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]/Cd(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;
517 /* > \brief \b ZTGSEN */
519 /* =========== DOCUMENTATION =========== */
521 /* Online html documentation available at */
522 /* http://www.netlib.org/lapack/explore-html/ */
525 /* > Download ZTGSEN + dependencies */
526 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsen.
529 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztgsen.
532 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztgsen.
540 /* SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, */
541 /* ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, */
542 /* WORK, LWORK, IWORK, LIWORK, INFO ) */
544 /* LOGICAL WANTQ, WANTZ */
545 /* INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, */
547 /* DOUBLE PRECISION PL, PR */
548 /* LOGICAL SELECT( * ) */
549 /* INTEGER IWORK( * ) */
550 /* DOUBLE PRECISION DIF( * ) */
551 /* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), */
552 /* $ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) */
555 /* > \par Purpose: */
560 /* > ZTGSEN reorders the generalized Schur decomposition of a complex */
561 /* > matrix pair (A, B) (in terms of an unitary equivalence trans- */
562 /* > formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues */
563 /* > appears in the leading diagonal blocks of the pair (A,B). The leading */
564 /* > columns of Q and Z form unitary bases of the corresponding left and */
565 /* > right eigenspaces (deflating subspaces). (A, B) must be in */
566 /* > generalized Schur canonical form, that is, A and B are both upper */
569 /* > ZTGSEN also computes the generalized eigenvalues */
571 /* > w(j)= ALPHA(j) / BETA(j) */
573 /* > of the reordered matrix pair (A, B). */
575 /* > Optionally, the routine computes estimates of reciprocal condition */
576 /* > numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */
577 /* > (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */
578 /* > between the matrix pairs (A11, B11) and (A22,B22) that correspond to */
579 /* > the selected cluster and the eigenvalues outside the cluster, resp., */
580 /* > and norms of "projections" onto left and right eigenspaces w.r.t. */
581 /* > the selected cluster in the (1,1)-block. */
588 /* > \param[in] IJOB */
590 /* > IJOB is INTEGER */
591 /* > Specifies whether condition numbers are required for the */
592 /* > cluster of eigenvalues (PL and PR) or the deflating subspaces */
593 /* > (Difu and Difl): */
594 /* > =0: Only reorder w.r.t. SELECT. No extras. */
595 /* > =1: Reciprocal of norms of "projections" onto left and right */
596 /* > eigenspaces w.r.t. the selected cluster (PL and PR). */
597 /* > =2: Upper bounds on Difu and Difl. F-norm-based estimate */
599 /* > =3: Estimate of Difu and Difl. 1-norm-based estimate */
601 /* > About 5 times as expensive as IJOB = 2. */
602 /* > =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */
603 /* > version to get it all. */
604 /* > =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */
607 /* > \param[in] WANTQ */
609 /* > WANTQ is LOGICAL */
610 /* > .TRUE. : update the left transformation matrix Q; */
611 /* > .FALSE.: do not update Q. */
614 /* > \param[in] WANTZ */
616 /* > WANTZ is LOGICAL */
617 /* > .TRUE. : update the right transformation matrix Z; */
618 /* > .FALSE.: do not update Z. */
621 /* > \param[in] SELECT */
623 /* > SELECT is LOGICAL array, dimension (N) */
624 /* > SELECT specifies the eigenvalues in the selected cluster. To */
625 /* > select an eigenvalue w(j), SELECT(j) must be set to */
632 /* > The order of the matrices A and B. N >= 0. */
635 /* > \param[in,out] A */
637 /* > A is COMPLEX*16 array, dimension(LDA,N) */
638 /* > On entry, the upper triangular matrix A, in generalized */
639 /* > Schur canonical form. */
640 /* > On exit, A is overwritten by the reordered matrix A. */
643 /* > \param[in] LDA */
645 /* > LDA is INTEGER */
646 /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */
649 /* > \param[in,out] B */
651 /* > B is COMPLEX*16 array, dimension(LDB,N) */
652 /* > On entry, the upper triangular matrix B, in generalized */
653 /* > Schur canonical form. */
654 /* > On exit, B is overwritten by the reordered matrix B. */
657 /* > \param[in] LDB */
659 /* > LDB is INTEGER */
660 /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */
663 /* > \param[out] ALPHA */
665 /* > ALPHA is COMPLEX*16 array, dimension (N) */
668 /* > \param[out] BETA */
670 /* > BETA is COMPLEX*16 array, dimension (N) */
672 /* > The diagonal elements of A and B, respectively, */
673 /* > when the pair (A,B) has been reduced to generalized Schur */
674 /* > form. ALPHA(i)/BETA(i) i=1,...,N are the generalized */
678 /* > \param[in,out] Q */
680 /* > Q is COMPLEX*16 array, dimension (LDQ,N) */
681 /* > On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */
682 /* > On exit, Q has been postmultiplied by the left unitary */
683 /* > transformation matrix which reorder (A, B); The leading M */
684 /* > columns of Q form orthonormal bases for the specified pair of */
685 /* > left eigenspaces (deflating subspaces). */
686 /* > If WANTQ = .FALSE., Q is not referenced. */
689 /* > \param[in] LDQ */
691 /* > LDQ is INTEGER */
692 /* > The leading dimension of the array Q. LDQ >= 1. */
693 /* > If WANTQ = .TRUE., LDQ >= N. */
696 /* > \param[in,out] Z */
698 /* > Z is COMPLEX*16 array, dimension (LDZ,N) */
699 /* > On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */
700 /* > On exit, Z has been postmultiplied by the left unitary */
701 /* > transformation matrix which reorder (A, B); The leading M */
702 /* > columns of Z form orthonormal bases for the specified pair of */
703 /* > left eigenspaces (deflating subspaces). */
704 /* > If WANTZ = .FALSE., Z is not referenced. */
707 /* > \param[in] LDZ */
709 /* > LDZ is INTEGER */
710 /* > The leading dimension of the array Z. LDZ >= 1. */
711 /* > If WANTZ = .TRUE., LDZ >= N. */
714 /* > \param[out] M */
717 /* > The dimension of the specified pair of left and right */
718 /* > eigenspaces, (deflating subspaces) 0 <= M <= N. */
721 /* > \param[out] PL */
723 /* > PL is DOUBLE PRECISION */
726 /* > \param[out] PR */
728 /* > PR is DOUBLE PRECISION */
730 /* > If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */
731 /* > reciprocal of the norm of "projections" onto left and right */
732 /* > eigenspace with respect to the selected cluster. */
733 /* > 0 < PL, PR <= 1. */
734 /* > If M = 0 or M = N, PL = PR = 1. */
735 /* > If IJOB = 0, 2 or 3 PL, PR are not referenced. */
738 /* > \param[out] DIF */
740 /* > DIF is DOUBLE PRECISION array, dimension (2). */
741 /* > If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */
742 /* > If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */
743 /* > Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */
744 /* > estimates of Difu and Difl, computed using reversed */
745 /* > communication with ZLACN2. */
746 /* > If M = 0 or N, DIF(1:2) = F-norm([A, B]). */
747 /* > If IJOB = 0 or 1, DIF is not referenced. */
750 /* > \param[out] WORK */
752 /* > WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) */
753 /* > On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
756 /* > \param[in] LWORK */
758 /* > LWORK is INTEGER */
759 /* > The dimension of the array WORK. LWORK >= 1 */
760 /* > If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) */
761 /* > If IJOB = 3 or 5, LWORK >= 4*M*(N-M) */
763 /* > If LWORK = -1, then a workspace query is assumed; the routine */
764 /* > only calculates the optimal size of the WORK array, returns */
765 /* > this value as the first entry of the WORK array, and no error */
766 /* > message related to LWORK is issued by XERBLA. */
769 /* > \param[out] IWORK */
771 /* > IWORK is INTEGER array, dimension (MAX(1,LIWORK)) */
772 /* > On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
775 /* > \param[in] LIWORK */
777 /* > LIWORK is INTEGER */
778 /* > The dimension of the array IWORK. LIWORK >= 1. */
779 /* > If IJOB = 1, 2 or 4, LIWORK >= N+2; */
780 /* > If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); */
782 /* > If LIWORK = -1, then a workspace query is assumed; the */
783 /* > routine only calculates the optimal size of the IWORK array, */
784 /* > returns this value as the first entry of the IWORK array, and */
785 /* > no error message related to LIWORK is issued by XERBLA. */
788 /* > \param[out] INFO */
790 /* > INFO is INTEGER */
791 /* > =0: Successful exit. */
792 /* > <0: If INFO = -i, the i-th argument had an illegal value. */
793 /* > =1: Reordering of (A, B) failed because the transformed */
794 /* > matrix pair (A, B) would be too far from generalized */
795 /* > Schur form; the problem is very ill-conditioned. */
796 /* > (A, B) may have been partially reordered. */
797 /* > If requested, 0 is returned in DIF(*), PL and PR. */
803 /* > \author Univ. of Tennessee */
804 /* > \author Univ. of California Berkeley */
805 /* > \author Univ. of Colorado Denver */
806 /* > \author NAG Ltd. */
808 /* > \date June 2016 */
810 /* > \ingroup complex16OTHERcomputational */
812 /* > \par Further Details: */
813 /* ===================== */
817 /* > ZTGSEN first collects the selected eigenvalues by computing unitary */
818 /* > U and W that move them to the top left corner of (A, B). In other */
819 /* > words, the selected eigenvalues are the eigenvalues of (A11, B11) in */
821 /* > U**H*(A, B)*W = (A11 A12) (B11 B12) n1 */
822 /* > ( 0 A22),( 0 B22) n2 */
825 /* > where N = n1+n2 and U**H means the conjugate transpose of U. The first */
826 /* > n1 columns of U and W span the specified pair of left and right */
827 /* > eigenspaces (deflating subspaces) of (A, B). */
829 /* > If (A, B) has been obtained from the generalized real Schur */
830 /* > decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the */
831 /* > reordered generalized Schur form of (C, D) is given by */
833 /* > (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H, */
835 /* > and the first n1 columns of Q*U and Z*W span the corresponding */
836 /* > deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */
838 /* > Note that if the selected eigenvalue is sufficiently ill-conditioned, */
839 /* > then its value may differ significantly from its value before */
842 /* > The reciprocal condition numbers of the left and right eigenspaces */
843 /* > spanned by the first n1 columns of U and W (or Q*U and Z*W) may */
844 /* > be returned in DIF(1:2), corresponding to Difu and Difl, resp. */
846 /* > The Difu and Difl are defined as: */
848 /* > Difu[(A11, B11), (A22, B22)] = sigma-f2cmin( Zu ) */
850 /* > Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */
852 /* > where sigma-f2cmin(Zu) is the smallest singular value of the */
853 /* > (2*n1*n2)-by-(2*n1*n2) matrix */
855 /* > Zu = [ kron(In2, A11) -kron(A22**H, In1) ] */
856 /* > [ kron(In2, B11) -kron(B22**H, In1) ]. */
858 /* > Here, Inx is the identity matrix of size nx and A22**H is the */
859 /* > conjugate transpose of A22. kron(X, Y) is the Kronecker product between */
860 /* > the matrices X and Y. */
862 /* > When DIF(2) is small, small changes in (A, B) can cause large changes */
863 /* > in the deflating subspace. An approximate (asymptotic) bound on the */
864 /* > maximum angular error in the computed deflating subspaces is */
866 /* > EPS * norm((A, B)) / DIF(2), */
868 /* > where EPS is the machine precision. */
870 /* > The reciprocal norm of the projectors on the left and right */
871 /* > eigenspaces associated with (A11, B11) may be returned in PL and PR. */
872 /* > They are computed as follows. First we compute L and R so that */
873 /* > P*(A, B)*Q is block diagonal, where */
875 /* > P = ( I -L ) n1 Q = ( I R ) n1 */
876 /* > ( 0 I ) n2 and ( 0 I ) n2 */
879 /* > and (L, R) is the solution to the generalized Sylvester equation */
881 /* > A11*R - L*A22 = -A12 */
882 /* > B11*R - L*B22 = -B12 */
884 /* > Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */
885 /* > An approximate (asymptotic) bound on the average absolute error of */
886 /* > the selected eigenvalues is */
888 /* > EPS * norm((A, B)) / PL. */
890 /* > There are also global error bounds which valid for perturbations up */
891 /* > to a certain restriction: A lower bound (x) on the smallest */
892 /* > F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */
893 /* > coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */
894 /* > (i.e. (A + E, B + F), is */
896 /* > x = f2cmin(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*f2cmax(1/PL,1/PR)). */
898 /* > An approximate bound on x can be computed from DIF(1:2), PL and PR. */
900 /* > If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */
901 /* > (L', R') and unperturbed (L, R) left and right deflating subspaces */
902 /* > associated with the selected cluster in the (1,1)-blocks can be */
905 /* > f2cmax-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */
906 /* > f2cmax-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */
908 /* > See LAPACK User's Guide section 4.11 or the following references */
909 /* > for more information. */
911 /* > Note that if the default method for computing the Frobenius-norm- */
912 /* > based estimate DIF is not wanted (see ZLATDF), then the parameter */
913 /* > IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF */
914 /* > (IJOB = 2 will be used)). See ZTGSYL for more details. */
917 /* > \par Contributors: */
918 /* ================== */
920 /* > Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
921 /* > Umea University, S-901 87 Umea, Sweden. */
923 /* > \par References: */
924 /* ================ */
926 /* > [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */
927 /* > Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */
928 /* > M.S. Moonen et al (eds), Linear Algebra for Large Scale and */
929 /* > Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */
931 /* > [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */
932 /* > Eigenvalues of a Regular Matrix Pair (A, B) and Condition */
933 /* > Estimation: Theory, Algorithms and Software, Report */
934 /* > UMINF - 94.04, Department of Computing Science, Umea University, */
935 /* > S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */
936 /* > To appear in Numerical Algorithms, 1996. */
938 /* > [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */
939 /* > for Solving the Generalized Sylvester Equation and Estimating the */
940 /* > Separation between Regular Matrix Pairs, Report UMINF - 93.23, */
941 /* > Department of Computing Science, Umea University, S-901 87 Umea, */
942 /* > Sweden, December 1993, Revised April 1994, Also as LAPACK working */
943 /* > Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */
946 /* ===================================================================== */
947 /* Subroutine */ int ztgsen_(integer *ijob, logical *wantq, logical *wantz,
948 logical *select, integer *n, doublecomplex *a, integer *lda,
949 doublecomplex *b, integer *ldb, doublecomplex *alpha, doublecomplex *
950 beta, doublecomplex *q, integer *ldq, doublecomplex *z__, integer *
951 ldz, integer *m, doublereal *pl, doublereal *pr, doublereal *dif,
952 doublecomplex *work, integer *lwork, integer *iwork, integer *liwork,
955 /* System generated locals */
956 integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
957 z_offset, i__1, i__2, i__3;
958 doublecomplex z__1, z__2;
960 /* Local variables */
964 doublecomplex temp1, temp2;
965 integer i__, k, isave[3];
966 extern /* Subroutine */ int zscal_(integer *, doublecomplex *,
967 doublecomplex *, integer *);
972 extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
973 doublecomplex *, doublereal *, integer *, integer *);
974 logical wantd1, wantd2;
975 extern doublereal dlamch_(char *);
978 doublereal rdscal, safmin;
979 extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
981 extern /* Subroutine */ int zlacpy_(char *, integer *, integer *,
982 doublecomplex *, integer *, doublecomplex *, integer *),
983 ztgexc_(logical *, logical *, integer *, doublecomplex *, integer
984 *, doublecomplex *, integer *, doublecomplex *, integer *,
985 doublecomplex *, integer *, integer *, integer *, integer *);
987 extern /* Subroutine */ int zlassq_(integer *, doublecomplex *, integer *,
988 doublereal *, doublereal *);
990 extern /* Subroutine */ int ztgsyl_(char *, integer *, integer *, integer
991 *, doublecomplex *, integer *, doublecomplex *, integer *,
992 doublecomplex *, integer *, doublecomplex *, integer *,
993 doublecomplex *, integer *, doublecomplex *, integer *,
994 doublereal *, doublereal *, doublecomplex *, integer *, integer *,
999 /* -- LAPACK computational routine (version 3.7.1) -- */
1000 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
1001 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
1005 /* ===================================================================== */
1008 /* Decode and test the input parameters */
1010 /* Parameter adjustments */
1013 a_offset = 1 + a_dim1 * 1;
1016 b_offset = 1 + b_dim1 * 1;
1021 q_offset = 1 + q_dim1 * 1;
1024 z_offset = 1 + z_dim1 * 1;
1032 lquery = *lwork == -1 || *liwork == -1;
1034 if (*ijob < 0 || *ijob > 5) {
1036 } else if (*n < 0) {
1038 } else if (*lda < f2cmax(1,*n)) {
1040 } else if (*ldb < f2cmax(1,*n)) {
1042 } else if (*ldq < 1 || *wantq && *ldq < *n) {
1044 } else if (*ldz < 1 || *wantz && *ldz < *n) {
1050 xerbla_("ZTGSEN", &i__1, (ftnlen)6);
1056 wantp = *ijob == 1 || *ijob >= 4;
1057 wantd1 = *ijob == 2 || *ijob == 4;
1058 wantd2 = *ijob == 3 || *ijob == 5;
1059 wantd = wantd1 || wantd2;
1061 /* Set M to the dimension of the specified pair of deflating */
1065 if (! lquery || *ijob != 0) {
1067 for (k = 1; k <= i__1; ++k) {
1069 i__3 = k + k * a_dim1;
1070 alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
1072 i__3 = k + k * b_dim1;
1073 beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
1087 if (*ijob == 1 || *ijob == 2 || *ijob == 4) {
1089 i__1 = 1, i__2 = (*m << 1) * (*n - *m);
1090 lwmin = f2cmax(i__1,i__2);
1092 i__1 = 1, i__2 = *n + 2;
1093 liwmin = f2cmax(i__1,i__2);
1094 } else if (*ijob == 3 || *ijob == 5) {
1096 i__1 = 1, i__2 = (*m << 2) * (*n - *m);
1097 lwmin = f2cmax(i__1,i__2);
1099 i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = f2cmax(i__1,i__2), i__2 =
1101 liwmin = f2cmax(i__1,i__2);
1107 work[1].r = (doublereal) lwmin, work[1].i = 0.;
1110 if (*lwork < lwmin && ! lquery) {
1112 } else if (*liwork < liwmin && ! lquery) {
1118 xerbla_("ZTGSEN", &i__1, (ftnlen)6);
1120 } else if (lquery) {
1124 /* Quick return if possible. */
1126 if (*m == *n || *m == 0) {
1135 for (i__ = 1; i__ <= i__1; ++i__) {
1136 zlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum);
1137 zlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum);
1140 dif[1] = dscale * sqrt(dsum);
1146 /* Get machine constant */
1148 safmin = dlamch_("S");
1150 /* Collect the selected blocks at the top-left corner of (A, B). */
1154 for (k = 1; k <= i__1; ++k) {
1159 /* Swap the K-th block to position KS. Compute unitary Q */
1160 /* and Z that will swap adjacent diagonal blocks in (A, B). */
1163 ztgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb,
1164 &q[q_offset], ldq, &z__[z_offset], ldz, &k, &ks, &
1170 /* Swap is rejected: exit. */
1188 /* Solve generalized Sylvester equation for R and L: */
1189 /* A11 * R - L * A22 = A12 */
1190 /* B11 * R - L * B22 = B12 */
1195 zlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1);
1196 zlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 +
1199 i__1 = *lwork - (n1 << 1) * n2;
1200 ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1]
1201 , lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ *
1202 b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &
1203 work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr);
1205 /* Estimate the reciprocal of norms of "projections" onto */
1206 /* left and right eigenspaces */
1211 zlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum);
1212 *pl = rdscal * sqrt(dsum);
1216 *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl));
1221 zlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum);
1222 *pr = rdscal * sqrt(dsum);
1226 *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr));
1231 /* Compute estimates Difu and Difl. */
1239 /* Frobenius norm-based Difu estimate. */
1241 i__1 = *lwork - (n1 << 1) * n2;
1242 ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ *
1243 a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ +
1244 i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &
1245 dif[1], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
1248 /* Frobenius norm-based Difl estimate. */
1250 i__1 = *lwork - (n1 << 1) * n2;
1251 ztgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[
1252 a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1],
1253 ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale,
1254 &dif[2], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
1258 /* Compute 1-norm-based estimates of Difu and Difl using */
1259 /* reversed communication with ZLACN2. In each step a */
1260 /* generalized Sylvester equation or a transposed variant */
1268 mn2 = (n1 << 1) * n2;
1270 /* 1-norm-based estimate of Difu. */
1273 zlacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[1], &kase, isave);
1277 /* Solve generalized Sylvester equation */
1279 i__1 = *lwork - (n1 << 1) * n2;
1280 ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
1281 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
1282 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1283 1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1284 1], &i__1, &iwork[1], &ierr);
1287 /* Solve the transposed variant. */
1289 i__1 = *lwork - (n1 << 1) * n2;
1290 ztgsyl_("C", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
1291 i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
1292 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1293 1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1294 1], &i__1, &iwork[1], &ierr);
1298 dif[1] = dscale / dif[1];
1300 /* 1-norm-based estimate of Difl. */
1303 zlacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[2], &kase, isave);
1307 /* Solve generalized Sylvester equation */
1309 i__1 = *lwork - (n1 << 1) * n2;
1310 ztgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
1311 &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ *
1312 b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 +
1313 1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1314 1], &i__1, &iwork[1], &ierr);
1317 /* Solve the transposed variant. */
1319 i__1 = *lwork - (n1 << 1) * n2;
1320 ztgsyl_("C", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
1321 &a[a_offset], lda, &work[1], &n2, &b[b_offset],
1322 ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1323 1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1324 1], &i__1, &iwork[1], &ierr);
1328 dif[2] = dscale / dif[2];
1332 /* If B(K,K) is complex, make it real and positive (normalization */
1333 /* of the generalized Schur form) and Store the generalized */
1334 /* eigenvalues of reordered pair (A, B) */
1337 for (k = 1; k <= i__1; ++k) {
1338 dscale = z_abs(&b[k + k * b_dim1]);
1339 if (dscale > safmin) {
1340 i__2 = k + k * b_dim1;
1341 z__2.r = b[i__2].r / dscale, z__2.i = b[i__2].i / dscale;
1342 d_cnjg(&z__1, &z__2);
1343 temp1.r = z__1.r, temp1.i = z__1.i;
1344 i__2 = k + k * b_dim1;
1345 z__1.r = b[i__2].r / dscale, z__1.i = b[i__2].i / dscale;
1346 temp2.r = z__1.r, temp2.i = z__1.i;
1347 i__2 = k + k * b_dim1;
1348 b[i__2].r = dscale, b[i__2].i = 0.;
1350 zscal_(&i__2, &temp1, &b[k + (k + 1) * b_dim1], ldb);
1352 zscal_(&i__2, &temp1, &a[k + k * a_dim1], lda);
1354 zscal_(n, &temp2, &q[k * q_dim1 + 1], &c__1);
1357 i__2 = k + k * b_dim1;
1358 b[i__2].r = 0., b[i__2].i = 0.;
1362 i__3 = k + k * a_dim1;
1363 alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
1365 i__3 = k + k * b_dim1;
1366 beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
1373 work[1].r = (doublereal) lwmin, work[1].i = 0.;