1122 lines
33 KiB
C
1122 lines
33 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_n1 = -1;
|
|
|
|
/* > \brief \b DTRSEN */
|
|
|
|
/* =========== DOCUMENTATION =========== */
|
|
|
|
/* Online html documentation available at */
|
|
/* http://www.netlib.org/lapack/explore-html/ */
|
|
|
|
/* > \htmlonly */
|
|
/* > Download DTRSEN + dependencies */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsen.
|
|
f"> */
|
|
/* > [TGZ]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsen.
|
|
f"> */
|
|
/* > [ZIP]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsen.
|
|
f"> */
|
|
/* > [TXT]</a> */
|
|
/* > \endhtmlonly */
|
|
|
|
/* Definition: */
|
|
/* =========== */
|
|
|
|
/* SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, */
|
|
/* M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) */
|
|
|
|
/* CHARACTER COMPQ, JOB */
|
|
/* INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N */
|
|
/* DOUBLE PRECISION S, SEP */
|
|
/* LOGICAL SELECT( * ) */
|
|
/* INTEGER IWORK( * ) */
|
|
/* DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), */
|
|
/* $ WR( * ) */
|
|
|
|
|
|
/* > \par Purpose: */
|
|
/* ============= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > DTRSEN reorders the real Schur factorization of a real matrix */
|
|
/* > A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in */
|
|
/* > the leading diagonal blocks of the upper quasi-triangular matrix T, */
|
|
/* > and the leading columns of Q form an orthonormal basis of the */
|
|
/* > corresponding right invariant subspace. */
|
|
/* > */
|
|
/* > Optionally the routine computes the reciprocal condition numbers of */
|
|
/* > the cluster of eigenvalues and/or the invariant subspace. */
|
|
/* > */
|
|
/* > T must be in Schur canonical form (as returned by DHSEQR), that is, */
|
|
/* > block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each */
|
|
/* > 2-by-2 diagonal block has its diagonal elements equal and its */
|
|
/* > off-diagonal elements of opposite sign. */
|
|
/* > \endverbatim */
|
|
|
|
/* Arguments: */
|
|
/* ========== */
|
|
|
|
/* > \param[in] JOB */
|
|
/* > \verbatim */
|
|
/* > JOB is CHARACTER*1 */
|
|
/* > Specifies whether condition numbers are required for the */
|
|
/* > cluster of eigenvalues (S) or the invariant subspace (SEP): */
|
|
/* > = 'N': none; */
|
|
/* > = 'E': for eigenvalues only (S); */
|
|
/* > = 'V': for invariant subspace only (SEP); */
|
|
/* > = 'B': for both eigenvalues and invariant subspace (S and */
|
|
/* > SEP). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] COMPQ */
|
|
/* > \verbatim */
|
|
/* > COMPQ is CHARACTER*1 */
|
|
/* > = 'V': update the matrix Q of Schur vectors; */
|
|
/* > = 'N': do not update Q. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] SELECT */
|
|
/* > \verbatim */
|
|
/* > SELECT is LOGICAL array, dimension (N) */
|
|
/* > SELECT specifies the eigenvalues in the selected cluster. To */
|
|
/* > select a real eigenvalue w(j), SELECT(j) must be set to */
|
|
/* > .TRUE.. To select a complex conjugate pair of eigenvalues */
|
|
/* > w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */
|
|
/* > either SELECT(j) or SELECT(j+1) or both must be set to */
|
|
/* > .TRUE.; a complex conjugate pair of eigenvalues must be */
|
|
/* > either both included in the cluster or both excluded. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] N */
|
|
/* > \verbatim */
|
|
/* > N is INTEGER */
|
|
/* > The order of the matrix T. N >= 0. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] T */
|
|
/* > \verbatim */
|
|
/* > T is DOUBLE PRECISION array, dimension (LDT,N) */
|
|
/* > On entry, the upper quasi-triangular matrix T, in Schur */
|
|
/* > canonical form. */
|
|
/* > On exit, T is overwritten by the reordered matrix T, again in */
|
|
/* > Schur canonical form, with the selected eigenvalues in the */
|
|
/* > leading diagonal blocks. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDT */
|
|
/* > \verbatim */
|
|
/* > LDT is INTEGER */
|
|
/* > The leading dimension of the array T. LDT >= f2cmax(1,N). */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in,out] Q */
|
|
/* > \verbatim */
|
|
/* > Q is DOUBLE PRECISION array, dimension (LDQ,N) */
|
|
/* > On entry, if COMPQ = 'V', the matrix Q of Schur vectors. */
|
|
/* > On exit, if COMPQ = 'V', Q has been postmultiplied by the */
|
|
/* > orthogonal transformation matrix which reorders T; the */
|
|
/* > leading M columns of Q form an orthonormal basis for the */
|
|
/* > specified invariant subspace. */
|
|
/* > If COMPQ = 'N', Q is not referenced. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDQ */
|
|
/* > \verbatim */
|
|
/* > LDQ is INTEGER */
|
|
/* > The leading dimension of the array Q. */
|
|
/* > LDQ >= 1; and if COMPQ = 'V', LDQ >= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WR */
|
|
/* > \verbatim */
|
|
/* > WR is DOUBLE PRECISION array, dimension (N) */
|
|
/* > \endverbatim */
|
|
/* > \param[out] WI */
|
|
/* > \verbatim */
|
|
/* > WI is DOUBLE PRECISION array, dimension (N) */
|
|
/* > */
|
|
/* > The real and imaginary parts, respectively, of the reordered */
|
|
/* > eigenvalues of T. The eigenvalues are stored in the same */
|
|
/* > order as on the diagonal of T, with WR(i) = T(i,i) and, if */
|
|
/* > T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and */
|
|
/* > WI(i+1) = -WI(i). Note that if a complex eigenvalue is */
|
|
/* > sufficiently ill-conditioned, then its value may differ */
|
|
/* > significantly from its value before reordering. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] M */
|
|
/* > \verbatim */
|
|
/* > M is INTEGER */
|
|
/* > The dimension of the specified invariant subspace. */
|
|
/* > 0 < = M <= N. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] S */
|
|
/* > \verbatim */
|
|
/* > S is DOUBLE PRECISION */
|
|
/* > If JOB = 'E' or 'B', S is a lower bound on the reciprocal */
|
|
/* > condition number for the selected cluster of eigenvalues. */
|
|
/* > S cannot underestimate the true reciprocal condition number */
|
|
/* > by more than a factor of sqrt(N). If M = 0 or N, S = 1. */
|
|
/* > If JOB = 'N' or 'V', S is not referenced. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] SEP */
|
|
/* > \verbatim */
|
|
/* > SEP is DOUBLE PRECISION */
|
|
/* > If JOB = 'V' or 'B', SEP is the estimated reciprocal */
|
|
/* > condition number of the specified invariant subspace. If */
|
|
/* > M = 0 or N, SEP = norm(T). */
|
|
/* > If JOB = 'N' or 'E', SEP is not referenced. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WORK */
|
|
/* > \verbatim */
|
|
/* > WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
|
|
/* > On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LWORK */
|
|
/* > \verbatim */
|
|
/* > LWORK is INTEGER */
|
|
/* > The dimension of the array WORK. */
|
|
/* > If JOB = 'N', LWORK >= f2cmax(1,N); */
|
|
/* > if JOB = 'E', LWORK >= f2cmax(1,M*(N-M)); */
|
|
/* > if JOB = 'V' or 'B', LWORK >= f2cmax(1,2*M*(N-M)). */
|
|
/* > */
|
|
/* > If LWORK = -1, then a workspace query is assumed; the routine */
|
|
/* > only calculates the optimal size of the WORK array, returns */
|
|
/* > this value as the first entry of the WORK array, and no error */
|
|
/* > message related to LWORK is issued by XERBLA. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] IWORK */
|
|
/* > \verbatim */
|
|
/* > IWORK is INTEGER array, dimension (MAX(1,LIWORK)) */
|
|
/* > On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LIWORK */
|
|
/* > \verbatim */
|
|
/* > LIWORK is INTEGER */
|
|
/* > The dimension of the array IWORK. */
|
|
/* > If JOB = 'N' or 'E', LIWORK >= 1; */
|
|
/* > if JOB = 'V' or 'B', LIWORK >= f2cmax(1,M*(N-M)). */
|
|
/* > */
|
|
/* > If LIWORK = -1, then a workspace query is assumed; the */
|
|
/* > routine only calculates the optimal size of the IWORK array, */
|
|
/* > returns this value as the first entry of the IWORK array, and */
|
|
/* > no error message related to LIWORK is issued by XERBLA. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] INFO */
|
|
/* > \verbatim */
|
|
/* > INFO is INTEGER */
|
|
/* > = 0: successful exit */
|
|
/* > < 0: if INFO = -i, the i-th argument had an illegal value */
|
|
/* > = 1: reordering of T failed because some eigenvalues are too */
|
|
/* > close to separate (the problem is very ill-conditioned); */
|
|
/* > T may have been partially reordered, and WR and WI */
|
|
/* > contain the eigenvalues in the same order as in T; S and */
|
|
/* > SEP (if requested) are set to zero. */
|
|
/* > \endverbatim */
|
|
|
|
/* Authors: */
|
|
/* ======== */
|
|
|
|
/* > \author Univ. of Tennessee */
|
|
/* > \author Univ. of California Berkeley */
|
|
/* > \author Univ. of Colorado Denver */
|
|
/* > \author NAG Ltd. */
|
|
|
|
/* > \date April 2012 */
|
|
|
|
/* > \ingroup doubleOTHERcomputational */
|
|
|
|
/* > \par Further Details: */
|
|
/* ===================== */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > DTRSEN first collects the selected eigenvalues by computing an */
|
|
/* > orthogonal transformation Z to move them to the top left corner of T. */
|
|
/* > In other words, the selected eigenvalues are the eigenvalues of T11 */
|
|
/* > in: */
|
|
/* > */
|
|
/* > Z**T * T * Z = ( T11 T12 ) n1 */
|
|
/* > ( 0 T22 ) n2 */
|
|
/* > n1 n2 */
|
|
/* > */
|
|
/* > where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns */
|
|
/* > of Z span the specified invariant subspace of T. */
|
|
/* > */
|
|
/* > If T has been obtained from the real Schur factorization of a matrix */
|
|
/* > A = Q*T*Q**T, then the reordered real Schur factorization of A is given */
|
|
/* > by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span */
|
|
/* > the corresponding invariant subspace of A. */
|
|
/* > */
|
|
/* > The reciprocal condition number of the average of the eigenvalues of */
|
|
/* > T11 may be returned in S. S lies between 0 (very badly conditioned) */
|
|
/* > and 1 (very well conditioned). It is computed as follows. First we */
|
|
/* > compute R so that */
|
|
/* > */
|
|
/* > P = ( I R ) n1 */
|
|
/* > ( 0 0 ) n2 */
|
|
/* > n1 n2 */
|
|
/* > */
|
|
/* > is the projector on the invariant subspace associated with T11. */
|
|
/* > R is the solution of the Sylvester equation: */
|
|
/* > */
|
|
/* > T11*R - R*T22 = T12. */
|
|
/* > */
|
|
/* > Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote */
|
|
/* > the two-norm of M. Then S is computed as the lower bound */
|
|
/* > */
|
|
/* > (1 + F-norm(R)**2)**(-1/2) */
|
|
/* > */
|
|
/* > on the reciprocal of 2-norm(P), the true reciprocal condition number. */
|
|
/* > S cannot underestimate 1 / 2-norm(P) by more than a factor of */
|
|
/* > sqrt(N). */
|
|
/* > */
|
|
/* > An approximate error bound for the computed average of the */
|
|
/* > eigenvalues of T11 is */
|
|
/* > */
|
|
/* > EPS * norm(T) / S */
|
|
/* > */
|
|
/* > where EPS is the machine precision. */
|
|
/* > */
|
|
/* > The reciprocal condition number of the right invariant subspace */
|
|
/* > spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. */
|
|
/* > SEP is defined as the separation of T11 and T22: */
|
|
/* > */
|
|
/* > sep( T11, T22 ) = sigma-f2cmin( C ) */
|
|
/* > */
|
|
/* > where sigma-f2cmin(C) is the smallest singular value of the */
|
|
/* > n1*n2-by-n1*n2 matrix */
|
|
/* > */
|
|
/* > C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) */
|
|
/* > */
|
|
/* > I(m) is an m by m identity matrix, and kprod denotes the Kronecker */
|
|
/* > product. We estimate sigma-f2cmin(C) by the reciprocal of an estimate of */
|
|
/* > the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) */
|
|
/* > cannot differ from sigma-f2cmin(C) by more than a factor of sqrt(n1*n2). */
|
|
/* > */
|
|
/* > When SEP is small, small changes in T can cause large changes in */
|
|
/* > the invariant subspace. An approximate bound on the maximum angular */
|
|
/* > error in the computed right invariant subspace is */
|
|
/* > */
|
|
/* > EPS * norm(T) / SEP */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* ===================================================================== */
|
|
/* Subroutine */ void dtrsen_(char *job, char *compq, logical *select, integer
|
|
*n, doublereal *t, integer *ldt, doublereal *q, integer *ldq,
|
|
doublereal *wr, doublereal *wi, integer *m, doublereal *s, doublereal
|
|
*sep, doublereal *work, integer *lwork, integer *iwork, integer *
|
|
liwork, integer *info)
|
|
{
|
|
/* System generated locals */
|
|
integer q_dim1, q_offset, t_dim1, t_offset, i__1, i__2;
|
|
doublereal d__1, d__2;
|
|
|
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/* Local variables */
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integer kase;
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logical pair;
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integer ierr;
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logical swap;
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integer k;
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doublereal scale;
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extern logical lsame_(char *, char *);
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integer isave[3], lwmin;
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logical wantq, wants;
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doublereal rnorm;
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integer n1, n2;
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extern /* Subroutine */ void dlacn2_(integer *, doublereal *, doublereal *,
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integer *, doublereal *, integer *, integer *);
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integer kk;
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extern doublereal dlange_(char *, integer *, integer *, doublereal *,
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integer *, doublereal *);
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integer nn, ks;
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extern /* Subroutine */ void dlacpy_(char *, integer *, integer *,
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doublereal *, integer *, doublereal *, integer *);
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extern int xerbla_(char *, integer *, ftnlen);
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logical wantbh;
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extern /* Subroutine */ void dtrexc_(char *, integer *, doublereal *,
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integer *, doublereal *, integer *, integer *, integer *,
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doublereal *, integer *);
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integer liwmin;
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logical wantsp, lquery;
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extern /* Subroutine */ void dtrsyl_(char *, char *, integer *, integer *,
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integer *, doublereal *, integer *, doublereal *, integer *,
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doublereal *, integer *, doublereal *, integer *);
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doublereal est;
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/* -- LAPACK computational routine (version 3.7.0) -- */
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/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
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/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
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/* April 2012 */
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/* ===================================================================== */
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/* Decode and test the input parameters */
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/* Parameter adjustments */
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--select;
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t_dim1 = *ldt;
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t_offset = 1 + t_dim1 * 1;
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t -= t_offset;
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q_dim1 = *ldq;
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q_offset = 1 + q_dim1 * 1;
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q -= q_offset;
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--wr;
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--wi;
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--work;
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--iwork;
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/* Function Body */
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wantbh = lsame_(job, "B");
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wants = lsame_(job, "E") || wantbh;
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wantsp = lsame_(job, "V") || wantbh;
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wantq = lsame_(compq, "V");
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*info = 0;
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lquery = *lwork == -1;
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if (! lsame_(job, "N") && ! wants && ! wantsp) {
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*info = -1;
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} else if (! lsame_(compq, "N") && ! wantq) {
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*info = -2;
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} else if (*n < 0) {
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*info = -4;
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} else if (*ldt < f2cmax(1,*n)) {
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*info = -6;
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} else if (*ldq < 1 || wantq && *ldq < *n) {
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*info = -8;
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} else {
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/* Set M to the dimension of the specified invariant subspace, */
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/* and test LWORK and LIWORK. */
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*m = 0;
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pair = FALSE_;
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i__1 = *n;
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for (k = 1; k <= i__1; ++k) {
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if (pair) {
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pair = FALSE_;
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} else {
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if (k < *n) {
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if (t[k + 1 + k * t_dim1] == 0.) {
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if (select[k]) {
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++(*m);
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}
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} else {
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pair = TRUE_;
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if (select[k] || select[k + 1]) {
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*m += 2;
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}
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}
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} else {
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if (select[*n]) {
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++(*m);
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}
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}
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}
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/* L10: */
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}
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n1 = *m;
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n2 = *n - *m;
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nn = n1 * n2;
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if (wantsp) {
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/* Computing MAX */
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i__1 = 1, i__2 = nn << 1;
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lwmin = f2cmax(i__1,i__2);
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liwmin = f2cmax(1,nn);
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} else if (lsame_(job, "N")) {
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lwmin = f2cmax(1,*n);
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liwmin = 1;
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} else if (lsame_(job, "E")) {
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lwmin = f2cmax(1,nn);
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liwmin = 1;
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}
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if (*lwork < lwmin && ! lquery) {
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*info = -15;
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} else if (*liwork < liwmin && ! lquery) {
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*info = -17;
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}
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}
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if (*info == 0) {
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work[1] = (doublereal) lwmin;
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iwork[1] = liwmin;
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_("DTRSEN", &i__1, (ftnlen)6);
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return;
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} else if (lquery) {
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return;
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}
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/* Quick return if possible. */
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if (*m == *n || *m == 0) {
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if (wants) {
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*s = 1.;
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}
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if (wantsp) {
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*sep = dlange_("1", n, n, &t[t_offset], ldt, &work[1]);
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}
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goto L40;
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}
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/* Collect the selected blocks at the top-left corner of T. */
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ks = 0;
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pair = FALSE_;
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i__1 = *n;
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for (k = 1; k <= i__1; ++k) {
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if (pair) {
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pair = FALSE_;
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} else {
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swap = select[k];
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if (k < *n) {
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if (t[k + 1 + k * t_dim1] != 0.) {
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pair = TRUE_;
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swap = swap || select[k + 1];
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}
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}
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if (swap) {
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++ks;
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/* Swap the K-th block to position KS. */
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ierr = 0;
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kk = k;
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if (k != ks) {
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dtrexc_(compq, n, &t[t_offset], ldt, &q[q_offset], ldq, &
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kk, &ks, &work[1], &ierr);
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}
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if (ierr == 1 || ierr == 2) {
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/* Blocks too close to swap: exit. */
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*info = 1;
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if (wants) {
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*s = 0.;
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}
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if (wantsp) {
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*sep = 0.;
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}
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goto L40;
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}
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if (pair) {
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++ks;
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}
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}
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}
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/* L20: */
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}
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if (wants) {
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/* Solve Sylvester equation for R: */
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/* T11*R - R*T22 = scale*T12 */
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dlacpy_("F", &n1, &n2, &t[(n1 + 1) * t_dim1 + 1], ldt, &work[1], &n1);
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dtrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 + 1 + (n1
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+ 1) * t_dim1], ldt, &work[1], &n1, &scale, &ierr);
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/* Estimate the reciprocal of the condition number of the cluster */
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/* of eigenvalues. */
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rnorm = dlange_("F", &n1, &n2, &work[1], &n1, &work[1]);
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if (rnorm == 0.) {
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*s = 1.;
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} else {
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*s = scale / (sqrt(scale * scale / rnorm + rnorm) * sqrt(rnorm));
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}
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}
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if (wantsp) {
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/* Estimate sep(T11,T22). */
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est = 0.;
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kase = 0;
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L30:
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dlacn2_(&nn, &work[nn + 1], &work[1], &iwork[1], &est, &kase, isave);
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if (kase != 0) {
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if (kase == 1) {
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/* Solve T11*R - R*T22 = scale*X. */
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dtrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
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1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
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ierr);
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} else {
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/* Solve T11**T*R - R*T22**T = scale*X. */
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dtrsyl_("T", "T", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
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1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
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ierr);
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}
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goto L30;
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}
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*sep = scale / est;
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}
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L40:
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/* Store the output eigenvalues in WR and WI. */
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i__1 = *n;
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for (k = 1; k <= i__1; ++k) {
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wr[k] = t[k + k * t_dim1];
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wi[k] = 0.;
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/* L50: */
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}
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i__1 = *n - 1;
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for (k = 1; k <= i__1; ++k) {
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if (t[k + 1 + k * t_dim1] != 0.) {
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wi[k] = sqrt((d__1 = t[k + (k + 1) * t_dim1], abs(d__1))) * sqrt((
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d__2 = t[k + 1 + k * t_dim1], abs(d__2)));
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wi[k + 1] = -wi[k];
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}
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/* L60: */
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}
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work[1] = (doublereal) lwmin;
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iwork[1] = liwmin;
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return;
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/* End of DTRSEN */
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} /* dtrsen_ */
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