14 typedef long long BLASLONG;
15 typedef unsigned long long BLASULONG;
17 typedef long BLASLONG;
18 typedef unsigned long BLASULONG;
22 typedef BLASLONG blasint;
24 #define blasabs(x) llabs(x)
26 #define blasabs(x) labs(x)
30 #define blasabs(x) abs(x)
33 typedef blasint integer;
35 typedef unsigned int uinteger;
36 typedef char *address;
37 typedef short int shortint;
39 typedef double doublereal;
40 typedef struct { real r, i; } complex;
41 typedef struct { doublereal r, i; } doublecomplex;
43 static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
44 static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
45 static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
46 static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
48 static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
49 static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
50 static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
51 static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
53 #define pCf(z) (*_pCf(z))
54 #define pCd(z) (*_pCd(z))
56 typedef short int shortlogical;
57 typedef char logical1;
58 typedef char integer1;
63 /* Extern is for use with -E */
74 /*external read, write*/
83 /*internal read, write*/
113 /*rewind, backspace, endfile*/
125 ftnint *inex; /*parameters in standard's order*/
151 union Multitype { /* for multiple entry points */
162 typedef union Multitype Multitype;
164 struct Vardesc { /* for Namelist */
170 typedef struct Vardesc Vardesc;
177 typedef struct Namelist Namelist;
179 #define abs(x) ((x) >= 0 ? (x) : -(x))
180 #define dabs(x) (fabs(x))
181 #define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
182 #define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
183 #define dmin(a,b) (f2cmin(a,b))
184 #define dmax(a,b) (f2cmax(a,b))
185 #define bit_test(a,b) ((a) >> (b) & 1)
186 #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
187 #define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
189 #define abort_() { sig_die("Fortran abort routine called", 1); }
190 #define c_abs(z) (cabsf(Cf(z)))
191 #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
193 #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]);}
194 #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]);}
196 #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
197 #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
199 #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
200 #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
201 #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
202 //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
203 #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
204 #define d_abs(x) (fabs(*(x)))
205 #define d_acos(x) (acos(*(x)))
206 #define d_asin(x) (asin(*(x)))
207 #define d_atan(x) (atan(*(x)))
208 #define d_atn2(x, y) (atan2(*(x),*(y)))
209 #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
210 #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
211 #define d_cos(x) (cos(*(x)))
212 #define d_cosh(x) (cosh(*(x)))
213 #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
214 #define d_exp(x) (exp(*(x)))
215 #define d_imag(z) (cimag(Cd(z)))
216 #define r_imag(z) (cimagf(Cf(z)))
217 #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
218 #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
219 #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
220 #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
221 #define d_log(x) (log(*(x)))
222 #define d_mod(x, y) (fmod(*(x), *(y)))
223 #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
224 #define d_nint(x) u_nint(*(x))
225 #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
226 #define d_sign(a,b) u_sign(*(a),*(b))
227 #define r_sign(a,b) u_sign(*(a),*(b))
228 #define d_sin(x) (sin(*(x)))
229 #define d_sinh(x) (sinh(*(x)))
230 #define d_sqrt(x) (sqrt(*(x)))
231 #define d_tan(x) (tan(*(x)))
232 #define d_tanh(x) (tanh(*(x)))
233 #define i_abs(x) abs(*(x))
234 #define i_dnnt(x) ((integer)u_nint(*(x)))
235 #define i_len(s, n) (n)
236 #define i_nint(x) ((integer)u_nint(*(x)))
237 #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
238 #define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
239 #define pow_si(B,E) spow_ui(*(B),*(E))
240 #define pow_ri(B,E) spow_ui(*(B),*(E))
241 #define pow_di(B,E) dpow_ui(*(B),*(E))
242 #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
243 #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
244 #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
245 #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++ = ' '; }
246 #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
247 #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]; }
248 #define sig_die(s, kill) { exit(1); }
249 #define s_stop(s, n) {exit(0);}
250 static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
251 #define z_abs(z) (cabs(Cd(z)))
252 #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
253 #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
254 #define myexit_() break;
255 #define mycycle() continue;
256 #define myceiling(w) {ceil(w)}
257 #define myhuge(w) {HUGE_VAL}
258 //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
259 #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)}
261 /* procedure parameter types for -A and -C++ */
263 #define F2C_proc_par_types 1
265 typedef logical (*L_fp)(...);
267 typedef logical (*L_fp)();
270 static float spow_ui(float x, integer n) {
271 float pow=1.0; unsigned long int u;
273 if(n < 0) n = -n, x = 1/x;
282 static double dpow_ui(double x, integer n) {
283 double pow=1.0; unsigned long int u;
285 if(n < 0) n = -n, x = 1/x;
295 static _Fcomplex cpow_ui(complex x, integer n) {
296 complex pow={1.0,0.0}; unsigned long int u;
298 if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
300 if(u & 01) pow.r *= x.r, pow.i *= x.i;
301 if(u >>= 1) x.r *= x.r, x.i *= x.i;
305 _Fcomplex p={pow.r, pow.i};
309 static _Complex float cpow_ui(_Complex float x, integer n) {
310 _Complex float pow=1.0; unsigned long int u;
312 if(n < 0) n = -n, x = 1/x;
323 static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
324 _Dcomplex pow={1.0,0.0}; unsigned long int u;
326 if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
328 if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
329 if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
333 _Dcomplex p = {pow._Val[0], pow._Val[1]};
337 static _Complex double zpow_ui(_Complex double x, integer n) {
338 _Complex double pow=1.0; unsigned long int u;
340 if(n < 0) n = -n, x = 1/x;
350 static integer pow_ii(integer x, integer n) {
351 integer pow; unsigned long int u;
353 if (n == 0 || x == 1) pow = 1;
354 else if (x != -1) pow = x == 0 ? 1/x : 0;
357 if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
367 static integer dmaxloc_(double *w, integer s, integer e, integer *n)
369 double m; integer i, mi;
370 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
371 if (w[i-1]>m) mi=i ,m=w[i-1];
374 static integer smaxloc_(float *w, integer s, integer e, integer *n)
376 float m; integer i, mi;
377 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
378 if (w[i-1]>m) mi=i ,m=w[i-1];
381 static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
382 integer n = *n_, incx = *incx_, incy = *incy_, i;
384 _Fcomplex zdotc = {0.0, 0.0};
385 if (incx == 1 && incy == 1) {
386 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
387 zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
388 zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
391 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
392 zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
393 zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
399 _Complex float zdotc = 0.0;
400 if (incx == 1 && incy == 1) {
401 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
402 zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
405 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
406 zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
412 static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
413 integer n = *n_, incx = *incx_, incy = *incy_, i;
415 _Dcomplex zdotc = {0.0, 0.0};
416 if (incx == 1 && incy == 1) {
417 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
418 zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
419 zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
422 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
423 zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
424 zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
430 _Complex double zdotc = 0.0;
431 if (incx == 1 && incy == 1) {
432 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
433 zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
436 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
437 zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
443 static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
444 integer n = *n_, incx = *incx_, incy = *incy_, i;
446 _Fcomplex zdotc = {0.0, 0.0};
447 if (incx == 1 && incy == 1) {
448 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
449 zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
450 zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
453 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
454 zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
455 zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
461 _Complex float zdotc = 0.0;
462 if (incx == 1 && incy == 1) {
463 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
464 zdotc += Cf(&x[i]) * Cf(&y[i]);
467 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
468 zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
474 static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
475 integer n = *n_, incx = *incx_, incy = *incy_, i;
477 _Dcomplex zdotc = {0.0, 0.0};
478 if (incx == 1 && incy == 1) {
479 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
480 zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
481 zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
484 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
485 zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
486 zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
492 _Complex double zdotc = 0.0;
493 if (incx == 1 && incy == 1) {
494 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
495 zdotc += Cd(&x[i]) * Cd(&y[i]);
498 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
499 zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
505 /* -- translated by f2c (version 20000121).
506 You must link the resulting object file with the libraries:
507 -lf2c -lm (in that order)
513 /* Table of constant values */
515 static integer c__1 = 1;
516 static logical c_true = TRUE_;
517 static logical c_false = FALSE_;
519 /* > \brief \b STRSNA */
521 /* =========== DOCUMENTATION =========== */
523 /* Online html documentation available at */
524 /* http://www.netlib.org/lapack/explore-html/ */
527 /* > Download STRSNA + dependencies */
528 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/strsna.
531 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/strsna.
534 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/strsna.
542 /* SUBROUTINE STRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, */
543 /* LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, */
546 /* CHARACTER HOWMNY, JOB */
547 /* INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N */
548 /* LOGICAL SELECT( * ) */
549 /* INTEGER IWORK( * ) */
550 /* REAL S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), */
551 /* $ VR( LDVR, * ), WORK( LDWORK, * ) */
554 /* > \par Purpose: */
559 /* > STRSNA estimates reciprocal condition numbers for specified */
560 /* > eigenvalues and/or right eigenvectors of a real upper */
561 /* > quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q */
564 /* > T must be in Schur canonical form (as returned by SHSEQR), that is, */
565 /* > block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each */
566 /* > 2-by-2 diagonal block has its diagonal elements equal and its */
567 /* > off-diagonal elements of opposite sign. */
573 /* > \param[in] JOB */
575 /* > JOB is CHARACTER*1 */
576 /* > Specifies whether condition numbers are required for */
577 /* > eigenvalues (S) or eigenvectors (SEP): */
578 /* > = 'E': for eigenvalues only (S); */
579 /* > = 'V': for eigenvectors only (SEP); */
580 /* > = 'B': for both eigenvalues and eigenvectors (S and SEP). */
583 /* > \param[in] HOWMNY */
585 /* > HOWMNY is CHARACTER*1 */
586 /* > = 'A': compute condition numbers for all eigenpairs; */
587 /* > = 'S': compute condition numbers for selected eigenpairs */
588 /* > specified by the array SELECT. */
591 /* > \param[in] SELECT */
593 /* > SELECT is LOGICAL array, dimension (N) */
594 /* > If HOWMNY = 'S', SELECT specifies the eigenpairs for which */
595 /* > condition numbers are required. To select condition numbers */
596 /* > for the eigenpair corresponding to a real eigenvalue w(j), */
597 /* > SELECT(j) must be set to .TRUE.. To select condition numbers */
598 /* > corresponding to a complex conjugate pair of eigenvalues w(j) */
599 /* > and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be */
600 /* > set to .TRUE.. */
601 /* > If HOWMNY = 'A', SELECT is not referenced. */
607 /* > The order of the matrix T. N >= 0. */
612 /* > T is REAL array, dimension (LDT,N) */
613 /* > The upper quasi-triangular matrix T, in Schur canonical form. */
616 /* > \param[in] LDT */
618 /* > LDT is INTEGER */
619 /* > The leading dimension of the array T. LDT >= f2cmax(1,N). */
622 /* > \param[in] VL */
624 /* > VL is REAL array, dimension (LDVL,M) */
625 /* > If JOB = 'E' or 'B', VL must contain left eigenvectors of T */
626 /* > (or of any Q*T*Q**T with Q orthogonal), corresponding to the */
627 /* > eigenpairs specified by HOWMNY and SELECT. The eigenvectors */
628 /* > must be stored in consecutive columns of VL, as returned by */
629 /* > SHSEIN or STREVC. */
630 /* > If JOB = 'V', VL is not referenced. */
633 /* > \param[in] LDVL */
635 /* > LDVL is INTEGER */
636 /* > The leading dimension of the array VL. */
637 /* > LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. */
640 /* > \param[in] VR */
642 /* > VR is REAL array, dimension (LDVR,M) */
643 /* > If JOB = 'E' or 'B', VR must contain right eigenvectors of T */
644 /* > (or of any Q*T*Q**T with Q orthogonal), corresponding to the */
645 /* > eigenpairs specified by HOWMNY and SELECT. The eigenvectors */
646 /* > must be stored in consecutive columns of VR, as returned by */
647 /* > SHSEIN or STREVC. */
648 /* > If JOB = 'V', VR is not referenced. */
651 /* > \param[in] LDVR */
653 /* > LDVR is INTEGER */
654 /* > The leading dimension of the array VR. */
655 /* > LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. */
658 /* > \param[out] S */
660 /* > S is REAL array, dimension (MM) */
661 /* > If JOB = 'E' or 'B', the reciprocal condition numbers of the */
662 /* > selected eigenvalues, stored in consecutive elements of the */
663 /* > array. For a complex conjugate pair of eigenvalues two */
664 /* > consecutive elements of S are set to the same value. Thus */
665 /* > S(j), SEP(j), and the j-th columns of VL and VR all */
666 /* > correspond to the same eigenpair (but not in general the */
667 /* > j-th eigenpair, unless all eigenpairs are selected). */
668 /* > If JOB = 'V', S is not referenced. */
671 /* > \param[out] SEP */
673 /* > SEP is REAL array, dimension (MM) */
674 /* > If JOB = 'V' or 'B', the estimated reciprocal condition */
675 /* > numbers of the selected eigenvectors, stored in consecutive */
676 /* > elements of the array. For a complex eigenvector two */
677 /* > consecutive elements of SEP are set to the same value. If */
678 /* > the eigenvalues cannot be reordered to compute SEP(j), SEP(j) */
679 /* > is set to 0; this can only occur when the true value would be */
680 /* > very small anyway. */
681 /* > If JOB = 'E', SEP is not referenced. */
684 /* > \param[in] MM */
686 /* > MM is INTEGER */
687 /* > The number of elements in the arrays S (if JOB = 'E' or 'B') */
688 /* > and/or SEP (if JOB = 'V' or 'B'). MM >= M. */
691 /* > \param[out] M */
694 /* > The number of elements of the arrays S and/or SEP actually */
695 /* > used to store the estimated condition numbers. */
696 /* > If HOWMNY = 'A', M is set to N. */
699 /* > \param[out] WORK */
701 /* > WORK is REAL array, dimension (LDWORK,N+6) */
702 /* > If JOB = 'E', WORK is not referenced. */
705 /* > \param[in] LDWORK */
707 /* > LDWORK is INTEGER */
708 /* > The leading dimension of the array WORK. */
709 /* > LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. */
712 /* > \param[out] IWORK */
714 /* > IWORK is INTEGER array, dimension (2*(N-1)) */
715 /* > If JOB = 'E', IWORK is not referenced. */
718 /* > \param[out] INFO */
720 /* > INFO is INTEGER */
721 /* > = 0: successful exit */
722 /* > < 0: if INFO = -i, the i-th argument had an illegal value */
728 /* > \author Univ. of Tennessee */
729 /* > \author Univ. of California Berkeley */
730 /* > \author Univ. of Colorado Denver */
731 /* > \author NAG Ltd. */
733 /* > \date December 2016 */
735 /* > \ingroup realOTHERcomputational */
737 /* > \par Further Details: */
738 /* ===================== */
742 /* > The reciprocal of the condition number of an eigenvalue lambda is */
745 /* > S(lambda) = |v**T*u| / (norm(u)*norm(v)) */
747 /* > where u and v are the right and left eigenvectors of T corresponding */
748 /* > to lambda; v**T denotes the transpose of v, and norm(u) */
749 /* > denotes the Euclidean norm. These reciprocal condition numbers always */
750 /* > lie between zero (very badly conditioned) and one (very well */
751 /* > conditioned). If n = 1, S(lambda) is defined to be 1. */
753 /* > An approximate error bound for a computed eigenvalue W(i) is given by */
755 /* > EPS * norm(T) / S(i) */
757 /* > where EPS is the machine precision. */
759 /* > The reciprocal of the condition number of the right eigenvector u */
760 /* > corresponding to lambda is defined as follows. Suppose */
762 /* > T = ( lambda c ) */
765 /* > Then the reciprocal condition number is */
767 /* > SEP( lambda, T22 ) = sigma-f2cmin( T22 - lambda*I ) */
769 /* > where sigma-f2cmin denotes the smallest singular value. We approximate */
770 /* > the smallest singular value by the reciprocal of an estimate of the */
771 /* > one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is */
772 /* > defined to be abs(T(1,1)). */
774 /* > An approximate error bound for a computed right eigenvector VR(i) */
777 /* > EPS * norm(T) / SEP(i) */
780 /* ===================================================================== */
781 /* Subroutine */ int strsna_(char *job, char *howmny, logical *select,
782 integer *n, real *t, integer *ldt, real *vl, integer *ldvl, real *vr,
783 integer *ldvr, real *s, real *sep, integer *mm, integer *m, real *
784 work, integer *ldwork, integer *iwork, integer *info)
786 /* System generated locals */
787 integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset,
788 work_dim1, work_offset, i__1, i__2;
791 /* Local variables */
799 extern real sdot_(integer *, real *, integer *, real *, integer *);
801 real rnrm, prod1, prod2;
802 extern real snrm2_(integer *, real *, integer *);
805 extern logical lsame_(char *, char *);
810 extern /* Subroutine */ int slacn2_(integer *, real *, real *, integer *,
811 real *, integer *, integer *);
812 extern real slapy2_(real *, real *);
814 extern /* Subroutine */ int slabad_(real *, real *);
817 extern real slamch_(char *);
818 extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
821 extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
822 integer *, real *, integer *);
824 extern /* Subroutine */ int slaqtr_(logical *, logical *, integer *, real
825 *, integer *, real *, real *, real *, real *, real *, integer *),
826 strexc_(char *, integer *, real *, integer *, real *, integer *,
827 integer *, integer *, real *, integer *);
833 /* -- LAPACK computational routine (version 3.7.0) -- */
834 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
835 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
839 /* ===================================================================== */
842 /* Decode and test the input parameters */
844 /* Parameter adjustments */
847 t_offset = 1 + t_dim1 * 1;
850 vl_offset = 1 + vl_dim1 * 1;
853 vr_offset = 1 + vr_dim1 * 1;
858 work_offset = 1 + work_dim1 * 1;
863 wantbh = lsame_(job, "B");
864 wants = lsame_(job, "E") || wantbh;
865 wantsp = lsame_(job, "V") || wantbh;
867 somcon = lsame_(howmny, "S");
870 if (! wants && ! wantsp) {
872 } else if (! lsame_(howmny, "A") && ! somcon) {
876 } else if (*ldt < f2cmax(1,*n)) {
878 } else if (*ldvl < 1 || wants && *ldvl < *n) {
880 } else if (*ldvr < 1 || wants && *ldvr < *n) {
884 /* Set M to the number of eigenpairs for which condition numbers */
885 /* are required, and test MM. */
891 for (k = 1; k <= i__1; ++k) {
896 if (t[k + 1 + k * t_dim1] == 0.f) {
902 if (select[k] || select[k + 1]) {
920 } else if (*ldwork < 1 || wantsp && *ldwork < *n) {
926 xerbla_("STRSNA", &i__1, (ftnlen)6);
930 /* Quick return if possible */
946 sep[1] = (r__1 = t[t_dim1 + 1], abs(r__1));
951 /* Get machine constants */
954 smlnum = slamch_("S") / eps;
955 bignum = 1.f / smlnum;
956 slabad_(&smlnum, &bignum);
961 for (k = 1; k <= i__1; ++k) {
963 /* Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block. */
970 pair = t[k + 1 + k * t_dim1] != 0.f;
974 /* Determine whether condition numbers are required for the k-th */
979 if (! select[k] && ! select[k + 1]) {
993 /* Compute the reciprocal condition number of the k-th */
998 /* Real eigenvalue. */
1000 prod = sdot_(n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks *
1001 vl_dim1 + 1], &c__1);
1002 rnrm = snrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
1003 lnrm = snrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
1004 s[ks] = abs(prod) / (rnrm * lnrm);
1007 /* Complex eigenvalue. */
1009 prod1 = sdot_(n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks *
1010 vl_dim1 + 1], &c__1);
1011 prod1 += sdot_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &vl[(ks
1012 + 1) * vl_dim1 + 1], &c__1);
1013 prod2 = sdot_(n, &vl[ks * vl_dim1 + 1], &c__1, &vr[(ks + 1) *
1014 vr_dim1 + 1], &c__1);
1015 prod2 -= sdot_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1, &vr[ks *
1016 vr_dim1 + 1], &c__1);
1017 r__1 = snrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
1018 r__2 = snrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1);
1019 rnrm = slapy2_(&r__1, &r__2);
1020 r__1 = snrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
1021 r__2 = snrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1);
1022 lnrm = slapy2_(&r__1, &r__2);
1023 cond = slapy2_(&prod1, &prod2) / (rnrm * lnrm);
1031 /* Estimate the reciprocal condition number of the k-th */
1034 /* Copy the matrix T to the array WORK and swap the diagonal */
1035 /* block beginning at T(k,k) to the (1,1) position. */
1037 slacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset],
1041 strexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, &
1042 ifst, &ilst, &work[(*n + 1) * work_dim1 + 1], &ierr);
1044 if (ierr == 1 || ierr == 2) {
1046 /* Could not swap because blocks not well separated */
1052 /* Reordering successful */
1054 if (work[work_dim1 + 2] == 0.f) {
1056 /* Form C = T22 - lambda*I in WORK(2:N,2:N). */
1059 for (i__ = 2; i__ <= i__2; ++i__) {
1060 work[i__ + i__ * work_dim1] -= work[work_dim1 + 1];
1067 /* Triangularize the 2 by 2 block by unitary */
1068 /* transformation U = [ cs i*ss ] */
1070 /* such that the (1,1) position of WORK is complex */
1071 /* eigenvalue lambda with positive imaginary part. (2,2) */
1072 /* position of WORK is the complex eigenvalue lambda */
1073 /* with negative imaginary part. */
1075 mu = sqrt((r__1 = work[(work_dim1 << 1) + 1], abs(r__1)))
1076 * sqrt((r__2 = work[work_dim1 + 2], abs(r__2)));
1077 delta = slapy2_(&mu, &work[work_dim1 + 2]);
1079 sn = -work[work_dim1 + 2] / delta;
1083 /* C**T = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] */
1088 /* where C**T is transpose of matrix C, */
1089 /* and RWORK is stored starting in the N+1-st column of */
1093 for (j = 3; j <= i__2; ++j) {
1094 work[j * work_dim1 + 2] = cs * work[j * work_dim1 + 2]
1096 work[j + j * work_dim1] -= work[work_dim1 + 1];
1099 work[(work_dim1 << 1) + 2] = 0.f;
1101 work[(*n + 1) * work_dim1 + 1] = mu * 2.f;
1103 for (i__ = 2; i__ <= i__2; ++i__) {
1104 work[i__ + (*n + 1) * work_dim1] = sn * work[(i__ + 1)
1112 /* Estimate norm(inv(C**T)) */
1117 slacn2_(&nn, &work[(*n + 2) * work_dim1 + 1], &work[(*n + 4) *
1118 work_dim1 + 1], &iwork[1], &est, &kase, isave);
1123 /* Real eigenvalue: solve C**T*x = scale*c. */
1126 slaqtr_(&c_true, &c_true, &i__2, &work[(work_dim1
1127 << 1) + 2], ldwork, dummy, &dumm, &scale,
1128 &work[(*n + 4) * work_dim1 + 1], &work[(*
1129 n + 6) * work_dim1 + 1], &ierr);
1132 /* Complex eigenvalue: solve */
1133 /* C**T*(p+iq) = scale*(c+id) in real arithmetic. */
1136 slaqtr_(&c_true, &c_false, &i__2, &work[(
1137 work_dim1 << 1) + 2], ldwork, &work[(*n +
1138 1) * work_dim1 + 1], &mu, &scale, &work[(*
1139 n + 4) * work_dim1 + 1], &work[(*n + 6) *
1140 work_dim1 + 1], &ierr);
1145 /* Real eigenvalue: solve C*x = scale*c. */
1148 slaqtr_(&c_false, &c_true, &i__2, &work[(
1149 work_dim1 << 1) + 2], ldwork, dummy, &
1150 dumm, &scale, &work[(*n + 4) * work_dim1
1151 + 1], &work[(*n + 6) * work_dim1 + 1], &
1155 /* Complex eigenvalue: solve */
1156 /* C*(p+iq) = scale*(c+id) in real arithmetic. */
1159 slaqtr_(&c_false, &c_false, &i__2, &work[(
1160 work_dim1 << 1) + 2], ldwork, &work[(*n +
1161 1) * work_dim1 + 1], &mu, &scale, &work[(*
1162 n + 4) * work_dim1 + 1], &work[(*n + 6) *
1163 work_dim1 + 1], &ierr);
1172 sep[ks] = scale / f2cmax(est,smlnum);
1174 sep[ks + 1] = sep[ks];