1 /* ./src_f77/dtgsna.f -- translated by f2c (version 20030320).
2 You must link the resulting object file with the libraries:
3 -lf2c -lm (in that order)
4 */
5
6 #include <punc/vf2c.h>
7
8 /* Table of constant values */
9
10 static integer c__1 = 1;
11 static doublereal c_b19 = 1.;
12 static doublereal c_b21 = 0.;
13 static integer c__2 = 2;
14 static logical c_false = FALSE_;
15 static integer c__3 = 3;
16
dtgsna_(char * job,char * howmny,logical * select,integer * n,doublereal * a,integer * lda,doublereal * b,integer * ldb,doublereal * vl,integer * ldvl,doublereal * vr,integer * ldvr,doublereal * s,doublereal * dif,integer * mm,integer * m,doublereal * work,integer * lwork,integer * iwork,integer * info,ftnlen job_len,ftnlen howmny_len)17 /* Subroutine */ int dtgsna_(char *job, char *howmny, logical *select,
18 integer *n, doublereal *a, integer *lda, doublereal *b, integer *ldb,
19 doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr,
20 doublereal *s, doublereal *dif, integer *mm, integer *m, doublereal *
21 work, integer *lwork, integer *iwork, integer *info, ftnlen job_len,
22 ftnlen howmny_len)
23 {
24 /* System generated locals */
25 integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
26 vr_offset, i__1, i__2;
27 doublereal d__1, d__2;
28
29 /* Builtin functions */
30 double sqrt(doublereal);
31
32 /* Local variables */
33 static integer i__, k;
34 static doublereal c1, c2;
35 static integer n1, n2, ks, iz;
36 static doublereal eps, beta, cond;
37 extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *,
38 integer *);
39 static logical pair;
40 static integer ierr;
41 static doublereal uhav, uhbv;
42 static integer ifst;
43 static doublereal lnrm;
44 static integer ilst;
45 static doublereal rnrm;
46 extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *,
47 integer *, doublereal *, doublereal *, doublereal *, doublereal *,
48 doublereal *, doublereal *);
49 extern doublereal dnrm2_(integer *, doublereal *, integer *);
50 static doublereal root1, root2, scale;
51 extern logical lsame_(char *, char *, ftnlen, ftnlen);
52 extern /* Subroutine */ int dgemv_(char *, integer *, integer *,
53 doublereal *, doublereal *, integer *, doublereal *, integer *,
54 doublereal *, doublereal *, integer *, ftnlen);
55 static doublereal uhavi, uhbvi, tmpii;
56 static integer lwmin;
57 static logical wants;
58 static doublereal tmpir, tmpri, dummy[1], tmprr;
59 extern doublereal dlapy2_(doublereal *, doublereal *);
60 static doublereal dummy1[1];
61 extern doublereal dlamch_(char *, ftnlen);
62 static doublereal alphai, alphar;
63 extern /* Subroutine */ int dlacpy_(char *, integer *, integer *,
64 doublereal *, integer *, doublereal *, integer *, ftnlen),
65 xerbla_(char *, integer *, ftnlen), dtgexc_(logical *, logical *,
66 integer *, doublereal *, integer *, doublereal *, integer *,
67 doublereal *, integer *, doublereal *, integer *, integer *,
68 integer *, doublereal *, integer *, integer *);
69 static logical wantbh, wantdf, somcon;
70 static doublereal alprqt;
71 extern /* Subroutine */ int dtgsyl_(char *, integer *, integer *, integer
72 *, doublereal *, integer *, doublereal *, integer *, doublereal *,
73 integer *, doublereal *, integer *, doublereal *, integer *,
74 doublereal *, integer *, doublereal *, doublereal *, doublereal *,
75 integer *, integer *, integer *, ftnlen);
76 static doublereal smlnum;
77 static logical lquery;
78
79
80 /* -- LAPACK routine (version 3.0) -- */
81 /* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
82 /* Courant Institute, Argonne National Lab, and Rice University */
83 /* June 30, 1999 */
84
85 /* .. Scalar Arguments .. */
86 /* .. */
87 /* .. Array Arguments .. */
88 /* .. */
89
90 /* Purpose */
91 /* ======= */
92
93 /* DTGSNA estimates reciprocal condition numbers for specified */
94 /* eigenvalues and/or eigenvectors of a matrix pair (A, B) in */
95 /* generalized real Schur canonical form (or of any matrix pair */
96 /* (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where */
97 /* Z' denotes the transpose of Z. */
98
99 /* (A, B) must be in generalized real Schur form (as returned by DGGES), */
100 /* i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal */
101 /* blocks. B is upper triangular. */
102
103
104 /* Arguments */
105 /* ========= */
106
107 /* JOB (input) CHARACTER*1 */
108 /* Specifies whether condition numbers are required for */
109 /* eigenvalues (S) or eigenvectors (DIF): */
110 /* = 'E': for eigenvalues only (S); */
111 /* = 'V': for eigenvectors only (DIF); */
112 /* = 'B': for both eigenvalues and eigenvectors (S and DIF). */
113
114 /* HOWMNY (input) CHARACTER*1 */
115 /* = 'A': compute condition numbers for all eigenpairs; */
116 /* = 'S': compute condition numbers for selected eigenpairs */
117 /* specified by the array SELECT. */
118
119 /* SELECT (input) LOGICAL array, dimension (N) */
120 /* If HOWMNY = 'S', SELECT specifies the eigenpairs for which */
121 /* condition numbers are required. To select condition numbers */
122 /* for the eigenpair corresponding to a real eigenvalue w(j), */
123 /* SELECT(j) must be set to .TRUE.. To select condition numbers */
124 /* corresponding to a complex conjugate pair of eigenvalues w(j) */
125 /* and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be */
126 /* set to .TRUE.. */
127 /* If HOWMNY = 'A', SELECT is not referenced. */
128
129 /* N (input) INTEGER */
130 /* The order of the square matrix pair (A, B). N >= 0. */
131
132 /* A (input) DOUBLE PRECISION array, dimension (LDA,N) */
133 /* The upper quasi-triangular matrix A in the pair (A,B). */
134
135 /* LDA (input) INTEGER */
136 /* The leading dimension of the array A. LDA >= max(1,N). */
137
138 /* B (input) DOUBLE PRECISION array, dimension (LDB,N) */
139 /* The upper triangular matrix B in the pair (A,B). */
140
141 /* LDB (input) INTEGER */
142 /* The leading dimension of the array B. LDB >= max(1,N). */
143
144 /* VL (input) DOUBLE PRECISION array, dimension (LDVL,M) */
145 /* If JOB = 'E' or 'B', VL must contain left eigenvectors of */
146 /* (A, B), corresponding to the eigenpairs specified by HOWMNY */
147 /* and SELECT. The eigenvectors must be stored in consecutive */
148 /* columns of VL, as returned by DTGEVC. */
149 /* If JOB = 'V', VL is not referenced. */
150
151 /* LDVL (input) INTEGER */
152 /* The leading dimension of the array VL. LDVL >= 1. */
153 /* If JOB = 'E' or 'B', LDVL >= N. */
154
155 /* VR (input) DOUBLE PRECISION array, dimension (LDVR,M) */
156 /* If JOB = 'E' or 'B', VR must contain right eigenvectors of */
157 /* (A, B), corresponding to the eigenpairs specified by HOWMNY */
158 /* and SELECT. The eigenvectors must be stored in consecutive */
159 /* columns ov VR, as returned by DTGEVC. */
160 /* If JOB = 'V', VR is not referenced. */
161
162 /* LDVR (input) INTEGER */
163 /* The leading dimension of the array VR. LDVR >= 1. */
164 /* If JOB = 'E' or 'B', LDVR >= N. */
165
166 /* S (output) DOUBLE PRECISION array, dimension (MM) */
167 /* If JOB = 'E' or 'B', the reciprocal condition numbers of the */
168 /* selected eigenvalues, stored in consecutive elements of the */
169 /* array. For a complex conjugate pair of eigenvalues two */
170 /* consecutive elements of S are set to the same value. Thus */
171 /* S(j), DIF(j), and the j-th columns of VL and VR all */
172 /* correspond to the same eigenpair (but not in general the */
173 /* j-th eigenpair, unless all eigenpairs are selected). */
174 /* If JOB = 'V', S is not referenced. */
175
176 /* DIF (output) DOUBLE PRECISION array, dimension (MM) */
177 /* If JOB = 'V' or 'B', the estimated reciprocal condition */
178 /* numbers of the selected eigenvectors, stored in consecutive */
179 /* elements of the array. For a complex eigenvector two */
180 /* consecutive elements of DIF are set to the same value. If */
181 /* the eigenvalues cannot be reordered to compute DIF(j), DIF(j) */
182 /* is set to 0; this can only occur when the true value would be */
183 /* very small anyway. */
184 /* If JOB = 'E', DIF is not referenced. */
185
186 /* MM (input) INTEGER */
187 /* The number of elements in the arrays S and DIF. MM >= M. */
188
189 /* M (output) INTEGER */
190 /* The number of elements of the arrays S and DIF used to store */
191 /* the specified condition numbers; for each selected real */
192 /* eigenvalue one element is used, and for each selected complex */
193 /* conjugate pair of eigenvalues, two elements are used. */
194 /* If HOWMNY = 'A', M is set to N. */
195
196 /* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) */
197 /* If JOB = 'E', WORK is not referenced. Otherwise, */
198 /* on exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
199
200 /* LWORK (input) INTEGER */
201 /* The dimension of the array WORK. LWORK >= N. */
202 /* If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. */
203
204 /* If LWORK = -1, then a workspace query is assumed; the routine */
205 /* only calculates the optimal size of the WORK array, returns */
206 /* this value as the first entry of the WORK array, and no error */
207 /* message related to LWORK is issued by XERBLA. */
208
209 /* IWORK (workspace) INTEGER array, dimension (N + 6) */
210 /* If JOB = 'E', IWORK is not referenced. */
211
212 /* INFO (output) INTEGER */
213 /* =0: Successful exit */
214 /* <0: If INFO = -i, the i-th argument had an illegal value */
215
216
217 /* Further Details */
218 /* =============== */
219
220 /* The reciprocal of the condition number of a generalized eigenvalue */
221 /* w = (a, b) is defined as */
222
223 /* S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v)) */
224
225 /* where u and v are the left and right eigenvectors of (A, B) */
226 /* corresponding to w; |z| denotes the absolute value of the complex */
227 /* number, and norm(u) denotes the 2-norm of the vector u. */
228 /* The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv) */
229 /* of the matrix pair (A, B). If both a and b equal zero, then (A B) is */
230 /* singular and S(I) = -1 is returned. */
231
232 /* An approximate error bound on the chordal distance between the i-th */
233 /* computed generalized eigenvalue w and the corresponding exact */
234 /* eigenvalue lambda is */
235
236 /* chord(w, lambda) <= EPS * norm(A, B) / S(I) */
237
238 /* where EPS is the machine precision. */
239
240 /* The reciprocal of the condition number DIF(i) of right eigenvector u */
241 /* and left eigenvector v corresponding to the generalized eigenvalue w */
242 /* is defined as follows: */
243
244 /* a) If the i-th eigenvalue w = (a,b) is real */
245
246 /* Suppose U and V are orthogonal transformations such that */
247
248 /* U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 */
249 /* ( 0 S22 ),( 0 T22 ) n-1 */
250 /* 1 n-1 1 n-1 */
251
252 /* Then the reciprocal condition number DIF(i) is */
253
254 /* Difl((a, b), (S22, T22)) = sigma-min( Zl ), */
255
256 /* where sigma-min(Zl) denotes the smallest singular value of the */
257 /* 2(n-1)-by-2(n-1) matrix */
258
259 /* Zl = [ kron(a, In-1) -kron(1, S22) ] */
260 /* [ kron(b, In-1) -kron(1, T22) ] . */
261
262 /* Here In-1 is the identity matrix of size n-1. kron(X, Y) is the */
263 /* Kronecker product between the matrices X and Y. */
264
265 /* Note that if the default method for computing DIF(i) is wanted */
266 /* (see DLATDF), then the parameter DIFDRI (see below) should be */
267 /* changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). */
268 /* See DTGSYL for more details. */
269
270 /* b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, */
271
272 /* Suppose U and V are orthogonal transformations such that */
273
274 /* U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 */
275 /* ( 0 S22 ),( 0 T22) n-2 */
276 /* 2 n-2 2 n-2 */
277
278 /* and (S11, T11) corresponds to the complex conjugate eigenvalue */
279 /* pair (w, conjg(w)). There exist unitary matrices U1 and V1 such */
280 /* that */
281
282 /* U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 ) */
283 /* ( 0 s22 ) ( 0 t22 ) */
284
285 /* where the generalized eigenvalues w = s11/t11 and */
286 /* conjg(w) = s22/t22. */
287
288 /* Then the reciprocal condition number DIF(i) is bounded by */
289
290 /* min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) */
291
292 /* where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where */
293 /* Z1 is the complex 2-by-2 matrix */
294
295 /* Z1 = [ s11 -s22 ] */
296 /* [ t11 -t22 ], */
297
298 /* This is done by computing (using real arithmetic) the */
299 /* roots of the characteristical polynomial det(Z1' * Z1 - lambda I), */
300 /* where Z1' denotes the conjugate transpose of Z1 and det(X) denotes */
301 /* the determinant of X. */
302
303 /* and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an */
304 /* upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) */
305
306 /* Z2 = [ kron(S11', In-2) -kron(I2, S22) ] */
307 /* [ kron(T11', In-2) -kron(I2, T22) ] */
308
309 /* Note that if the default method for computing DIF is wanted (see */
310 /* DLATDF), then the parameter DIFDRI (see below) should be changed */
311 /* from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL */
312 /* for more details. */
313
314 /* For each eigenvalue/vector specified by SELECT, DIF stores a */
315 /* Frobenius norm-based estimate of Difl. */
316
317 /* An approximate error bound for the i-th computed eigenvector VL(i) or */
318 /* VR(i) is given by */
319
320 /* EPS * norm(A, B) / DIF(i). */
321
322 /* See ref. [2-3] for more details and further references. */
323
324 /* Based on contributions by */
325 /* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
326 /* Umea University, S-901 87 Umea, Sweden. */
327
328 /* References */
329 /* ========== */
330
331 /* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */
332 /* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */
333 /* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */
334 /* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */
335
336 /* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */
337 /* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */
338 /* Estimation: Theory, Algorithms and Software, */
339 /* Report UMINF - 94.04, Department of Computing Science, Umea */
340 /* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */
341 /* Note 87. To appear in Numerical Algorithms, 1996. */
342
343 /* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */
344 /* for Solving the Generalized Sylvester Equation and Estimating the */
345 /* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */
346 /* Department of Computing Science, Umea University, S-901 87 Umea, */
347 /* Sweden, December 1993, Revised April 1994, Also as LAPACK Working */
348 /* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, */
349 /* No 1, 1996. */
350
351 /* ===================================================================== */
352
353 /* .. Parameters .. */
354 /* .. */
355 /* .. Local Scalars .. */
356 /* .. */
357 /* .. Local Arrays .. */
358 /* .. */
359 /* .. External Functions .. */
360 /* .. */
361 /* .. External Subroutines .. */
362 /* .. */
363 /* .. Intrinsic Functions .. */
364 /* .. */
365 /* .. Executable Statements .. */
366
367 /* Decode and test the input parameters */
368
369 /* Parameter adjustments */
370 --select;
371 a_dim1 = *lda;
372 a_offset = 1 + a_dim1;
373 a -= a_offset;
374 b_dim1 = *ldb;
375 b_offset = 1 + b_dim1;
376 b -= b_offset;
377 vl_dim1 = *ldvl;
378 vl_offset = 1 + vl_dim1;
379 vl -= vl_offset;
380 vr_dim1 = *ldvr;
381 vr_offset = 1 + vr_dim1;
382 vr -= vr_offset;
383 --s;
384 --dif;
385 --work;
386 --iwork;
387
388 /* Function Body */
389 wantbh = lsame_(job, "B", (ftnlen)1, (ftnlen)1);
390 wants = lsame_(job, "E", (ftnlen)1, (ftnlen)1) || wantbh;
391 wantdf = lsame_(job, "V", (ftnlen)1, (ftnlen)1) || wantbh;
392
393 somcon = lsame_(howmny, "S", (ftnlen)1, (ftnlen)1);
394
395 *info = 0;
396 lquery = *lwork == -1;
397
398 if (lsame_(job, "V", (ftnlen)1, (ftnlen)1) || lsame_(job, "B", (ftnlen)1,
399 (ftnlen)1)) {
400 /* Computing MAX */
401 i__1 = 1, i__2 = (*n << 1) * (*n + 2) + 16;
402 lwmin = max(i__1,i__2);
403 } else {
404 lwmin = 1;
405 }
406
407 if (! wants && ! wantdf) {
408 *info = -1;
409 } else if (! lsame_(howmny, "A", (ftnlen)1, (ftnlen)1) && ! somcon) {
410 *info = -2;
411 } else if (*n < 0) {
412 *info = -4;
413 } else if (*lda < max(1,*n)) {
414 *info = -6;
415 } else if (*ldb < max(1,*n)) {
416 *info = -8;
417 } else if (wants && *ldvl < *n) {
418 *info = -10;
419 } else if (wants && *ldvr < *n) {
420 *info = -12;
421 } else {
422
423 /* Set M to the number of eigenpairs for which condition numbers */
424 /* are required, and test MM. */
425
426 if (somcon) {
427 *m = 0;
428 pair = FALSE_;
429 i__1 = *n;
430 for (k = 1; k <= i__1; ++k) {
431 if (pair) {
432 pair = FALSE_;
433 } else {
434 if (k < *n) {
435 if (a[k + 1 + k * a_dim1] == 0.) {
436 if (select[k]) {
437 ++(*m);
438 }
439 } else {
440 pair = TRUE_;
441 if (select[k] || select[k + 1]) {
442 *m += 2;
443 }
444 }
445 } else {
446 if (select[*n]) {
447 ++(*m);
448 }
449 }
450 }
451 /* L10: */
452 }
453 } else {
454 *m = *n;
455 }
456
457 if (*mm < *m) {
458 *info = -15;
459 } else if (*lwork < lwmin && ! lquery) {
460 *info = -18;
461 /* ELSE IF( WANTDF .AND. LWORK.LT.2*N*( N+2 )+16 ) THEN */
462 /* INFO = -18 */
463 }
464 }
465
466 if (*info == 0) {
467 work[1] = (doublereal) lwmin;
468 }
469
470 if (*info != 0) {
471 i__1 = -(*info);
472 xerbla_("DTGSNA", &i__1, (ftnlen)6);
473 return 0;
474 } else if (lquery) {
475 return 0;
476 }
477
478 /* Quick return if possible */
479
480 if (*n == 0) {
481 return 0;
482 }
483
484 /* Get machine constants */
485
486 eps = dlamch_("P", (ftnlen)1);
487 smlnum = dlamch_("S", (ftnlen)1) / eps;
488 ks = 0;
489 pair = FALSE_;
490
491 i__1 = *n;
492 for (k = 1; k <= i__1; ++k) {
493
494 /* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block. */
495
496 if (pair) {
497 pair = FALSE_;
498 goto L20;
499 } else {
500 if (k < *n) {
501 pair = a[k + 1 + k * a_dim1] != 0.;
502 }
503 }
504
505 /* Determine whether condition numbers are required for the k-th */
506 /* eigenpair. */
507
508 if (somcon) {
509 if (pair) {
510 if (! select[k] && ! select[k + 1]) {
511 goto L20;
512 }
513 } else {
514 if (! select[k]) {
515 goto L20;
516 }
517 }
518 }
519
520 ++ks;
521
522 if (wants) {
523
524 /* Compute the reciprocal condition number of the k-th */
525 /* eigenvalue. */
526
527 if (pair) {
528
529 /* Complex eigenvalue pair. */
530
531 d__1 = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
532 d__2 = dnrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1);
533 rnrm = dlapy2_(&d__1, &d__2);
534 d__1 = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
535 d__2 = dnrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1);
536 lnrm = dlapy2_(&d__1, &d__2);
537 dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1
538 + 1], &c__1, &c_b21, &work[1], &c__1, (ftnlen)1);
539 tmprr = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &
540 c__1);
541 tmpri = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1],
542 &c__1);
543 dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[(ks + 1) *
544 vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1, (ftnlen)
545 1);
546 tmpii = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1],
547 &c__1);
548 tmpir = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &
549 c__1);
550 uhav = tmprr + tmpii;
551 uhavi = tmpir - tmpri;
552 dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1
553 + 1], &c__1, &c_b21, &work[1], &c__1, (ftnlen)1);
554 tmprr = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &
555 c__1);
556 tmpri = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1],
557 &c__1);
558 dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[(ks + 1) *
559 vr_dim1 + 1], &c__1, &c_b21, &work[1], &c__1, (ftnlen)
560 1);
561 tmpii = ddot_(n, &work[1], &c__1, &vl[(ks + 1) * vl_dim1 + 1],
562 &c__1);
563 tmpir = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &
564 c__1);
565 uhbv = tmprr + tmpii;
566 uhbvi = tmpir - tmpri;
567 uhav = dlapy2_(&uhav, &uhavi);
568 uhbv = dlapy2_(&uhbv, &uhbvi);
569 cond = dlapy2_(&uhav, &uhbv);
570 s[ks] = cond / (rnrm * lnrm);
571 s[ks + 1] = s[ks];
572
573 } else {
574
575 /* Real eigenvalue. */
576
577 rnrm = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
578 lnrm = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
579 dgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1
580 + 1], &c__1, &c_b21, &work[1], &c__1, (ftnlen)1);
581 uhav = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1)
582 ;
583 dgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1
584 + 1], &c__1, &c_b21, &work[1], &c__1, (ftnlen)1);
585 uhbv = ddot_(n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1)
586 ;
587 cond = dlapy2_(&uhav, &uhbv);
588 if (cond == 0.) {
589 s[ks] = -1.;
590 } else {
591 s[ks] = cond / (rnrm * lnrm);
592 }
593 }
594 }
595
596 if (wantdf) {
597 if (*n == 1) {
598 dif[ks] = dlapy2_(&a[a_dim1 + 1], &b[b_dim1 + 1]);
599 goto L20;
600 }
601
602 /* Estimate the reciprocal condition number of the k-th */
603 /* eigenvectors. */
604 if (pair) {
605
606 /* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)). */
607 /* Compute the eigenvalue(s) at position K. */
608
609 work[1] = a[k + k * a_dim1];
610 work[2] = a[k + 1 + k * a_dim1];
611 work[3] = a[k + (k + 1) * a_dim1];
612 work[4] = a[k + 1 + (k + 1) * a_dim1];
613 work[5] = b[k + k * b_dim1];
614 work[6] = b[k + 1 + k * b_dim1];
615 work[7] = b[k + (k + 1) * b_dim1];
616 work[8] = b[k + 1 + (k + 1) * b_dim1];
617 d__1 = smlnum * eps;
618 dlag2_(&work[1], &c__2, &work[5], &c__2, &d__1, &beta, dummy1,
619 &alphar, dummy, &alphai);
620 alprqt = 1.;
621 c1 = (alphar * alphar + alphai * alphai + beta * beta) * 2.;
622 c2 = beta * 4. * beta * alphai * alphai;
623 root1 = c1 + sqrt(c1 * c1 - c2 * 4.);
624 root2 = c2 / root1;
625 root1 /= 2.;
626 /* Computing MIN */
627 d__1 = sqrt(root1), d__2 = sqrt(root2);
628 cond = min(d__1,d__2);
629 }
630
631 /* Copy the matrix (A, B) to the array WORK and swap the */
632 /* diagonal block beginning at A(k,k) to the (1,1) position. */
633
634 dlacpy_("Full", n, n, &a[a_offset], lda, &work[1], n, (ftnlen)4);
635 dlacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], n, (
636 ftnlen)4);
637 ifst = k;
638 ilst = 1;
639
640 i__2 = *lwork - (*n << 1) * *n;
641 dtgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1], n,
642 dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &work[(*n * *
643 n << 1) + 1], &i__2, &ierr);
644
645 if (ierr > 0) {
646
647 /* Ill-conditioned problem - swap rejected. */
648
649 dif[ks] = 0.;
650 } else {
651
652 /* Reordering successful, solve generalized Sylvester */
653 /* equation for R and L, */
654 /* A22 * R - L * A11 = A12 */
655 /* B22 * R - L * B11 = B12, */
656 /* and compute estimate of Difl((A11,B11), (A22, B22)). */
657
658 n1 = 1;
659 if (work[2] != 0.) {
660 n1 = 2;
661 }
662 n2 = *n - n1;
663 if (n2 == 0) {
664 dif[ks] = cond;
665 } else {
666 i__ = *n * *n + 1;
667 iz = (*n << 1) * *n + 1;
668 i__2 = *lwork - (*n << 1) * *n;
669 dtgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n,
670 &work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1
671 + i__], n, &work[i__], n, &work[n1 + i__], n, &
672 scale, &dif[ks], &work[iz + 1], &i__2, &iwork[1],
673 &ierr, (ftnlen)1);
674
675 if (pair) {
676 /* Computing MIN */
677 d__1 = max(1.,alprqt) * dif[ks];
678 dif[ks] = min(d__1,cond);
679 }
680 }
681 }
682 if (pair) {
683 dif[ks + 1] = dif[ks];
684 }
685 }
686 if (pair) {
687 ++ks;
688 }
689
690 L20:
691 ;
692 }
693 work[1] = (doublereal) lwmin;
694 return 0;
695
696 /* End of DTGSNA */
697
698 } /* dtgsna_ */
699
700