1 SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
2 $ LDZ, J1, N1, N2, WORK, LWORK, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 LOGICAL WANTQ, WANTZ
11 INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
15 $ WORK( * ), Z( LDZ, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
22 * of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
23 * (A, B) by an orthogonal equivalence transformation.
24 *
25 * (A, B) must be in generalized real Schur canonical form (as returned
26 * by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
27 * diagonal blocks. B is upper triangular.
28 *
29 * Optionally, the matrices Q and Z of generalized Schur vectors are
30 * updated.
31 *
32 * Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
33 * Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
34 *
35 *
36 * Arguments
37 * =========
38 *
39 * WANTQ (input) LOGICAL
40 * .TRUE. : update the left transformation matrix Q;
41 * .FALSE.: do not update Q.
42 *
43 * WANTZ (input) LOGICAL
44 * .TRUE. : update the right transformation matrix Z;
45 * .FALSE.: do not update Z.
46 *
47 * N (input) INTEGER
48 * The order of the matrices A and B. N >= 0.
49 *
50 * A (input/output) DOUBLE PRECISION array, dimensions (LDA,N)
51 * On entry, the matrix A in the pair (A, B).
52 * On exit, the updated matrix A.
53 *
54 * LDA (input) INTEGER
55 * The leading dimension of the array A. LDA >= max(1,N).
56 *
57 * B (input/output) DOUBLE PRECISION array, dimensions (LDB,N)
58 * On entry, the matrix B in the pair (A, B).
59 * On exit, the updated matrix B.
60 *
61 * LDB (input) INTEGER
62 * The leading dimension of the array B. LDB >= max(1,N).
63 *
64 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
65 * On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
66 * On exit, the updated matrix Q.
67 * Not referenced if WANTQ = .FALSE..
68 *
69 * LDQ (input) INTEGER
70 * The leading dimension of the array Q. LDQ >= 1.
71 * If WANTQ = .TRUE., LDQ >= N.
72 *
73 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
74 * On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
75 * On exit, the updated matrix Z.
76 * Not referenced if WANTZ = .FALSE..
77 *
78 * LDZ (input) INTEGER
79 * The leading dimension of the array Z. LDZ >= 1.
80 * If WANTZ = .TRUE., LDZ >= N.
81 *
82 * J1 (input) INTEGER
83 * The index to the first block (A11, B11). 1 <= J1 <= N.
84 *
85 * N1 (input) INTEGER
86 * The order of the first block (A11, B11). N1 = 0, 1 or 2.
87 *
88 * N2 (input) INTEGER
89 * The order of the second block (A22, B22). N2 = 0, 1 or 2.
90 *
91 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
92 *
93 * LWORK (input) INTEGER
94 * The dimension of the array WORK.
95 * LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
96 *
97 * INFO (output) INTEGER
98 * =0: Successful exit
99 * >0: If INFO = 1, the transformed matrix (A, B) would be
100 * too far from generalized Schur form; the blocks are
101 * not swapped and (A, B) and (Q, Z) are unchanged.
102 * The problem of swapping is too ill-conditioned.
103 * <0: If INFO = -16: LWORK is too small. Appropriate value
104 * for LWORK is returned in WORK(1).
105 *
106 * Further Details
107 * ===============
108 *
109 * Based on contributions by
110 * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
111 * Umea University, S-901 87 Umea, Sweden.
112 *
113 * In the current code both weak and strong stability tests are
114 * performed. The user can omit the strong stability test by changing
115 * the internal logical parameter WANDS to .FALSE.. See ref. [2] for
116 * details.
117 *
118 * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
119 * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
120 * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
121 * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
122 *
123 * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
124 * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
125 * Estimation: Theory, Algorithms and Software,
126 * Report UMINF - 94.04, Department of Computing Science, Umea
127 * University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
128 * Note 87. To appear in Numerical Algorithms, 1996.
129 *
130 * =====================================================================
131 * Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
132 * loops. Sven Hammarling, 1/5/02.
133 *
134 * .. Parameters ..
135 DOUBLE PRECISION ZERO, ONE
136 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
137 DOUBLE PRECISION TWENTY
138 PARAMETER ( TWENTY = 2.0D+01 )
139 INTEGER LDST
140 PARAMETER ( LDST = 4 )
141 LOGICAL WANDS
142 PARAMETER ( WANDS = .TRUE. )
143 * ..
144 * .. Local Scalars ..
145 LOGICAL DTRONG, WEAK
146 INTEGER I, IDUM, LINFO, M
147 DOUBLE PRECISION BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS,
148 $ F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS
149 * ..
150 * .. Local Arrays ..
151 INTEGER IWORK( LDST )
152 DOUBLE PRECISION AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
153 $ IRCOP( LDST, LDST ), LI( LDST, LDST ),
154 $ LICOP( LDST, LDST ), S( LDST, LDST ),
155 $ SCPY( LDST, LDST ), T( LDST, LDST ),
156 $ TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
157 * ..
158 * .. External Functions ..
159 DOUBLE PRECISION DLAMCH
160 EXTERNAL DLAMCH
161 * ..
162 * .. External Subroutines ..
163 EXTERNAL DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2, DLARTG,
164 $ DLASET, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
165 $ DROT, DSCAL, DTGSY2
166 * ..
167 * .. Intrinsic Functions ..
168 INTRINSIC ABS, MAX, SQRT
169 * ..
170 * .. Executable Statements ..
171 *
172 INFO = 0
173 *
174 * Quick return if possible
175 *
176 IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
177 $ RETURN
178 IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
179 $ RETURN
180 M = N1 + N2
181 IF( LWORK.LT.MAX( 1, N*M, M*M*2 ) ) THEN
182 INFO = -16
183 WORK( 1 ) = MAX( 1, N*M, M*M*2 )
184 RETURN
185 END IF
186 *
187 WEAK = .FALSE.
188 DTRONG = .FALSE.
189 *
190 * Make a local copy of selected block
191 *
192 CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, LI, LDST )
193 CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, IR, LDST )
194 CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
195 CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
196 *
197 * Compute threshold for testing acceptance of swapping.
198 *
199 EPS = DLAMCH( 'P' )
200 SMLNUM = DLAMCH( 'S' ) / EPS
201 DSCALE = ZERO
202 DSUM = ONE
203 CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
204 CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
205 CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
206 CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
207 DNORM = DSCALE*SQRT( DSUM )
208 *
209 * THRES has been changed from
210 * THRESH = MAX( TEN*EPS*SA, SMLNUM )
211 * to
212 * THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
213 * on 04/01/10.
214 * "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
215 * Jim Demmel and Guillaume Revy. See forum post 1783.
216 *
217 THRESH = MAX( TWENTY*EPS*DNORM, SMLNUM )
218 *
219 IF( M.EQ.2 ) THEN
220 *
221 * CASE 1: Swap 1-by-1 and 1-by-1 blocks.
222 *
223 * Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
224 * using Givens rotations and perform the swap tentatively.
225 *
226 F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
227 G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
228 SB = ABS( T( 2, 2 ) )
229 SA = ABS( S( 2, 2 ) )
230 CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
231 IR( 2, 1 ) = -IR( 1, 2 )
232 IR( 2, 2 ) = IR( 1, 1 )
233 CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
234 $ IR( 2, 1 ) )
235 CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
236 $ IR( 2, 1 ) )
237 IF( SA.GE.SB ) THEN
238 CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
239 $ DDUM )
240 ELSE
241 CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
242 $ DDUM )
243 END IF
244 CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
245 $ LI( 2, 1 ) )
246 CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
247 $ LI( 2, 1 ) )
248 LI( 2, 2 ) = LI( 1, 1 )
249 LI( 1, 2 ) = -LI( 2, 1 )
250 *
251 * Weak stability test:
252 * |S21| + |T21| <= O(EPS * F-norm((S, T)))
253 *
254 WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
255 WEAK = WS.LE.THRESH
256 IF( .NOT.WEAK )
257 $ GO TO 70
258 *
259 IF( WANDS ) THEN
260 *
261 * Strong stability test:
262 * F-norm((A-QL**T*S*QR, B-QL**T*T*QR)) <= O(EPS*F-norm((A,B)))
263 *
264 CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
265 $ M )
266 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
267 $ WORK, M )
268 CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
269 $ WORK( M*M+1 ), M )
270 DSCALE = ZERO
271 DSUM = ONE
272 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
273 *
274 CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
275 $ M )
276 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
277 $ WORK, M )
278 CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
279 $ WORK( M*M+1 ), M )
280 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
281 SS = DSCALE*SQRT( DSUM )
282 DTRONG = SS.LE.THRESH
283 IF( .NOT.DTRONG )
284 $ GO TO 70
285 END IF
286 *
287 * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
288 * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
289 *
290 CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
291 $ IR( 2, 1 ) )
292 CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
293 $ IR( 2, 1 ) )
294 CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
295 $ LI( 1, 1 ), LI( 2, 1 ) )
296 CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
297 $ LI( 1, 1 ), LI( 2, 1 ) )
298 *
299 * Set N1-by-N2 (2,1) - blocks to ZERO.
300 *
301 A( J1+1, J1 ) = ZERO
302 B( J1+1, J1 ) = ZERO
303 *
304 * Accumulate transformations into Q and Z if requested.
305 *
306 IF( WANTZ )
307 $ CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
308 $ IR( 2, 1 ) )
309 IF( WANTQ )
310 $ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
311 $ LI( 2, 1 ) )
312 *
313 * Exit with INFO = 0 if swap was successfully performed.
314 *
315 RETURN
316 *
317 ELSE
318 *
319 * CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
320 * and 2-by-2 blocks.
321 *
322 * Solve the generalized Sylvester equation
323 * S11 * R - L * S22 = SCALE * S12
324 * T11 * R - L * T22 = SCALE * T12
325 * for R and L. Solutions in LI and IR.
326 *
327 CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
328 CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
329 $ IR( N2+1, N1+1 ), LDST )
330 CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
331 $ IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
332 $ LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
333 $ LINFO )
334 *
335 * Compute orthogonal matrix QL:
336 *
337 * QL**T * LI = [ TL ]
338 * [ 0 ]
339 * where
340 * LI = [ -L ]
341 * [ SCALE * identity(N2) ]
342 *
343 DO 10 I = 1, N2
344 CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
345 LI( N1+I, I ) = SCALE
346 10 CONTINUE
347 CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
348 IF( LINFO.NE.0 )
349 $ GO TO 70
350 CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
351 IF( LINFO.NE.0 )
352 $ GO TO 70
353 *
354 * Compute orthogonal matrix RQ:
355 *
356 * IR * RQ**T = [ 0 TR],
357 *
358 * where IR = [ SCALE * identity(N1), R ]
359 *
360 DO 20 I = 1, N1
361 IR( N2+I, I ) = SCALE
362 20 CONTINUE
363 CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
364 IF( LINFO.NE.0 )
365 $ GO TO 70
366 CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
367 IF( LINFO.NE.0 )
368 $ GO TO 70
369 *
370 * Perform the swapping tentatively:
371 *
372 CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
373 $ WORK, M )
374 CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, S,
375 $ LDST )
376 CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
377 $ WORK, M )
378 CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, T,
379 $ LDST )
380 CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
381 CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
382 CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
383 CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
384 *
385 * Triangularize the B-part by an RQ factorization.
386 * Apply transformation (from left) to A-part, giving S.
387 *
388 CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
389 IF( LINFO.NE.0 )
390 $ GO TO 70
391 CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST, WORK,
392 $ LINFO )
393 IF( LINFO.NE.0 )
394 $ GO TO 70
395 CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST, WORK,
396 $ LINFO )
397 IF( LINFO.NE.0 )
398 $ GO TO 70
399 *
400 * Compute F-norm(S21) in BRQA21. (T21 is 0.)
401 *
402 DSCALE = ZERO
403 DSUM = ONE
404 DO 30 I = 1, N2
405 CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
406 30 CONTINUE
407 BRQA21 = DSCALE*SQRT( DSUM )
408 *
409 * Triangularize the B-part by a QR factorization.
410 * Apply transformation (from right) to A-part, giving S.
411 *
412 CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
413 IF( LINFO.NE.0 )
414 $ GO TO 70
415 CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY, LDST,
416 $ WORK, INFO )
417 CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP, LDST,
418 $ WORK, INFO )
419 IF( LINFO.NE.0 )
420 $ GO TO 70
421 *
422 * Compute F-norm(S21) in BQRA21. (T21 is 0.)
423 *
424 DSCALE = ZERO
425 DSUM = ONE
426 DO 40 I = 1, N2
427 CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
428 40 CONTINUE
429 BQRA21 = DSCALE*SQRT( DSUM )
430 *
431 * Decide which method to use.
432 * Weak stability test:
433 * F-norm(S21) <= O(EPS * F-norm((S, T)))
434 *
435 IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN
436 CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
437 CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
438 CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
439 CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
440 ELSE IF( BRQA21.GE.THRESH ) THEN
441 GO TO 70
442 END IF
443 *
444 * Set lower triangle of B-part to zero
445 *
446 CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
447 *
448 IF( WANDS ) THEN
449 *
450 * Strong stability test:
451 * F-norm((A-QL*S*QR**T, B-QL*T*QR**T)) <= O(EPS*F-norm((A,B)))
452 *
453 CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
454 $ M )
455 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
456 $ WORK, M )
457 CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
458 $ WORK( M*M+1 ), M )
459 DSCALE = ZERO
460 DSUM = ONE
461 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
462 *
463 CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
464 $ M )
465 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
466 $ WORK, M )
467 CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
468 $ WORK( M*M+1 ), M )
469 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
470 SS = DSCALE*SQRT( DSUM )
471 DTRONG = ( SS.LE.THRESH )
472 IF( .NOT.DTRONG )
473 $ GO TO 70
474 *
475 END IF
476 *
477 * If the swap is accepted ("weakly" and "strongly"), apply the
478 * transformations and set N1-by-N2 (2,1)-block to zero.
479 *
480 CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
481 *
482 * copy back M-by-M diagonal block starting at index J1 of (A, B)
483 *
484 CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
485 CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
486 CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
487 *
488 * Standardize existing 2-by-2 blocks.
489 *
490 DO 50 I = 1, M*M
491 WORK(I) = ZERO
492 50 CONTINUE
493 WORK( 1 ) = ONE
494 T( 1, 1 ) = ONE
495 IDUM = LWORK - M*M - 2
496 IF( N2.GT.1 ) THEN
497 CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI, BE,
498 $ WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
499 WORK( M+1 ) = -WORK( 2 )
500 WORK( M+2 ) = WORK( 1 )
501 T( N2, N2 ) = T( 1, 1 )
502 T( 1, 2 ) = -T( 2, 1 )
503 END IF
504 WORK( M*M ) = ONE
505 T( M, M ) = ONE
506 *
507 IF( N1.GT.1 ) THEN
508 CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ), LDB,
509 $ TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
510 $ WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
511 $ T( M, M-1 ) )
512 WORK( M*M ) = WORK( N2*M+N2+1 )
513 WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
514 T( M, M ) = T( N2+1, N2+1 )
515 T( M-1, M ) = -T( M, M-1 )
516 END IF
517 CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1, J1+N2 ),
518 $ LDA, ZERO, WORK( M*M+1 ), N2 )
519 CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1, J1+N2 ),
520 $ LDA )
521 CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1, J1+N2 ),
522 $ LDB, ZERO, WORK( M*M+1 ), N2 )
523 CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1, J1+N2 ),
524 $ LDB )
525 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
526 $ WORK( M*M+1 ), M )
527 CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
528 CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
529 $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
530 CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
531 CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
532 $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
533 CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
534 CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
535 $ WORK, M )
536 CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
537 *
538 * Accumulate transformations into Q and Z if requested.
539 *
540 IF( WANTQ ) THEN
541 CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
542 $ LDST, ZERO, WORK, N )
543 CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
544 *
545 END IF
546 *
547 IF( WANTZ ) THEN
548 CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
549 $ LDST, ZERO, WORK, N )
550 CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
551 *
552 END IF
553 *
554 * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
555 * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
556 *
557 I = J1 + M
558 IF( I.LE.N ) THEN
559 CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
560 $ A( J1, I ), LDA, ZERO, WORK, M )
561 CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
562 CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
563 $ B( J1, I ), LDA, ZERO, WORK, M )
564 CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
565 END IF
566 I = J1 - 1
567 IF( I.GT.0 ) THEN
568 CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
569 $ LDST, ZERO, WORK, I )
570 CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
571 CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
572 $ LDST, ZERO, WORK, I )
573 CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
574 END IF
575 *
576 * Exit with INFO = 0 if swap was successfully performed.
577 *
578 RETURN
579 *
580 END IF
581 *
582 * Exit with INFO = 1 if swap was rejected.
583 *
584 70 CONTINUE
585 *
586 INFO = 1
587 RETURN
588 *
589 * End of DTGEX2
590 *
591 END
2 $ LDZ, J1, N1, N2, WORK, LWORK, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 LOGICAL WANTQ, WANTZ
11 INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
15 $ WORK( * ), Z( LDZ, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
22 * of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
23 * (A, B) by an orthogonal equivalence transformation.
24 *
25 * (A, B) must be in generalized real Schur canonical form (as returned
26 * by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
27 * diagonal blocks. B is upper triangular.
28 *
29 * Optionally, the matrices Q and Z of generalized Schur vectors are
30 * updated.
31 *
32 * Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
33 * Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
34 *
35 *
36 * Arguments
37 * =========
38 *
39 * WANTQ (input) LOGICAL
40 * .TRUE. : update the left transformation matrix Q;
41 * .FALSE.: do not update Q.
42 *
43 * WANTZ (input) LOGICAL
44 * .TRUE. : update the right transformation matrix Z;
45 * .FALSE.: do not update Z.
46 *
47 * N (input) INTEGER
48 * The order of the matrices A and B. N >= 0.
49 *
50 * A (input/output) DOUBLE PRECISION array, dimensions (LDA,N)
51 * On entry, the matrix A in the pair (A, B).
52 * On exit, the updated matrix A.
53 *
54 * LDA (input) INTEGER
55 * The leading dimension of the array A. LDA >= max(1,N).
56 *
57 * B (input/output) DOUBLE PRECISION array, dimensions (LDB,N)
58 * On entry, the matrix B in the pair (A, B).
59 * On exit, the updated matrix B.
60 *
61 * LDB (input) INTEGER
62 * The leading dimension of the array B. LDB >= max(1,N).
63 *
64 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
65 * On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
66 * On exit, the updated matrix Q.
67 * Not referenced if WANTQ = .FALSE..
68 *
69 * LDQ (input) INTEGER
70 * The leading dimension of the array Q. LDQ >= 1.
71 * If WANTQ = .TRUE., LDQ >= N.
72 *
73 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
74 * On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
75 * On exit, the updated matrix Z.
76 * Not referenced if WANTZ = .FALSE..
77 *
78 * LDZ (input) INTEGER
79 * The leading dimension of the array Z. LDZ >= 1.
80 * If WANTZ = .TRUE., LDZ >= N.
81 *
82 * J1 (input) INTEGER
83 * The index to the first block (A11, B11). 1 <= J1 <= N.
84 *
85 * N1 (input) INTEGER
86 * The order of the first block (A11, B11). N1 = 0, 1 or 2.
87 *
88 * N2 (input) INTEGER
89 * The order of the second block (A22, B22). N2 = 0, 1 or 2.
90 *
91 * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
92 *
93 * LWORK (input) INTEGER
94 * The dimension of the array WORK.
95 * LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
96 *
97 * INFO (output) INTEGER
98 * =0: Successful exit
99 * >0: If INFO = 1, the transformed matrix (A, B) would be
100 * too far from generalized Schur form; the blocks are
101 * not swapped and (A, B) and (Q, Z) are unchanged.
102 * The problem of swapping is too ill-conditioned.
103 * <0: If INFO = -16: LWORK is too small. Appropriate value
104 * for LWORK is returned in WORK(1).
105 *
106 * Further Details
107 * ===============
108 *
109 * Based on contributions by
110 * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
111 * Umea University, S-901 87 Umea, Sweden.
112 *
113 * In the current code both weak and strong stability tests are
114 * performed. The user can omit the strong stability test by changing
115 * the internal logical parameter WANDS to .FALSE.. See ref. [2] for
116 * details.
117 *
118 * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
119 * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
120 * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
121 * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
122 *
123 * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
124 * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
125 * Estimation: Theory, Algorithms and Software,
126 * Report UMINF - 94.04, Department of Computing Science, Umea
127 * University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
128 * Note 87. To appear in Numerical Algorithms, 1996.
129 *
130 * =====================================================================
131 * Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
132 * loops. Sven Hammarling, 1/5/02.
133 *
134 * .. Parameters ..
135 DOUBLE PRECISION ZERO, ONE
136 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
137 DOUBLE PRECISION TWENTY
138 PARAMETER ( TWENTY = 2.0D+01 )
139 INTEGER LDST
140 PARAMETER ( LDST = 4 )
141 LOGICAL WANDS
142 PARAMETER ( WANDS = .TRUE. )
143 * ..
144 * .. Local Scalars ..
145 LOGICAL DTRONG, WEAK
146 INTEGER I, IDUM, LINFO, M
147 DOUBLE PRECISION BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS,
148 $ F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS
149 * ..
150 * .. Local Arrays ..
151 INTEGER IWORK( LDST )
152 DOUBLE PRECISION AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
153 $ IRCOP( LDST, LDST ), LI( LDST, LDST ),
154 $ LICOP( LDST, LDST ), S( LDST, LDST ),
155 $ SCPY( LDST, LDST ), T( LDST, LDST ),
156 $ TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
157 * ..
158 * .. External Functions ..
159 DOUBLE PRECISION DLAMCH
160 EXTERNAL DLAMCH
161 * ..
162 * .. External Subroutines ..
163 EXTERNAL DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2, DLARTG,
164 $ DLASET, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
165 $ DROT, DSCAL, DTGSY2
166 * ..
167 * .. Intrinsic Functions ..
168 INTRINSIC ABS, MAX, SQRT
169 * ..
170 * .. Executable Statements ..
171 *
172 INFO = 0
173 *
174 * Quick return if possible
175 *
176 IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
177 $ RETURN
178 IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
179 $ RETURN
180 M = N1 + N2
181 IF( LWORK.LT.MAX( 1, N*M, M*M*2 ) ) THEN
182 INFO = -16
183 WORK( 1 ) = MAX( 1, N*M, M*M*2 )
184 RETURN
185 END IF
186 *
187 WEAK = .FALSE.
188 DTRONG = .FALSE.
189 *
190 * Make a local copy of selected block
191 *
192 CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, LI, LDST )
193 CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, IR, LDST )
194 CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
195 CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
196 *
197 * Compute threshold for testing acceptance of swapping.
198 *
199 EPS = DLAMCH( 'P' )
200 SMLNUM = DLAMCH( 'S' ) / EPS
201 DSCALE = ZERO
202 DSUM = ONE
203 CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
204 CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
205 CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
206 CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
207 DNORM = DSCALE*SQRT( DSUM )
208 *
209 * THRES has been changed from
210 * THRESH = MAX( TEN*EPS*SA, SMLNUM )
211 * to
212 * THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
213 * on 04/01/10.
214 * "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
215 * Jim Demmel and Guillaume Revy. See forum post 1783.
216 *
217 THRESH = MAX( TWENTY*EPS*DNORM, SMLNUM )
218 *
219 IF( M.EQ.2 ) THEN
220 *
221 * CASE 1: Swap 1-by-1 and 1-by-1 blocks.
222 *
223 * Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
224 * using Givens rotations and perform the swap tentatively.
225 *
226 F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
227 G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
228 SB = ABS( T( 2, 2 ) )
229 SA = ABS( S( 2, 2 ) )
230 CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
231 IR( 2, 1 ) = -IR( 1, 2 )
232 IR( 2, 2 ) = IR( 1, 1 )
233 CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
234 $ IR( 2, 1 ) )
235 CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
236 $ IR( 2, 1 ) )
237 IF( SA.GE.SB ) THEN
238 CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
239 $ DDUM )
240 ELSE
241 CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
242 $ DDUM )
243 END IF
244 CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
245 $ LI( 2, 1 ) )
246 CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
247 $ LI( 2, 1 ) )
248 LI( 2, 2 ) = LI( 1, 1 )
249 LI( 1, 2 ) = -LI( 2, 1 )
250 *
251 * Weak stability test:
252 * |S21| + |T21| <= O(EPS * F-norm((S, T)))
253 *
254 WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
255 WEAK = WS.LE.THRESH
256 IF( .NOT.WEAK )
257 $ GO TO 70
258 *
259 IF( WANDS ) THEN
260 *
261 * Strong stability test:
262 * F-norm((A-QL**T*S*QR, B-QL**T*T*QR)) <= O(EPS*F-norm((A,B)))
263 *
264 CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
265 $ M )
266 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
267 $ WORK, M )
268 CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
269 $ WORK( M*M+1 ), M )
270 DSCALE = ZERO
271 DSUM = ONE
272 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
273 *
274 CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
275 $ M )
276 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
277 $ WORK, M )
278 CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
279 $ WORK( M*M+1 ), M )
280 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
281 SS = DSCALE*SQRT( DSUM )
282 DTRONG = SS.LE.THRESH
283 IF( .NOT.DTRONG )
284 $ GO TO 70
285 END IF
286 *
287 * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
288 * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
289 *
290 CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
291 $ IR( 2, 1 ) )
292 CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
293 $ IR( 2, 1 ) )
294 CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
295 $ LI( 1, 1 ), LI( 2, 1 ) )
296 CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
297 $ LI( 1, 1 ), LI( 2, 1 ) )
298 *
299 * Set N1-by-N2 (2,1) - blocks to ZERO.
300 *
301 A( J1+1, J1 ) = ZERO
302 B( J1+1, J1 ) = ZERO
303 *
304 * Accumulate transformations into Q and Z if requested.
305 *
306 IF( WANTZ )
307 $ CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
308 $ IR( 2, 1 ) )
309 IF( WANTQ )
310 $ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
311 $ LI( 2, 1 ) )
312 *
313 * Exit with INFO = 0 if swap was successfully performed.
314 *
315 RETURN
316 *
317 ELSE
318 *
319 * CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
320 * and 2-by-2 blocks.
321 *
322 * Solve the generalized Sylvester equation
323 * S11 * R - L * S22 = SCALE * S12
324 * T11 * R - L * T22 = SCALE * T12
325 * for R and L. Solutions in LI and IR.
326 *
327 CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
328 CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
329 $ IR( N2+1, N1+1 ), LDST )
330 CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
331 $ IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
332 $ LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
333 $ LINFO )
334 *
335 * Compute orthogonal matrix QL:
336 *
337 * QL**T * LI = [ TL ]
338 * [ 0 ]
339 * where
340 * LI = [ -L ]
341 * [ SCALE * identity(N2) ]
342 *
343 DO 10 I = 1, N2
344 CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
345 LI( N1+I, I ) = SCALE
346 10 CONTINUE
347 CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
348 IF( LINFO.NE.0 )
349 $ GO TO 70
350 CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
351 IF( LINFO.NE.0 )
352 $ GO TO 70
353 *
354 * Compute orthogonal matrix RQ:
355 *
356 * IR * RQ**T = [ 0 TR],
357 *
358 * where IR = [ SCALE * identity(N1), R ]
359 *
360 DO 20 I = 1, N1
361 IR( N2+I, I ) = SCALE
362 20 CONTINUE
363 CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
364 IF( LINFO.NE.0 )
365 $ GO TO 70
366 CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
367 IF( LINFO.NE.0 )
368 $ GO TO 70
369 *
370 * Perform the swapping tentatively:
371 *
372 CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
373 $ WORK, M )
374 CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, S,
375 $ LDST )
376 CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
377 $ WORK, M )
378 CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, T,
379 $ LDST )
380 CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
381 CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
382 CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
383 CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
384 *
385 * Triangularize the B-part by an RQ factorization.
386 * Apply transformation (from left) to A-part, giving S.
387 *
388 CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
389 IF( LINFO.NE.0 )
390 $ GO TO 70
391 CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST, WORK,
392 $ LINFO )
393 IF( LINFO.NE.0 )
394 $ GO TO 70
395 CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST, WORK,
396 $ LINFO )
397 IF( LINFO.NE.0 )
398 $ GO TO 70
399 *
400 * Compute F-norm(S21) in BRQA21. (T21 is 0.)
401 *
402 DSCALE = ZERO
403 DSUM = ONE
404 DO 30 I = 1, N2
405 CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
406 30 CONTINUE
407 BRQA21 = DSCALE*SQRT( DSUM )
408 *
409 * Triangularize the B-part by a QR factorization.
410 * Apply transformation (from right) to A-part, giving S.
411 *
412 CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
413 IF( LINFO.NE.0 )
414 $ GO TO 70
415 CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY, LDST,
416 $ WORK, INFO )
417 CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP, LDST,
418 $ WORK, INFO )
419 IF( LINFO.NE.0 )
420 $ GO TO 70
421 *
422 * Compute F-norm(S21) in BQRA21. (T21 is 0.)
423 *
424 DSCALE = ZERO
425 DSUM = ONE
426 DO 40 I = 1, N2
427 CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
428 40 CONTINUE
429 BQRA21 = DSCALE*SQRT( DSUM )
430 *
431 * Decide which method to use.
432 * Weak stability test:
433 * F-norm(S21) <= O(EPS * F-norm((S, T)))
434 *
435 IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN
436 CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
437 CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
438 CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
439 CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
440 ELSE IF( BRQA21.GE.THRESH ) THEN
441 GO TO 70
442 END IF
443 *
444 * Set lower triangle of B-part to zero
445 *
446 CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
447 *
448 IF( WANDS ) THEN
449 *
450 * Strong stability test:
451 * F-norm((A-QL*S*QR**T, B-QL*T*QR**T)) <= O(EPS*F-norm((A,B)))
452 *
453 CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
454 $ M )
455 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
456 $ WORK, M )
457 CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
458 $ WORK( M*M+1 ), M )
459 DSCALE = ZERO
460 DSUM = ONE
461 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
462 *
463 CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
464 $ M )
465 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
466 $ WORK, M )
467 CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
468 $ WORK( M*M+1 ), M )
469 CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
470 SS = DSCALE*SQRT( DSUM )
471 DTRONG = ( SS.LE.THRESH )
472 IF( .NOT.DTRONG )
473 $ GO TO 70
474 *
475 END IF
476 *
477 * If the swap is accepted ("weakly" and "strongly"), apply the
478 * transformations and set N1-by-N2 (2,1)-block to zero.
479 *
480 CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
481 *
482 * copy back M-by-M diagonal block starting at index J1 of (A, B)
483 *
484 CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
485 CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
486 CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
487 *
488 * Standardize existing 2-by-2 blocks.
489 *
490 DO 50 I = 1, M*M
491 WORK(I) = ZERO
492 50 CONTINUE
493 WORK( 1 ) = ONE
494 T( 1, 1 ) = ONE
495 IDUM = LWORK - M*M - 2
496 IF( N2.GT.1 ) THEN
497 CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI, BE,
498 $ WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
499 WORK( M+1 ) = -WORK( 2 )
500 WORK( M+2 ) = WORK( 1 )
501 T( N2, N2 ) = T( 1, 1 )
502 T( 1, 2 ) = -T( 2, 1 )
503 END IF
504 WORK( M*M ) = ONE
505 T( M, M ) = ONE
506 *
507 IF( N1.GT.1 ) THEN
508 CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ), LDB,
509 $ TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
510 $ WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
511 $ T( M, M-1 ) )
512 WORK( M*M ) = WORK( N2*M+N2+1 )
513 WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
514 T( M, M ) = T( N2+1, N2+1 )
515 T( M-1, M ) = -T( M, M-1 )
516 END IF
517 CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1, J1+N2 ),
518 $ LDA, ZERO, WORK( M*M+1 ), N2 )
519 CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1, J1+N2 ),
520 $ LDA )
521 CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1, J1+N2 ),
522 $ LDB, ZERO, WORK( M*M+1 ), N2 )
523 CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1, J1+N2 ),
524 $ LDB )
525 CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
526 $ WORK( M*M+1 ), M )
527 CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
528 CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
529 $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
530 CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
531 CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
532 $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
533 CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
534 CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
535 $ WORK, M )
536 CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
537 *
538 * Accumulate transformations into Q and Z if requested.
539 *
540 IF( WANTQ ) THEN
541 CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
542 $ LDST, ZERO, WORK, N )
543 CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
544 *
545 END IF
546 *
547 IF( WANTZ ) THEN
548 CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
549 $ LDST, ZERO, WORK, N )
550 CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
551 *
552 END IF
553 *
554 * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
555 * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
556 *
557 I = J1 + M
558 IF( I.LE.N ) THEN
559 CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
560 $ A( J1, I ), LDA, ZERO, WORK, M )
561 CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
562 CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
563 $ B( J1, I ), LDA, ZERO, WORK, M )
564 CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
565 END IF
566 I = J1 - 1
567 IF( I.GT.0 ) THEN
568 CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
569 $ LDST, ZERO, WORK, I )
570 CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
571 CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
572 $ LDST, ZERO, WORK, I )
573 CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
574 END IF
575 *
576 * Exit with INFO = 0 if swap was successfully performed.
577 *
578 RETURN
579 *
580 END IF
581 *
582 * Exit with INFO = 1 if swap was rejected.
583 *
584 70 CONTINUE
585 *
586 INFO = 1
587 RETURN
588 *
589 * End of DTGEX2
590 *
591 END