1: *> \brief \b DTGEX2
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DTGEX2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgex2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgex2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgex2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
22: * LDZ, J1, N1, N2, WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * LOGICAL WANTQ, WANTZ
26: * INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
30: * $ WORK( * ), Z( LDZ, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
40: *> of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
41: *> (A, B) by an orthogonal equivalence transformation.
42: *>
43: *> (A, B) must be in generalized real Schur canonical form (as returned
44: *> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
45: *> diagonal blocks. B is upper triangular.
46: *>
47: *> Optionally, the matrices Q and Z of generalized Schur vectors are
48: *> updated.
49: *>
50: *> Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
51: *> Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
52: *>
53: *> \endverbatim
54: *
55: * Arguments:
56: * ==========
57: *
58: *> \param[in] WANTQ
59: *> \verbatim
60: *> WANTQ is LOGICAL
61: *> .TRUE. : update the left transformation matrix Q;
62: *> .FALSE.: do not update Q.
63: *> \endverbatim
64: *>
65: *> \param[in] WANTZ
66: *> \verbatim
67: *> WANTZ is LOGICAL
68: *> .TRUE. : update the right transformation matrix Z;
69: *> .FALSE.: do not update Z.
70: *> \endverbatim
71: *>
72: *> \param[in] N
73: *> \verbatim
74: *> N is INTEGER
75: *> The order of the matrices A and B. N >= 0.
76: *> \endverbatim
77: *>
78: *> \param[in,out] A
79: *> \verbatim
80: *> A is DOUBLE PRECISION array, dimensions (LDA,N)
81: *> On entry, the matrix A in the pair (A, B).
82: *> On exit, the updated matrix A.
83: *> \endverbatim
84: *>
85: *> \param[in] LDA
86: *> \verbatim
87: *> LDA is INTEGER
88: *> The leading dimension of the array A. LDA >= max(1,N).
89: *> \endverbatim
90: *>
91: *> \param[in,out] B
92: *> \verbatim
93: *> B is DOUBLE PRECISION array, dimensions (LDB,N)
94: *> On entry, the matrix B in the pair (A, B).
95: *> On exit, the updated matrix B.
96: *> \endverbatim
97: *>
98: *> \param[in] LDB
99: *> \verbatim
100: *> LDB is INTEGER
101: *> The leading dimension of the array B. LDB >= max(1,N).
102: *> \endverbatim
103: *>
104: *> \param[in,out] Q
105: *> \verbatim
106: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
107: *> On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
108: *> On exit, the updated matrix Q.
109: *> Not referenced if WANTQ = .FALSE..
110: *> \endverbatim
111: *>
112: *> \param[in] LDQ
113: *> \verbatim
114: *> LDQ is INTEGER
115: *> The leading dimension of the array Q. LDQ >= 1.
116: *> If WANTQ = .TRUE., LDQ >= N.
117: *> \endverbatim
118: *>
119: *> \param[in,out] Z
120: *> \verbatim
121: *> Z is DOUBLE PRECISION array, dimension (LDZ,N)
122: *> On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
123: *> On exit, the updated matrix Z.
124: *> Not referenced if WANTZ = .FALSE..
125: *> \endverbatim
126: *>
127: *> \param[in] LDZ
128: *> \verbatim
129: *> LDZ is INTEGER
130: *> The leading dimension of the array Z. LDZ >= 1.
131: *> If WANTZ = .TRUE., LDZ >= N.
132: *> \endverbatim
133: *>
134: *> \param[in] J1
135: *> \verbatim
136: *> J1 is INTEGER
137: *> The index to the first block (A11, B11). 1 <= J1 <= N.
138: *> \endverbatim
139: *>
140: *> \param[in] N1
141: *> \verbatim
142: *> N1 is INTEGER
143: *> The order of the first block (A11, B11). N1 = 0, 1 or 2.
144: *> \endverbatim
145: *>
146: *> \param[in] N2
147: *> \verbatim
148: *> N2 is INTEGER
149: *> The order of the second block (A22, B22). N2 = 0, 1 or 2.
150: *> \endverbatim
151: *>
152: *> \param[out] WORK
153: *> \verbatim
154: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
155: *> \endverbatim
156: *>
157: *> \param[in] LWORK
158: *> \verbatim
159: *> LWORK is INTEGER
160: *> The dimension of the array WORK.
161: *> LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
162: *> \endverbatim
163: *>
164: *> \param[out] INFO
165: *> \verbatim
166: *> INFO is INTEGER
167: *> =0: Successful exit
168: *> >0: If INFO = 1, the transformed matrix (A, B) would be
169: *> too far from generalized Schur form; the blocks are
170: *> not swapped and (A, B) and (Q, Z) are unchanged.
171: *> The problem of swapping is too ill-conditioned.
172: *> <0: If INFO = -16: LWORK is too small. Appropriate value
173: *> for LWORK is returned in WORK(1).
174: *> \endverbatim
175: *
176: * Authors:
177: * ========
178: *
179: *> \author Univ. of Tennessee
180: *> \author Univ. of California Berkeley
181: *> \author Univ. of Colorado Denver
182: *> \author NAG Ltd.
183: *
184: *> \date November 2011
185: *
186: *> \ingroup doubleGEauxiliary
187: *
188: *> \par Further Details:
189: * =====================
190: *>
191: *> In the current code both weak and strong stability tests are
192: *> performed. The user can omit the strong stability test by changing
193: *> the internal logical parameter WANDS to .FALSE.. See ref. [2] for
194: *> details.
195: *
196: *> \par Contributors:
197: * ==================
198: *>
199: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
200: *> Umea University, S-901 87 Umea, Sweden.
201: *
202: *> \par References:
203: * ================
204: *>
205: *> \verbatim
206: *>
207: *> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
208: *> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
209: *> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
210: *> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
211: *>
212: *> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
213: *> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
214: *> Estimation: Theory, Algorithms and Software,
215: *> Report UMINF - 94.04, Department of Computing Science, Umea
216: *> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
217: *> Note 87. To appear in Numerical Algorithms, 1996.
218: *> \endverbatim
219: *>
220: * =====================================================================
221: SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
222: $ LDZ, J1, N1, N2, WORK, LWORK, INFO )
223: *
224: * -- LAPACK auxiliary routine (version 3.4.0) --
225: * -- LAPACK is a software package provided by Univ. of Tennessee, --
226: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
227: * November 2011
228: *
229: * .. Scalar Arguments ..
230: LOGICAL WANTQ, WANTZ
231: INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
232: * ..
233: * .. Array Arguments ..
234: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
235: $ WORK( * ), Z( LDZ, * )
236: * ..
237: *
238: * =====================================================================
239: * Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
240: * loops. Sven Hammarling, 1/5/02.
241: *
242: * .. Parameters ..
243: DOUBLE PRECISION ZERO, ONE
244: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
245: DOUBLE PRECISION TWENTY
246: PARAMETER ( TWENTY = 2.0D+01 )
247: INTEGER LDST
248: PARAMETER ( LDST = 4 )
249: LOGICAL WANDS
250: PARAMETER ( WANDS = .TRUE. )
251: * ..
252: * .. Local Scalars ..
253: LOGICAL DTRONG, WEAK
254: INTEGER I, IDUM, LINFO, M
255: DOUBLE PRECISION BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS,
256: $ F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS
257: * ..
258: * .. Local Arrays ..
259: INTEGER IWORK( LDST )
260: DOUBLE PRECISION AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
261: $ IRCOP( LDST, LDST ), LI( LDST, LDST ),
262: $ LICOP( LDST, LDST ), S( LDST, LDST ),
263: $ SCPY( LDST, LDST ), T( LDST, LDST ),
264: $ TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
265: * ..
266: * .. External Functions ..
267: DOUBLE PRECISION DLAMCH
268: EXTERNAL DLAMCH
269: * ..
270: * .. External Subroutines ..
271: EXTERNAL DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2, DLARTG,
272: $ DLASET, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
273: $ DROT, DSCAL, DTGSY2
274: * ..
275: * .. Intrinsic Functions ..
276: INTRINSIC ABS, MAX, SQRT
277: * ..
278: * .. Executable Statements ..
279: *
280: INFO = 0
281: *
282: * Quick return if possible
283: *
284: IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
285: $ RETURN
286: IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
287: $ RETURN
288: M = N1 + N2
289: IF( LWORK.LT.MAX( 1, N*M, M*M*2 ) ) THEN
290: INFO = -16
291: WORK( 1 ) = MAX( 1, N*M, M*M*2 )
292: RETURN
293: END IF
294: *
295: WEAK = .FALSE.
296: DTRONG = .FALSE.
297: *
298: * Make a local copy of selected block
299: *
300: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, LI, LDST )
301: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, IR, LDST )
302: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
303: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
304: *
305: * Compute threshold for testing acceptance of swapping.
306: *
307: EPS = DLAMCH( 'P' )
308: SMLNUM = DLAMCH( 'S' ) / EPS
309: DSCALE = ZERO
310: DSUM = ONE
311: CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
312: CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
313: CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
314: CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
315: DNORM = DSCALE*SQRT( DSUM )
316: *
317: * THRES has been changed from
318: * THRESH = MAX( TEN*EPS*SA, SMLNUM )
319: * to
320: * THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
321: * on 04/01/10.
322: * "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
323: * Jim Demmel and Guillaume Revy. See forum post 1783.
324: *
325: THRESH = MAX( TWENTY*EPS*DNORM, SMLNUM )
326: *
327: IF( M.EQ.2 ) THEN
328: *
329: * CASE 1: Swap 1-by-1 and 1-by-1 blocks.
330: *
331: * Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
332: * using Givens rotations and perform the swap tentatively.
333: *
334: F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
335: G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
336: SB = ABS( T( 2, 2 ) )
337: SA = ABS( S( 2, 2 ) )
338: CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
339: IR( 2, 1 ) = -IR( 1, 2 )
340: IR( 2, 2 ) = IR( 1, 1 )
341: CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
342: $ IR( 2, 1 ) )
343: CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
344: $ IR( 2, 1 ) )
345: IF( SA.GE.SB ) THEN
346: CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
347: $ DDUM )
348: ELSE
349: CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
350: $ DDUM )
351: END IF
352: CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
353: $ LI( 2, 1 ) )
354: CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
355: $ LI( 2, 1 ) )
356: LI( 2, 2 ) = LI( 1, 1 )
357: LI( 1, 2 ) = -LI( 2, 1 )
358: *
359: * Weak stability test:
360: * |S21| + |T21| <= O(EPS * F-norm((S, T)))
361: *
362: WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
363: WEAK = WS.LE.THRESH
364: IF( .NOT.WEAK )
365: $ GO TO 70
366: *
367: IF( WANDS ) THEN
368: *
369: * Strong stability test:
370: * F-norm((A-QL**T*S*QR, B-QL**T*T*QR)) <= O(EPS*F-norm((A,B)))
371: *
372: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
373: $ M )
374: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
375: $ WORK, M )
376: CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
377: $ WORK( M*M+1 ), M )
378: DSCALE = ZERO
379: DSUM = ONE
380: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
381: *
382: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
383: $ M )
384: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
385: $ WORK, M )
386: CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
387: $ WORK( M*M+1 ), M )
388: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
389: SS = DSCALE*SQRT( DSUM )
390: DTRONG = SS.LE.THRESH
391: IF( .NOT.DTRONG )
392: $ GO TO 70
393: END IF
394: *
395: * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
396: * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
397: *
398: CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
399: $ IR( 2, 1 ) )
400: CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
401: $ IR( 2, 1 ) )
402: CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
403: $ LI( 1, 1 ), LI( 2, 1 ) )
404: CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
405: $ LI( 1, 1 ), LI( 2, 1 ) )
406: *
407: * Set N1-by-N2 (2,1) - blocks to ZERO.
408: *
409: A( J1+1, J1 ) = ZERO
410: B( J1+1, J1 ) = ZERO
411: *
412: * Accumulate transformations into Q and Z if requested.
413: *
414: IF( WANTZ )
415: $ CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
416: $ IR( 2, 1 ) )
417: IF( WANTQ )
418: $ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
419: $ LI( 2, 1 ) )
420: *
421: * Exit with INFO = 0 if swap was successfully performed.
422: *
423: RETURN
424: *
425: ELSE
426: *
427: * CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
428: * and 2-by-2 blocks.
429: *
430: * Solve the generalized Sylvester equation
431: * S11 * R - L * S22 = SCALE * S12
432: * T11 * R - L * T22 = SCALE * T12
433: * for R and L. Solutions in LI and IR.
434: *
435: CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
436: CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
437: $ IR( N2+1, N1+1 ), LDST )
438: CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
439: $ IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
440: $ LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
441: $ LINFO )
442: *
443: * Compute orthogonal matrix QL:
444: *
445: * QL**T * LI = [ TL ]
446: * [ 0 ]
447: * where
448: * LI = [ -L ]
449: * [ SCALE * identity(N2) ]
450: *
451: DO 10 I = 1, N2
452: CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
453: LI( N1+I, I ) = SCALE
454: 10 CONTINUE
455: CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
456: IF( LINFO.NE.0 )
457: $ GO TO 70
458: CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
459: IF( LINFO.NE.0 )
460: $ GO TO 70
461: *
462: * Compute orthogonal matrix RQ:
463: *
464: * IR * RQ**T = [ 0 TR],
465: *
466: * where IR = [ SCALE * identity(N1), R ]
467: *
468: DO 20 I = 1, N1
469: IR( N2+I, I ) = SCALE
470: 20 CONTINUE
471: CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
472: IF( LINFO.NE.0 )
473: $ GO TO 70
474: CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
475: IF( LINFO.NE.0 )
476: $ GO TO 70
477: *
478: * Perform the swapping tentatively:
479: *
480: CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
481: $ WORK, M )
482: CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, S,
483: $ LDST )
484: CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
485: $ WORK, M )
486: CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, T,
487: $ LDST )
488: CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
489: CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
490: CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
491: CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
492: *
493: * Triangularize the B-part by an RQ factorization.
494: * Apply transformation (from left) to A-part, giving S.
495: *
496: CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
497: IF( LINFO.NE.0 )
498: $ GO TO 70
499: CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST, WORK,
500: $ LINFO )
501: IF( LINFO.NE.0 )
502: $ GO TO 70
503: CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST, WORK,
504: $ LINFO )
505: IF( LINFO.NE.0 )
506: $ GO TO 70
507: *
508: * Compute F-norm(S21) in BRQA21. (T21 is 0.)
509: *
510: DSCALE = ZERO
511: DSUM = ONE
512: DO 30 I = 1, N2
513: CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
514: 30 CONTINUE
515: BRQA21 = DSCALE*SQRT( DSUM )
516: *
517: * Triangularize the B-part by a QR factorization.
518: * Apply transformation (from right) to A-part, giving S.
519: *
520: CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
521: IF( LINFO.NE.0 )
522: $ GO TO 70
523: CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY, LDST,
524: $ WORK, INFO )
525: CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP, LDST,
526: $ WORK, INFO )
527: IF( LINFO.NE.0 )
528: $ GO TO 70
529: *
530: * Compute F-norm(S21) in BQRA21. (T21 is 0.)
531: *
532: DSCALE = ZERO
533: DSUM = ONE
534: DO 40 I = 1, N2
535: CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
536: 40 CONTINUE
537: BQRA21 = DSCALE*SQRT( DSUM )
538: *
539: * Decide which method to use.
540: * Weak stability test:
541: * F-norm(S21) <= O(EPS * F-norm((S, T)))
542: *
543: IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN
544: CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
545: CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
546: CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
547: CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
548: ELSE IF( BRQA21.GE.THRESH ) THEN
549: GO TO 70
550: END IF
551: *
552: * Set lower triangle of B-part to zero
553: *
554: CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
555: *
556: IF( WANDS ) THEN
557: *
558: * Strong stability test:
559: * F-norm((A-QL*S*QR**T, B-QL*T*QR**T)) <= O(EPS*F-norm((A,B)))
560: *
561: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
562: $ M )
563: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
564: $ WORK, M )
565: CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
566: $ WORK( M*M+1 ), M )
567: DSCALE = ZERO
568: DSUM = ONE
569: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
570: *
571: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
572: $ M )
573: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
574: $ WORK, M )
575: CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
576: $ WORK( M*M+1 ), M )
577: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
578: SS = DSCALE*SQRT( DSUM )
579: DTRONG = ( SS.LE.THRESH )
580: IF( .NOT.DTRONG )
581: $ GO TO 70
582: *
583: END IF
584: *
585: * If the swap is accepted ("weakly" and "strongly"), apply the
586: * transformations and set N1-by-N2 (2,1)-block to zero.
587: *
588: CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
589: *
590: * copy back M-by-M diagonal block starting at index J1 of (A, B)
591: *
592: CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
593: CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
594: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
595: *
596: * Standardize existing 2-by-2 blocks.
597: *
598: DO 50 I = 1, M*M
599: WORK(I) = ZERO
600: 50 CONTINUE
601: WORK( 1 ) = ONE
602: T( 1, 1 ) = ONE
603: IDUM = LWORK - M*M - 2
604: IF( N2.GT.1 ) THEN
605: CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI, BE,
606: $ WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
607: WORK( M+1 ) = -WORK( 2 )
608: WORK( M+2 ) = WORK( 1 )
609: T( N2, N2 ) = T( 1, 1 )
610: T( 1, 2 ) = -T( 2, 1 )
611: END IF
612: WORK( M*M ) = ONE
613: T( M, M ) = ONE
614: *
615: IF( N1.GT.1 ) THEN
616: CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ), LDB,
617: $ TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
618: $ WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
619: $ T( M, M-1 ) )
620: WORK( M*M ) = WORK( N2*M+N2+1 )
621: WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
622: T( M, M ) = T( N2+1, N2+1 )
623: T( M-1, M ) = -T( M, M-1 )
624: END IF
625: CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1, J1+N2 ),
626: $ LDA, ZERO, WORK( M*M+1 ), N2 )
627: CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1, J1+N2 ),
628: $ LDA )
629: CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1, J1+N2 ),
630: $ LDB, ZERO, WORK( M*M+1 ), N2 )
631: CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1, J1+N2 ),
632: $ LDB )
633: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
634: $ WORK( M*M+1 ), M )
635: CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
636: CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
637: $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
638: CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
639: CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
640: $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
641: CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
642: CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
643: $ WORK, M )
644: CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
645: *
646: * Accumulate transformations into Q and Z if requested.
647: *
648: IF( WANTQ ) THEN
649: CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
650: $ LDST, ZERO, WORK, N )
651: CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
652: *
653: END IF
654: *
655: IF( WANTZ ) THEN
656: CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
657: $ LDST, ZERO, WORK, N )
658: CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
659: *
660: END IF
661: *
662: * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
663: * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
664: *
665: I = J1 + M
666: IF( I.LE.N ) THEN
667: CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
668: $ A( J1, I ), LDA, ZERO, WORK, M )
669: CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
670: CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
671: $ B( J1, I ), LDA, ZERO, WORK, M )
672: CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
673: END IF
674: I = J1 - 1
675: IF( I.GT.0 ) THEN
676: CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
677: $ LDST, ZERO, WORK, I )
678: CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
679: CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
680: $ LDST, ZERO, WORK, I )
681: CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
682: END IF
683: *
684: * Exit with INFO = 0 if swap was successfully performed.
685: *
686: RETURN
687: *
688: END IF
689: *
690: * Exit with INFO = 1 if swap was rejected.
691: *
692: 70 CONTINUE
693: *
694: INFO = 1
695: RETURN
696: *
697: * End of DTGEX2
698: *
699: END
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