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) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
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 arrays, 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 arrays, 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 (LDZ,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 TEN
138: PARAMETER ( TEN = 1.0D+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: THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
209: *
210: IF( M.EQ.2 ) THEN
211: *
212: * CASE 1: Swap 1-by-1 and 1-by-1 blocks.
213: *
214: * Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
215: * using Givens rotations and perform the swap tentatively.
216: *
217: F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
218: G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
219: SB = ABS( T( 2, 2 ) )
220: SA = ABS( S( 2, 2 ) )
221: CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
222: IR( 2, 1 ) = -IR( 1, 2 )
223: IR( 2, 2 ) = IR( 1, 1 )
224: CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
225: $ IR( 2, 1 ) )
226: CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
227: $ IR( 2, 1 ) )
228: IF( SA.GE.SB ) THEN
229: CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
230: $ DDUM )
231: ELSE
232: CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
233: $ DDUM )
234: END IF
235: CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
236: $ LI( 2, 1 ) )
237: CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
238: $ LI( 2, 1 ) )
239: LI( 2, 2 ) = LI( 1, 1 )
240: LI( 1, 2 ) = -LI( 2, 1 )
241: *
242: * Weak stability test:
243: * |S21| + |T21| <= O(EPS * F-norm((S, T)))
244: *
245: WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
246: WEAK = WS.LE.THRESH
247: IF( .NOT.WEAK )
248: $ GO TO 70
249: *
250: IF( WANDS ) THEN
251: *
252: * Strong stability test:
253: * F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B)))
254: *
255: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
256: $ M )
257: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
258: $ WORK, M )
259: CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
260: $ WORK( M*M+1 ), M )
261: DSCALE = ZERO
262: DSUM = ONE
263: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
264: *
265: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
266: $ M )
267: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
268: $ WORK, M )
269: CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
270: $ WORK( M*M+1 ), M )
271: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
272: SS = DSCALE*SQRT( DSUM )
273: DTRONG = SS.LE.THRESH
274: IF( .NOT.DTRONG )
275: $ GO TO 70
276: END IF
277: *
278: * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
279: * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
280: *
281: CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
282: $ IR( 2, 1 ) )
283: CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
284: $ IR( 2, 1 ) )
285: CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
286: $ LI( 1, 1 ), LI( 2, 1 ) )
287: CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
288: $ LI( 1, 1 ), LI( 2, 1 ) )
289: *
290: * Set N1-by-N2 (2,1) - blocks to ZERO.
291: *
292: A( J1+1, J1 ) = ZERO
293: B( J1+1, J1 ) = ZERO
294: *
295: * Accumulate transformations into Q and Z if requested.
296: *
297: IF( WANTZ )
298: $ CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ),
299: $ IR( 2, 1 ) )
300: IF( WANTQ )
301: $ CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ),
302: $ LI( 2, 1 ) )
303: *
304: * Exit with INFO = 0 if swap was successfully performed.
305: *
306: RETURN
307: *
308: ELSE
309: *
310: * CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
311: * and 2-by-2 blocks.
312: *
313: * Solve the generalized Sylvester equation
314: * S11 * R - L * S22 = SCALE * S12
315: * T11 * R - L * T22 = SCALE * T12
316: * for R and L. Solutions in LI and IR.
317: *
318: CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST )
319: CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST,
320: $ IR( N2+1, N1+1 ), LDST )
321: CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST,
322: $ IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
323: $ LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
324: $ LINFO )
325: *
326: * Compute orthogonal matrix QL:
327: *
328: * QL' * LI = [ TL ]
329: * [ 0 ]
330: * where
331: * LI = [ -L ]
332: * [ SCALE * identity(N2) ]
333: *
334: DO 10 I = 1, N2
335: CALL DSCAL( N1, -ONE, LI( 1, I ), 1 )
336: LI( N1+I, I ) = SCALE
337: 10 CONTINUE
338: CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO )
339: IF( LINFO.NE.0 )
340: $ GO TO 70
341: CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO )
342: IF( LINFO.NE.0 )
343: $ GO TO 70
344: *
345: * Compute orthogonal matrix RQ:
346: *
347: * IR * RQ' = [ 0 TR],
348: *
349: * where IR = [ SCALE * identity(N1), R ]
350: *
351: DO 20 I = 1, N1
352: IR( N2+I, I ) = SCALE
353: 20 CONTINUE
354: CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO )
355: IF( LINFO.NE.0 )
356: $ GO TO 70
357: CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO )
358: IF( LINFO.NE.0 )
359: $ GO TO 70
360: *
361: * Perform the swapping tentatively:
362: *
363: CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
364: $ WORK, M )
365: CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, S,
366: $ LDST )
367: CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
368: $ WORK, M )
369: CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, T,
370: $ LDST )
371: CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST )
372: CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST )
373: CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST )
374: CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST )
375: *
376: * Triangularize the B-part by an RQ factorization.
377: * Apply transformation (from left) to A-part, giving S.
378: *
379: CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO )
380: IF( LINFO.NE.0 )
381: $ GO TO 70
382: CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST, WORK,
383: $ LINFO )
384: IF( LINFO.NE.0 )
385: $ GO TO 70
386: CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST, WORK,
387: $ LINFO )
388: IF( LINFO.NE.0 )
389: $ GO TO 70
390: *
391: * Compute F-norm(S21) in BRQA21. (T21 is 0.)
392: *
393: DSCALE = ZERO
394: DSUM = ONE
395: DO 30 I = 1, N2
396: CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM )
397: 30 CONTINUE
398: BRQA21 = DSCALE*SQRT( DSUM )
399: *
400: * Triangularize the B-part by a QR factorization.
401: * Apply transformation (from right) to A-part, giving S.
402: *
403: CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO )
404: IF( LINFO.NE.0 )
405: $ GO TO 70
406: CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY, LDST,
407: $ WORK, INFO )
408: CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP, LDST,
409: $ WORK, INFO )
410: IF( LINFO.NE.0 )
411: $ GO TO 70
412: *
413: * Compute F-norm(S21) in BQRA21. (T21 is 0.)
414: *
415: DSCALE = ZERO
416: DSUM = ONE
417: DO 40 I = 1, N2
418: CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM )
419: 40 CONTINUE
420: BQRA21 = DSCALE*SQRT( DSUM )
421: *
422: * Decide which method to use.
423: * Weak stability test:
424: * F-norm(S21) <= O(EPS * F-norm((S, T)))
425: *
426: IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN
427: CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
428: CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
429: CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
430: CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
431: ELSE IF( BRQA21.GE.THRESH ) THEN
432: GO TO 70
433: END IF
434: *
435: * Set lower triangle of B-part to zero
436: *
437: CALL DLASET( 'Lower', M-1, M-1, ZERO, ZERO, T(2,1), LDST )
438: *
439: IF( WANDS ) THEN
440: *
441: * Strong stability test:
442: * F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B)))
443: *
444: CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
445: $ M )
446: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
447: $ WORK, M )
448: CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
449: $ WORK( M*M+1 ), M )
450: DSCALE = ZERO
451: DSUM = ONE
452: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
453: *
454: CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
455: $ M )
456: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
457: $ WORK, M )
458: CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
459: $ WORK( M*M+1 ), M )
460: CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
461: SS = DSCALE*SQRT( DSUM )
462: DTRONG = ( SS.LE.THRESH )
463: IF( .NOT.DTRONG )
464: $ GO TO 70
465: *
466: END IF
467: *
468: * If the swap is accepted ("weakly" and "strongly"), apply the
469: * transformations and set N1-by-N2 (2,1)-block to zero.
470: *
471: CALL DLASET( 'Full', N1, N2, ZERO, ZERO, S(N2+1,1), LDST )
472: *
473: * copy back M-by-M diagonal block starting at index J1 of (A, B)
474: *
475: CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA )
476: CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB )
477: CALL DLASET( 'Full', LDST, LDST, ZERO, ZERO, T, LDST )
478: *
479: * Standardize existing 2-by-2 blocks.
480: *
481: DO 50 I = 1, M*M
482: WORK(I) = ZERO
483: 50 CONTINUE
484: WORK( 1 ) = ONE
485: T( 1, 1 ) = ONE
486: IDUM = LWORK - M*M - 2
487: IF( N2.GT.1 ) THEN
488: CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI, BE,
489: $ WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) )
490: WORK( M+1 ) = -WORK( 2 )
491: WORK( M+2 ) = WORK( 1 )
492: T( N2, N2 ) = T( 1, 1 )
493: T( 1, 2 ) = -T( 2, 1 )
494: END IF
495: WORK( M*M ) = ONE
496: T( M, M ) = ONE
497: *
498: IF( N1.GT.1 ) THEN
499: CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ), LDB,
500: $ TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ),
501: $ WORK( N2*M+N2+2 ), T( N2+1, N2+1 ),
502: $ T( M, M-1 ) )
503: WORK( M*M ) = WORK( N2*M+N2+1 )
504: WORK( M*M-1 ) = -WORK( N2*M+N2+2 )
505: T( M, M ) = T( N2+1, N2+1 )
506: T( M-1, M ) = -T( M, M-1 )
507: END IF
508: CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1, J1+N2 ),
509: $ LDA, ZERO, WORK( M*M+1 ), N2 )
510: CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1, J1+N2 ),
511: $ LDA )
512: CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1, J1+N2 ),
513: $ LDB, ZERO, WORK( M*M+1 ), N2 )
514: CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1, J1+N2 ),
515: $ LDB )
516: CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO,
517: $ WORK( M*M+1 ), M )
518: CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST )
519: CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA,
520: $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
521: CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA )
522: CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDB,
523: $ T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 )
524: CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB )
525: CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO,
526: $ WORK, M )
527: CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST )
528: *
529: * Accumulate transformations into Q and Z if requested.
530: *
531: IF( WANTQ ) THEN
532: CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI,
533: $ LDST, ZERO, WORK, N )
534: CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ )
535: *
536: END IF
537: *
538: IF( WANTZ ) THEN
539: CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR,
540: $ LDST, ZERO, WORK, N )
541: CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ )
542: *
543: END IF
544: *
545: * Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
546: * (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
547: *
548: I = J1 + M
549: IF( I.LE.N ) THEN
550: CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
551: $ A( J1, I ), LDA, ZERO, WORK, M )
552: CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
553: CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
554: $ B( J1, I ), LDA, ZERO, WORK, M )
555: CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
556: END IF
557: I = J1 - 1
558: IF( I.GT.0 ) THEN
559: CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR,
560: $ LDST, ZERO, WORK, I )
561: CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA )
562: CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR,
563: $ LDST, ZERO, WORK, I )
564: CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB )
565: END IF
566: *
567: * Exit with INFO = 0 if swap was successfully performed.
568: *
569: RETURN
570: *
571: END IF
572: *
573: * Exit with INFO = 1 if swap was rejected.
574: *
575: 70 CONTINUE
576: *
577: INFO = 1
578: RETURN
579: *
580: * End of DTGEX2
581: *
582: END
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