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 3.2.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * June 2010
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)' = Q(out) * A(out) * Z(out)'
33: * Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
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'*S*QR, B-QL'*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' * 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' = [ 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', B-QL*T*QR')) <= 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
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