1: SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
2: $ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF,
3: $ WORK, LWORK, IWORK, LIWORK, INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * January 2007
9: *
10: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
11: *
12: * .. Scalar Arguments ..
13: LOGICAL WANTQ, WANTZ
14: INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
15: $ M, N
16: DOUBLE PRECISION PL, PR
17: * ..
18: * .. Array Arguments ..
19: LOGICAL SELECT( * )
20: INTEGER IWORK( * )
21: DOUBLE PRECISION DIF( * )
22: COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
23: $ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
24: * ..
25: *
26: * Purpose
27: * =======
28: *
29: * ZTGSEN reorders the generalized Schur decomposition of a complex
30: * matrix pair (A, B) (in terms of an unitary equivalence trans-
31: * formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
32: * appears in the leading diagonal blocks of the pair (A,B). The leading
33: * columns of Q and Z form unitary bases of the corresponding left and
34: * right eigenspaces (deflating subspaces). (A, B) must be in
35: * generalized Schur canonical form, that is, A and B are both upper
36: * triangular.
37: *
38: * ZTGSEN also computes the generalized eigenvalues
39: *
40: * w(j)= ALPHA(j) / BETA(j)
41: *
42: * of the reordered matrix pair (A, B).
43: *
44: * Optionally, the routine computes estimates of reciprocal condition
45: * numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
46: * (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
47: * between the matrix pairs (A11, B11) and (A22,B22) that correspond to
48: * the selected cluster and the eigenvalues outside the cluster, resp.,
49: * and norms of "projections" onto left and right eigenspaces w.r.t.
50: * the selected cluster in the (1,1)-block.
51: *
52: *
53: * Arguments
54: * =========
55: *
56: * IJOB (input) integer
57: * Specifies whether condition numbers are required for the
58: * cluster of eigenvalues (PL and PR) or the deflating subspaces
59: * (Difu and Difl):
60: * =0: Only reorder w.r.t. SELECT. No extras.
61: * =1: Reciprocal of norms of "projections" onto left and right
62: * eigenspaces w.r.t. the selected cluster (PL and PR).
63: * =2: Upper bounds on Difu and Difl. F-norm-based estimate
64: * (DIF(1:2)).
65: * =3: Estimate of Difu and Difl. 1-norm-based estimate
66: * (DIF(1:2)).
67: * About 5 times as expensive as IJOB = 2.
68: * =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
69: * version to get it all.
70: * =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
71: *
72: * WANTQ (input) LOGICAL
73: * .TRUE. : update the left transformation matrix Q;
74: * .FALSE.: do not update Q.
75: *
76: * WANTZ (input) LOGICAL
77: * .TRUE. : update the right transformation matrix Z;
78: * .FALSE.: do not update Z.
79: *
80: * SELECT (input) LOGICAL array, dimension (N)
81: * SELECT specifies the eigenvalues in the selected cluster. To
82: * select an eigenvalue w(j), SELECT(j) must be set to
83: * .TRUE..
84: *
85: * N (input) INTEGER
86: * The order of the matrices A and B. N >= 0.
87: *
88: * A (input/output) COMPLEX*16 array, dimension(LDA,N)
89: * On entry, the upper triangular matrix A, in generalized
90: * Schur canonical form.
91: * On exit, A is overwritten by the reordered matrix A.
92: *
93: * LDA (input) INTEGER
94: * The leading dimension of the array A. LDA >= max(1,N).
95: *
96: * B (input/output) COMPLEX*16 array, dimension(LDB,N)
97: * On entry, the upper triangular matrix B, in generalized
98: * Schur canonical form.
99: * On exit, B is overwritten by the reordered matrix B.
100: *
101: * LDB (input) INTEGER
102: * The leading dimension of the array B. LDB >= max(1,N).
103: *
104: * ALPHA (output) COMPLEX*16 array, dimension (N)
105: * BETA (output) COMPLEX*16 array, dimension (N)
106: * The diagonal elements of A and B, respectively,
107: * when the pair (A,B) has been reduced to generalized Schur
108: * form. ALPHA(i)/BETA(i) i=1,...,N are the generalized
109: * eigenvalues.
110: *
111: * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
112: * On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
113: * On exit, Q has been postmultiplied by the left unitary
114: * transformation matrix which reorder (A, B); The leading M
115: * columns of Q form orthonormal bases for the specified pair of
116: * left eigenspaces (deflating subspaces).
117: * If WANTQ = .FALSE., Q is not referenced.
118: *
119: * LDQ (input) INTEGER
120: * The leading dimension of the array Q. LDQ >= 1.
121: * If WANTQ = .TRUE., LDQ >= N.
122: *
123: * Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
124: * On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
125: * On exit, Z has been postmultiplied by the left unitary
126: * transformation matrix which reorder (A, B); The leading M
127: * columns of Z form orthonormal bases for the specified pair of
128: * left eigenspaces (deflating subspaces).
129: * If WANTZ = .FALSE., Z is not referenced.
130: *
131: * LDZ (input) INTEGER
132: * The leading dimension of the array Z. LDZ >= 1.
133: * If WANTZ = .TRUE., LDZ >= N.
134: *
135: * M (output) INTEGER
136: * The dimension of the specified pair of left and right
137: * eigenspaces, (deflating subspaces) 0 <= M <= N.
138: *
139: * PL (output) DOUBLE PRECISION
140: * PR (output) DOUBLE PRECISION
141: * If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
142: * reciprocal of the norm of "projections" onto left and right
143: * eigenspace with respect to the selected cluster.
144: * 0 < PL, PR <= 1.
145: * If M = 0 or M = N, PL = PR = 1.
146: * If IJOB = 0, 2 or 3 PL, PR are not referenced.
147: *
148: * DIF (output) DOUBLE PRECISION array, dimension (2).
149: * If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
150: * If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
151: * Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
152: * estimates of Difu and Difl, computed using reversed
153: * communication with ZLACN2.
154: * If M = 0 or N, DIF(1:2) = F-norm([A, B]).
155: * If IJOB = 0 or 1, DIF is not referenced.
156: *
157: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
158: * IF IJOB = 0, WORK is not referenced. Otherwise,
159: * on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
160: *
161: * LWORK (input) INTEGER
162: * The dimension of the array WORK. LWORK >= 1
163: * If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M)
164: * If IJOB = 3 or 5, LWORK >= 4*M*(N-M)
165: *
166: * If LWORK = -1, then a workspace query is assumed; the routine
167: * only calculates the optimal size of the WORK array, returns
168: * this value as the first entry of the WORK array, and no error
169: * message related to LWORK is issued by XERBLA.
170: *
171: * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
172: * IF IJOB = 0, IWORK is not referenced. Otherwise,
173: * on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
174: *
175: * LIWORK (input) INTEGER
176: * The dimension of the array IWORK. LIWORK >= 1.
177: * If IJOB = 1, 2 or 4, LIWORK >= N+2;
178: * If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
179: *
180: * If LIWORK = -1, then a workspace query is assumed; the
181: * routine only calculates the optimal size of the IWORK array,
182: * returns this value as the first entry of the IWORK array, and
183: * no error message related to LIWORK is issued by XERBLA.
184: *
185: * INFO (output) INTEGER
186: * =0: Successful exit.
187: * <0: If INFO = -i, the i-th argument had an illegal value.
188: * =1: Reordering of (A, B) failed because the transformed
189: * matrix pair (A, B) would be too far from generalized
190: * Schur form; the problem is very ill-conditioned.
191: * (A, B) may have been partially reordered.
192: * If requested, 0 is returned in DIF(*), PL and PR.
193: *
194: *
195: * Further Details
196: * ===============
197: *
198: * ZTGSEN first collects the selected eigenvalues by computing unitary
199: * U and W that move them to the top left corner of (A, B). In other
200: * words, the selected eigenvalues are the eigenvalues of (A11, B11) in
201: *
202: * U'*(A, B)*W = (A11 A12) (B11 B12) n1
203: * ( 0 A22),( 0 B22) n2
204: * n1 n2 n1 n2
205: *
206: * where N = n1+n2 and U' means the conjugate transpose of U. The first
207: * n1 columns of U and W span the specified pair of left and right
208: * eigenspaces (deflating subspaces) of (A, B).
209: *
210: * If (A, B) has been obtained from the generalized real Schur
211: * decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
212: * reordered generalized Schur form of (C, D) is given by
213: *
214: * (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
215: *
216: * and the first n1 columns of Q*U and Z*W span the corresponding
217: * deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
218: *
219: * Note that if the selected eigenvalue is sufficiently ill-conditioned,
220: * then its value may differ significantly from its value before
221: * reordering.
222: *
223: * The reciprocal condition numbers of the left and right eigenspaces
224: * spanned by the first n1 columns of U and W (or Q*U and Z*W) may
225: * be returned in DIF(1:2), corresponding to Difu and Difl, resp.
226: *
227: * The Difu and Difl are defined as:
228: *
229: * Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
230: * and
231: * Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
232: *
233: * where sigma-min(Zu) is the smallest singular value of the
234: * (2*n1*n2)-by-(2*n1*n2) matrix
235: *
236: * Zu = [ kron(In2, A11) -kron(A22', In1) ]
237: * [ kron(In2, B11) -kron(B22', In1) ].
238: *
239: * Here, Inx is the identity matrix of size nx and A22' is the
240: * transpose of A22. kron(X, Y) is the Kronecker product between
241: * the matrices X and Y.
242: *
243: * When DIF(2) is small, small changes in (A, B) can cause large changes
244: * in the deflating subspace. An approximate (asymptotic) bound on the
245: * maximum angular error in the computed deflating subspaces is
246: *
247: * EPS * norm((A, B)) / DIF(2),
248: *
249: * where EPS is the machine precision.
250: *
251: * The reciprocal norm of the projectors on the left and right
252: * eigenspaces associated with (A11, B11) may be returned in PL and PR.
253: * They are computed as follows. First we compute L and R so that
254: * P*(A, B)*Q is block diagonal, where
255: *
256: * P = ( I -L ) n1 Q = ( I R ) n1
257: * ( 0 I ) n2 and ( 0 I ) n2
258: * n1 n2 n1 n2
259: *
260: * and (L, R) is the solution to the generalized Sylvester equation
261: *
262: * A11*R - L*A22 = -A12
263: * B11*R - L*B22 = -B12
264: *
265: * Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
266: * An approximate (asymptotic) bound on the average absolute error of
267: * the selected eigenvalues is
268: *
269: * EPS * norm((A, B)) / PL.
270: *
271: * There are also global error bounds which valid for perturbations up
272: * to a certain restriction: A lower bound (x) on the smallest
273: * F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
274: * coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
275: * (i.e. (A + E, B + F), is
276: *
277: * x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
278: *
279: * An approximate bound on x can be computed from DIF(1:2), PL and PR.
280: *
281: * If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
282: * (L', R') and unperturbed (L, R) left and right deflating subspaces
283: * associated with the selected cluster in the (1,1)-blocks can be
284: * bounded as
285: *
286: * max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
287: * max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
288: *
289: * See LAPACK User's Guide section 4.11 or the following references
290: * for more information.
291: *
292: * Note that if the default method for computing the Frobenius-norm-
293: * based estimate DIF is not wanted (see ZLATDF), then the parameter
294: * IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF
295: * (IJOB = 2 will be used)). See ZTGSYL for more details.
296: *
297: * Based on contributions by
298: * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
299: * Umea University, S-901 87 Umea, Sweden.
300: *
301: * References
302: * ==========
303: *
304: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
305: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
306: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
307: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
308: *
309: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
310: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
311: * Estimation: Theory, Algorithms and Software, Report
312: * UMINF - 94.04, Department of Computing Science, Umea University,
313: * S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
314: * To appear in Numerical Algorithms, 1996.
315: *
316: * [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
317: * for Solving the Generalized Sylvester Equation and Estimating the
318: * Separation between Regular Matrix Pairs, Report UMINF - 93.23,
319: * Department of Computing Science, Umea University, S-901 87 Umea,
320: * Sweden, December 1993, Revised April 1994, Also as LAPACK working
321: * Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
322: * 1996.
323: *
324: * =====================================================================
325: *
326: * .. Parameters ..
327: INTEGER IDIFJB
328: PARAMETER ( IDIFJB = 3 )
329: DOUBLE PRECISION ZERO, ONE
330: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
331: * ..
332: * .. Local Scalars ..
333: LOGICAL LQUERY, SWAP, WANTD, WANTD1, WANTD2, WANTP
334: INTEGER I, IERR, IJB, K, KASE, KS, LIWMIN, LWMIN, MN2,
335: $ N1, N2
336: DOUBLE PRECISION DSCALE, DSUM, RDSCAL, SAFMIN
337: COMPLEX*16 TEMP1, TEMP2
338: * ..
339: * .. Local Arrays ..
340: INTEGER ISAVE( 3 )
341: * ..
342: * .. External Subroutines ..
343: EXTERNAL XERBLA, ZLACN2, ZLACPY, ZLASSQ, ZSCAL, ZTGEXC,
344: $ ZTGSYL
345: * ..
346: * .. Intrinsic Functions ..
347: INTRINSIC ABS, DCMPLX, DCONJG, MAX, SQRT
348: * ..
349: * .. External Functions ..
350: DOUBLE PRECISION DLAMCH
351: EXTERNAL DLAMCH
352: * ..
353: * .. Executable Statements ..
354: *
355: * Decode and test the input parameters
356: *
357: INFO = 0
358: LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
359: *
360: IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN
361: INFO = -1
362: ELSE IF( N.LT.0 ) THEN
363: INFO = -5
364: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
365: INFO = -7
366: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
367: INFO = -9
368: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
369: INFO = -13
370: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
371: INFO = -15
372: END IF
373: *
374: IF( INFO.NE.0 ) THEN
375: CALL XERBLA( 'ZTGSEN', -INFO )
376: RETURN
377: END IF
378: *
379: IERR = 0
380: *
381: WANTP = IJOB.EQ.1 .OR. IJOB.GE.4
382: WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4
383: WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5
384: WANTD = WANTD1 .OR. WANTD2
385: *
386: * Set M to the dimension of the specified pair of deflating
387: * subspaces.
388: *
389: M = 0
390: DO 10 K = 1, N
391: ALPHA( K ) = A( K, K )
392: BETA( K ) = B( K, K )
393: IF( K.LT.N ) THEN
394: IF( SELECT( K ) )
395: $ M = M + 1
396: ELSE
397: IF( SELECT( N ) )
398: $ M = M + 1
399: END IF
400: 10 CONTINUE
401: *
402: IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
403: LWMIN = MAX( 1, 2*M*( N-M ) )
404: LIWMIN = MAX( 1, N+2 )
405: ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN
406: LWMIN = MAX( 1, 4*M*( N-M ) )
407: LIWMIN = MAX( 1, 2*M*( N-M ), N+2 )
408: ELSE
409: LWMIN = 1
410: LIWMIN = 1
411: END IF
412: *
413: WORK( 1 ) = LWMIN
414: IWORK( 1 ) = LIWMIN
415: *
416: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
417: INFO = -21
418: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
419: INFO = -23
420: END IF
421: *
422: IF( INFO.NE.0 ) THEN
423: CALL XERBLA( 'ZTGSEN', -INFO )
424: RETURN
425: ELSE IF( LQUERY ) THEN
426: RETURN
427: END IF
428: *
429: * Quick return if possible.
430: *
431: IF( M.EQ.N .OR. M.EQ.0 ) THEN
432: IF( WANTP ) THEN
433: PL = ONE
434: PR = ONE
435: END IF
436: IF( WANTD ) THEN
437: DSCALE = ZERO
438: DSUM = ONE
439: DO 20 I = 1, N
440: CALL ZLASSQ( N, A( 1, I ), 1, DSCALE, DSUM )
441: CALL ZLASSQ( N, B( 1, I ), 1, DSCALE, DSUM )
442: 20 CONTINUE
443: DIF( 1 ) = DSCALE*SQRT( DSUM )
444: DIF( 2 ) = DIF( 1 )
445: END IF
446: GO TO 70
447: END IF
448: *
449: * Get machine constant
450: *
451: SAFMIN = DLAMCH( 'S' )
452: *
453: * Collect the selected blocks at the top-left corner of (A, B).
454: *
455: KS = 0
456: DO 30 K = 1, N
457: SWAP = SELECT( K )
458: IF( SWAP ) THEN
459: KS = KS + 1
460: *
461: * Swap the K-th block to position KS. Compute unitary Q
462: * and Z that will swap adjacent diagonal blocks in (A, B).
463: *
464: IF( K.NE.KS )
465: $ CALL ZTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
466: $ LDZ, K, KS, IERR )
467: *
468: IF( IERR.GT.0 ) THEN
469: *
470: * Swap is rejected: exit.
471: *
472: INFO = 1
473: IF( WANTP ) THEN
474: PL = ZERO
475: PR = ZERO
476: END IF
477: IF( WANTD ) THEN
478: DIF( 1 ) = ZERO
479: DIF( 2 ) = ZERO
480: END IF
481: GO TO 70
482: END IF
483: END IF
484: 30 CONTINUE
485: IF( WANTP ) THEN
486: *
487: * Solve generalized Sylvester equation for R and L:
488: * A11 * R - L * A22 = A12
489: * B11 * R - L * B22 = B12
490: *
491: N1 = M
492: N2 = N - M
493: I = N1 + 1
494: CALL ZLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 )
495: CALL ZLACPY( 'Full', N1, N2, B( 1, I ), LDB, WORK( N1*N2+1 ),
496: $ N1 )
497: IJB = 0
498: CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
499: $ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1,
500: $ DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
501: $ LWORK-2*N1*N2, IWORK, IERR )
502: *
503: * Estimate the reciprocal of norms of "projections" onto
504: * left and right eigenspaces
505: *
506: RDSCAL = ZERO
507: DSUM = ONE
508: CALL ZLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM )
509: PL = RDSCAL*SQRT( DSUM )
510: IF( PL.EQ.ZERO ) THEN
511: PL = ONE
512: ELSE
513: PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) )
514: END IF
515: RDSCAL = ZERO
516: DSUM = ONE
517: CALL ZLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM )
518: PR = RDSCAL*SQRT( DSUM )
519: IF( PR.EQ.ZERO ) THEN
520: PR = ONE
521: ELSE
522: PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) )
523: END IF
524: END IF
525: IF( WANTD ) THEN
526: *
527: * Compute estimates Difu and Difl.
528: *
529: IF( WANTD1 ) THEN
530: N1 = M
531: N2 = N - M
532: I = N1 + 1
533: IJB = IDIFJB
534: *
535: * Frobenius norm-based Difu estimate.
536: *
537: CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
538: $ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ),
539: $ N1, DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
540: $ LWORK-2*N1*N2, IWORK, IERR )
541: *
542: * Frobenius norm-based Difl estimate.
543: *
544: CALL ZTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, WORK,
545: $ N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ),
546: $ N2, DSCALE, DIF( 2 ), WORK( N1*N2*2+1 ),
547: $ LWORK-2*N1*N2, IWORK, IERR )
548: ELSE
549: *
550: * Compute 1-norm-based estimates of Difu and Difl using
551: * reversed communication with ZLACN2. In each step a
552: * generalized Sylvester equation or a transposed variant
553: * is solved.
554: *
555: KASE = 0
556: N1 = M
557: N2 = N - M
558: I = N1 + 1
559: IJB = 0
560: MN2 = 2*N1*N2
561: *
562: * 1-norm-based estimate of Difu.
563: *
564: 40 CONTINUE
565: CALL ZLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 1 ), KASE,
566: $ ISAVE )
567: IF( KASE.NE.0 ) THEN
568: IF( KASE.EQ.1 ) THEN
569: *
570: * Solve generalized Sylvester equation
571: *
572: CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA,
573: $ WORK, N1, B, LDB, B( I, I ), LDB,
574: $ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
575: $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
576: $ IERR )
577: ELSE
578: *
579: * Solve the transposed variant.
580: *
581: CALL ZTGSYL( 'C', IJB, N1, N2, A, LDA, A( I, I ), LDA,
582: $ WORK, N1, B, LDB, B( I, I ), LDB,
583: $ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
584: $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
585: $ IERR )
586: END IF
587: GO TO 40
588: END IF
589: DIF( 1 ) = DSCALE / DIF( 1 )
590: *
591: * 1-norm-based estimate of Difl.
592: *
593: 50 CONTINUE
594: CALL ZLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 2 ), KASE,
595: $ ISAVE )
596: IF( KASE.NE.0 ) THEN
597: IF( KASE.EQ.1 ) THEN
598: *
599: * Solve generalized Sylvester equation
600: *
601: CALL ZTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA,
602: $ WORK, N2, B( I, I ), LDB, B, LDB,
603: $ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
604: $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
605: $ IERR )
606: ELSE
607: *
608: * Solve the transposed variant.
609: *
610: CALL ZTGSYL( 'C', IJB, N2, N1, A( I, I ), LDA, A, LDA,
611: $ WORK, N2, B, LDB, B( I, I ), LDB,
612: $ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
613: $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
614: $ IERR )
615: END IF
616: GO TO 50
617: END IF
618: DIF( 2 ) = DSCALE / DIF( 2 )
619: END IF
620: END IF
621: *
622: * If B(K,K) is complex, make it real and positive (normalization
623: * of the generalized Schur form) and Store the generalized
624: * eigenvalues of reordered pair (A, B)
625: *
626: DO 60 K = 1, N
627: DSCALE = ABS( B( K, K ) )
628: IF( DSCALE.GT.SAFMIN ) THEN
629: TEMP1 = DCONJG( B( K, K ) / DSCALE )
630: TEMP2 = B( K, K ) / DSCALE
631: B( K, K ) = DSCALE
632: CALL ZSCAL( N-K, TEMP1, B( K, K+1 ), LDB )
633: CALL ZSCAL( N-K+1, TEMP1, A( K, K ), LDA )
634: IF( WANTQ )
635: $ CALL ZSCAL( N, TEMP2, Q( 1, K ), 1 )
636: ELSE
637: B( K, K ) = DCMPLX( ZERO, ZERO )
638: END IF
639: *
640: ALPHA( K ) = A( K, K )
641: BETA( K ) = B( K, K )
642: *
643: 60 CONTINUE
644: *
645: 70 CONTINUE
646: *
647: WORK( 1 ) = LWMIN
648: IWORK( 1 ) = LIWMIN
649: *
650: RETURN
651: *
652: * End of ZTGSEN
653: *
654: END
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