORK should be used for */
|
|
/* computing the gaps since they exhibit even very small */
|
|
/* differences in the eigenvalues, as opposed to the */
|
|
/* entries in W which might "look" the same. */
|
|
if (k == 1) {
|
|
/* In the case RANGE='I' and with not much initial */
|
|
/* accuracy in LAMBDA and VL, the formula */
|
|
/* LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA ) */
|
|
/* can lead to an overestimation of the left gap and */
|
|
/* thus to inadequately early RQI 'convergence'. */
|
|
/* Prevent this by forcing a small left gap. */
|
|
/* Computing MAX */
|
|
d__1 = abs(left), d__2 = abs(right);
|
|
lgap = eps * f2cmax(d__1,d__2);
|
|
} else {
|
|
lgap = wgap[windmn];
|
|
}
|
|
if (k == im) {
|
|
/* In the case RANGE='I' and with not much initial */
|
|
/* accuracy in LAMBDA and VU, the formula */
|
|
/* can lead to an overestimation of the right gap and */
|
|
/* thus to inadequately early RQI 'convergence'. */
|
|
/* Prevent this by forcing a small right gap. */
|
|
/* Computing MAX */
|
|
d__1 = abs(left), d__2 = abs(right);
|
|
rgap = eps * f2cmax(d__1,d__2);
|
|
} else {
|
|
rgap = wgap[windex];
|
|
}
|
|
gap = f2cmin(lgap,rgap);
|
|
if (k == 1 || k == im) {
|
|
/* The eigenvector support can become wrong */
|
|
/* because significant entries could be cut off due to a */
|
|
/* large GAPTOL parameter in LAR1V. Prevent this. */
|
|
gaptol = 0.;
|
|
} else {
|
|
gaptol = gap * eps;
|
|
}
|
|
isupmn = in;
|
|
isupmx = 1;
|
|
/* Update WGAP so that it holds the minimum gap */
|
|
/* to the left or the right. This is crucial in the */
|
|
/* case where bisection is used to ensure that the */
|
|
/* eigenvalue is refined up to the required precision. */
|
|
/* The correct value is restored afterwards. */
|
|
savgap = wgap[windex];
|
|
wgap[windex] = gap;
|
|
/* We want to use the Rayleigh Quotient Correction */
|
|
/* as often as possible since it converges quadratically */
|
|
/* when we are close enough to the desired eigenvalue. */
|
|
/* However, the Rayleigh Quotient can have the wrong sign */
|
|
/* and lead us away from the desired eigenvalue. In this */
|
|
/* case, the best we can do is to use bisection. */
|
|
usedbs = FALSE_;
|
|
usedrq = FALSE_;
|
|
/* Bisection is initially turned off unless it is forced */
|
|
needbs = ! tryrqc;
|
|
L120:
|
|
/* Check if bisection should be used to refine eigenvalue */
|
|
if (needbs) {
|
|
/* Take the bisection as new iterate */
|
|
usedbs = TRUE_;
|
|
itmp1 = iwork[iindr + windex];
|
|
offset = indexw[wbegin] - 1;
|
|
d__1 = eps * 2.;
|
|
dlarrb_(&in, &d__[ibegin], &work[indlld + ibegin
|
|
- 1], &indeig, &indeig, &c_b5, &d__1, &
|
|
offset, &work[wbegin], &wgap[wbegin], &
|
|
werr[wbegin], &work[indwrk], &iwork[
|
|
iindwk], pivmin, &spdiam, &itmp1, &iinfo);
|
|
if (iinfo != 0) {
|
|
*info = -3;
|
|
return;
|
|
}
|
|
lambda = work[windex];
|
|
/* Reset twist index from inaccurate LAMBDA to */
|
|
/* force computation of true MINGMA */
|
|
iwork[iindr + windex] = 0;
|
|
}
|
|
/* Given LAMBDA, compute the eigenvector. */
|
|
L__1 = ! usedbs;
|
|
dlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin], &l[
|
|
ibegin], &work[indld + ibegin - 1], &work[
|
|
indlld + ibegin - 1], pivmin, &gaptol, &z__[
|
|
ibegin + windex * z_dim1], &L__1, &negcnt, &
|
|
ztz, &mingma, &iwork[iindr + windex], &isuppz[
|
|
(windex << 1) - 1], &nrminv, &resid, &rqcorr,
|
|
&work[indwrk]);
|
|
if (iter == 0) {
|
|
bstres = resid;
|
|
bstw = lambda;
|
|
} else if (resid < bstres) {
|
|
bstres = resid;
|
|
bstw = lambda;
|
|
}
|
|
/* Computing MIN */
|
|
i__4 = isupmn, i__5 = isuppz[(windex << 1) - 1];
|
|
isupmn = f2cmin(i__4,i__5);
|
|
/* Computing MAX */
|
|
i__4 = isupmx, i__5 = isuppz[windex * 2];
|
|
isupmx = f2cmax(i__4,i__5);
|
|
++iter;
|
|
/* sin alpha <= |resid|/gap */
|
|
/* Note that both the residual and the gap are */
|
|
/* proportional to the matrix, so ||T|| doesn't play */
|
|
/* a role in the quotient */
|
|
|
|
/* Convergence test for Rayleigh-Quotient iteration */
|
|
/* (omitted when Bisection has been used) */
|
|
|
|
if (resid > tol * gap && abs(rqcorr) > rqtol * abs(
|
|
lambda) && ! usedbs) {
|
|
/* We need to check that the RQCORR update doesn't */
|
|
/* move the eigenvalue away from the desired one and */
|
|
/* towards a neighbor. -> protection with bisection */
|
|
if (indeig <= negcnt) {
|
|
/* The wanted eigenvalue lies to the left */
|
|
sgndef = -1.;
|
|
} else {
|
|
/* The wanted eigenvalue lies to the right */
|
|
sgndef = 1.;
|
|
}
|
|
/* We only use the RQCORR if it improves the */
|
|
/* the iterate reasonably. */
|
|
if (rqcorr * sgndef >= 0. && lambda + rqcorr <=
|
|
right && lambda + rqcorr >= left) {
|
|
usedrq = TRUE_;
|
|
/* Store new midpoint of bisection interval in WORK */
|
|
if (sgndef == 1.) {
|
|
/* The current LAMBDA is on the left of the true */
|
|
/* eigenvalue */
|
|
left = lambda;
|
|
/* We prefer to assume that the error estimate */
|
|
/* is correct. We could make the interval not */
|
|
/* as a bracket but to be modified if the RQCORR */
|
|
/* chooses to. In this case, the RIGHT side should */
|
|
/* be modified as follows: */
|
|
/* RIGHT = MAX(RIGHT, LAMBDA + RQCORR) */
|
|
} else {
|
|
/* The current LAMBDA is on the right of the true */
|
|
/* eigenvalue */
|
|
right = lambda;
|
|
/* See comment about assuming the error estimate is */
|
|
/* correct above. */
|
|
/* LEFT = MIN(LEFT, LAMBDA + RQCORR) */
|
|
}
|
|
work[windex] = (right + left) * .5;
|
|
/* Take RQCORR since it has the correct sign and */
|
|
/* improves the iterate reasonably */
|
|
lambda += rqcorr;
|
|
/* Update width of error interval */
|
|
werr[windex] = (right - left) * .5;
|
|
} else {
|
|
needbs = TRUE_;
|
|
}
|
|
if (right - left < rqtol * abs(lambda)) {
|
|
/* The eigenvalue is computed to bisection accuracy */
|
|
/* compute eigenvector and stop */
|
|
usedbs = TRUE_;
|
|
goto L120;
|
|
} else if (iter < 10) {
|
|
goto L120;
|
|
} else if (iter == 10) {
|
|
needbs = TRUE_;
|
|
goto L120;
|
|
} else {
|
|
*info = 5;
|
|
return;
|
|
}
|
|
} else {
|
|
stp2ii = FALSE_;
|
|
if (usedrq && usedbs && bstres <= resid) {
|
|
lambda = bstw;
|
|
stp2ii = TRUE_;
|
|
}
|
|
if (stp2ii) {
|
|
/* improve error angle by second step */
|
|
L__1 = ! usedbs;
|
|
dlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin]
|
|
, &l[ibegin], &work[indld + ibegin -
|
|
1], &work[indlld + ibegin - 1],
|
|
pivmin, &gaptol, &z__[ibegin + windex
|
|
* z_dim1], &L__1, &negcnt, &ztz, &
|
|
mingma, &iwork[iindr + windex], &
|
|
isuppz[(windex << 1) - 1], &nrminv, &
|
|
resid, &rqcorr, &work[indwrk]);
|
|
}
|
|
work[windex] = lambda;
|
|
}
|
|
|
|
/* Compute FP-vector support w.r.t. whole matrix */
|
|
|
|
isuppz[(windex << 1) - 1] += oldien;
|
|
isuppz[windex * 2] += oldien;
|
|
zfrom = isuppz[(windex << 1) - 1];
|
|
zto = isuppz[windex * 2];
|
|
isupmn += oldien;
|
|
isupmx += oldien;
|
|
/* Ensure vector is ok if support in the RQI has changed */
|
|
if (isupmn < zfrom) {
|
|
i__4 = zfrom - 1;
|
|
for (ii = isupmn; ii <= i__4; ++ii) {
|
|
z__[ii + windex * z_dim1] = 0.;
|
|
/* L122: */
|
|
}
|
|
}
|
|
if (isupmx > zto) {
|
|
i__4 = isupmx;
|
|
for (ii = zto + 1; ii <= i__4; ++ii) {
|
|
z__[ii + windex * z_dim1] = 0.;
|
|
/* L123: */
|
|
}
|
|
}
|
|
i__4 = zto - zfrom + 1;
|
|
dscal_(&i__4, &nrminv, &z__[zfrom + windex * z_dim1],
|
|
&c__1);
|
|
L125:
|
|
/* Update W */
|
|
w[windex] = lambda + sigma;
|
|
/* Recompute the gaps on the left and right */
|
|
/* But only allow them to become larger and not */
|
|
/* smaller (which can only happen through "bad" */
|
|
/* cancellation and doesn't reflect the theory */
|
|
/* where the initial gaps are underestimated due */
|
|
/* to WERR being too crude.) */
|
|
if (! eskip) {
|
|
if (k > 1) {
|
|
/* Computing MAX */
|
|
d__1 = wgap[windmn], d__2 = w[windex] - werr[
|
|
windex] - w[windmn] - werr[windmn];
|
|
wgap[windmn] = f2cmax(d__1,d__2);
|
|
}
|
|
if (windex < wend) {
|
|
/* Computing MAX */
|
|
d__1 = savgap, d__2 = w[windpl] - werr[windpl]
|
|
- w[windex] - werr[windex];
|
|
wgap[windex] = f2cmax(d__1,d__2);
|
|
}
|
|
}
|
|
++idone;
|
|
}
|
|
/* here ends the code for the current child */
|
|
|
|
L139:
|
|
/* Proceed to any remaining child nodes */
|
|
newfst = j + 1;
|
|
L140:
|
|
;
|
|
}
|
|
/* L150: */
|
|
}
|
|
++ndepth;
|
|
goto L40;
|
|
}
|
|
ibegin = iend + 1;
|
|
wbegin = wend + 1;
|
|
L170:
|
|
;
|
|
}
|
|
|
|
return;
|
|
|
|
/* End of DLARRV */
|
|
|
|
} /* dlarrv_ */
|
|
|