1: *> \brief \b DTGSYL
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DTGSYL + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsyl.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsyl.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsyl.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
22: * LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
23: * IWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER TRANS
27: * INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
28: * $ LWORK, M, N
29: * DOUBLE PRECISION DIF, SCALE
30: * ..
31: * .. Array Arguments ..
32: * INTEGER IWORK( * )
33: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
34: * $ D( LDD, * ), E( LDE, * ), F( LDF, * ),
35: * $ WORK( * )
36: * ..
37: *
38: *
39: *> \par Purpose:
40: * =============
41: *>
42: *> \verbatim
43: *>
44: *> DTGSYL solves the generalized Sylvester equation:
45: *>
46: *> A * R - L * B = scale * C (1)
47: *> D * R - L * E = scale * F
48: *>
49: *> where R and L are unknown m-by-n matrices, (A, D), (B, E) and
50: *> (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
51: *> respectively, with real entries. (A, D) and (B, E) must be in
52: *> generalized (real) Schur canonical form, i.e. A, B are upper quasi
53: *> triangular and D, E are upper triangular.
54: *>
55: *> The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
56: *> scaling factor chosen to avoid overflow.
57: *>
58: *> In matrix notation (1) is equivalent to solve Zx = scale b, where
59: *> Z is defined as
60: *>
61: *> Z = [ kron(In, A) -kron(B**T, Im) ] (2)
62: *> [ kron(In, D) -kron(E**T, Im) ].
63: *>
64: *> Here Ik is the identity matrix of size k and X**T is the transpose of
65: *> X. kron(X, Y) is the Kronecker product between the matrices X and Y.
66: *>
67: *> If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
68: *> which is equivalent to solve for R and L in
69: *>
70: *> A**T * R + D**T * L = scale * C (3)
71: *> R * B**T + L * E**T = scale * -F
72: *>
73: *> This case (TRANS = 'T') is used to compute an one-norm-based estimate
74: *> of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
75: *> and (B,E), using DLACON.
76: *>
77: *> If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
78: *> of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
79: *> reciprocal of the smallest singular value of Z. See [1-2] for more
80: *> information.
81: *>
82: *> This is a level 3 BLAS algorithm.
83: *> \endverbatim
84: *
85: * Arguments:
86: * ==========
87: *
88: *> \param[in] TRANS
89: *> \verbatim
90: *> TRANS is CHARACTER*1
91: *> = 'N': solve the generalized Sylvester equation (1).
92: *> = 'T': solve the 'transposed' system (3).
93: *> \endverbatim
94: *>
95: *> \param[in] IJOB
96: *> \verbatim
97: *> IJOB is INTEGER
98: *> Specifies what kind of functionality to be performed.
99: *> = 0: solve (1) only.
100: *> = 1: The functionality of 0 and 3.
101: *> = 2: The functionality of 0 and 4.
102: *> = 3: Only an estimate of Dif[(A,D), (B,E)] is computed.
103: *> (look ahead strategy IJOB = 1 is used).
104: *> = 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
105: *> ( DGECON on sub-systems is used ).
106: *> Not referenced if TRANS = 'T'.
107: *> \endverbatim
108: *>
109: *> \param[in] M
110: *> \verbatim
111: *> M is INTEGER
112: *> The order of the matrices A and D, and the row dimension of
113: *> the matrices C, F, R and L.
114: *> \endverbatim
115: *>
116: *> \param[in] N
117: *> \verbatim
118: *> N is INTEGER
119: *> The order of the matrices B and E, and the column dimension
120: *> of the matrices C, F, R and L.
121: *> \endverbatim
122: *>
123: *> \param[in] A
124: *> \verbatim
125: *> A is DOUBLE PRECISION array, dimension (LDA, M)
126: *> The upper quasi triangular matrix A.
127: *> \endverbatim
128: *>
129: *> \param[in] LDA
130: *> \verbatim
131: *> LDA is INTEGER
132: *> The leading dimension of the array A. LDA >= max(1, M).
133: *> \endverbatim
134: *>
135: *> \param[in] B
136: *> \verbatim
137: *> B is DOUBLE PRECISION array, dimension (LDB, N)
138: *> The upper quasi triangular matrix B.
139: *> \endverbatim
140: *>
141: *> \param[in] LDB
142: *> \verbatim
143: *> LDB is INTEGER
144: *> The leading dimension of the array B. LDB >= max(1, N).
145: *> \endverbatim
146: *>
147: *> \param[in,out] C
148: *> \verbatim
149: *> C is DOUBLE PRECISION array, dimension (LDC, N)
150: *> On entry, C contains the right-hand-side of the first matrix
151: *> equation in (1) or (3).
152: *> On exit, if IJOB = 0, 1 or 2, C has been overwritten by
153: *> the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
154: *> the solution achieved during the computation of the
155: *> Dif-estimate.
156: *> \endverbatim
157: *>
158: *> \param[in] LDC
159: *> \verbatim
160: *> LDC is INTEGER
161: *> The leading dimension of the array C. LDC >= max(1, M).
162: *> \endverbatim
163: *>
164: *> \param[in] D
165: *> \verbatim
166: *> D is DOUBLE PRECISION array, dimension (LDD, M)
167: *> The upper triangular matrix D.
168: *> \endverbatim
169: *>
170: *> \param[in] LDD
171: *> \verbatim
172: *> LDD is INTEGER
173: *> The leading dimension of the array D. LDD >= max(1, M).
174: *> \endverbatim
175: *>
176: *> \param[in] E
177: *> \verbatim
178: *> E is DOUBLE PRECISION array, dimension (LDE, N)
179: *> The upper triangular matrix E.
180: *> \endverbatim
181: *>
182: *> \param[in] LDE
183: *> \verbatim
184: *> LDE is INTEGER
185: *> The leading dimension of the array E. LDE >= max(1, N).
186: *> \endverbatim
187: *>
188: *> \param[in,out] F
189: *> \verbatim
190: *> F is DOUBLE PRECISION array, dimension (LDF, N)
191: *> On entry, F contains the right-hand-side of the second matrix
192: *> equation in (1) or (3).
193: *> On exit, if IJOB = 0, 1 or 2, F has been overwritten by
194: *> the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
195: *> the solution achieved during the computation of the
196: *> Dif-estimate.
197: *> \endverbatim
198: *>
199: *> \param[in] LDF
200: *> \verbatim
201: *> LDF is INTEGER
202: *> The leading dimension of the array F. LDF >= max(1, M).
203: *> \endverbatim
204: *>
205: *> \param[out] DIF
206: *> \verbatim
207: *> DIF is DOUBLE PRECISION
208: *> On exit DIF is the reciprocal of a lower bound of the
209: *> reciprocal of the Dif-function, i.e. DIF is an upper bound of
210: *> Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
211: *> IF IJOB = 0 or TRANS = 'T', DIF is not touched.
212: *> \endverbatim
213: *>
214: *> \param[out] SCALE
215: *> \verbatim
216: *> SCALE is DOUBLE PRECISION
217: *> On exit SCALE is the scaling factor in (1) or (3).
218: *> If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
219: *> to a slightly perturbed system but the input matrices A, B, D
220: *> and E have not been changed. If SCALE = 0, C and F hold the
221: *> solutions R and L, respectively, to the homogeneous system
222: *> with C = F = 0. Normally, SCALE = 1.
223: *> \endverbatim
224: *>
225: *> \param[out] WORK
226: *> \verbatim
227: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
228: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
229: *> \endverbatim
230: *>
231: *> \param[in] LWORK
232: *> \verbatim
233: *> LWORK is INTEGER
234: *> The dimension of the array WORK. LWORK > = 1.
235: *> If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
236: *>
237: *> If LWORK = -1, then a workspace query is assumed; the routine
238: *> only calculates the optimal size of the WORK array, returns
239: *> this value as the first entry of the WORK array, and no error
240: *> message related to LWORK is issued by XERBLA.
241: *> \endverbatim
242: *>
243: *> \param[out] IWORK
244: *> \verbatim
245: *> IWORK is INTEGER array, dimension (M+N+6)
246: *> \endverbatim
247: *>
248: *> \param[out] INFO
249: *> \verbatim
250: *> INFO is INTEGER
251: *> =0: successful exit
252: *> <0: If INFO = -i, the i-th argument had an illegal value.
253: *> >0: (A, D) and (B, E) have common or close eigenvalues.
254: *> \endverbatim
255: *
256: * Authors:
257: * ========
258: *
259: *> \author Univ. of Tennessee
260: *> \author Univ. of California Berkeley
261: *> \author Univ. of Colorado Denver
262: *> \author NAG Ltd.
263: *
264: *> \ingroup doubleSYcomputational
265: *
266: *> \par Contributors:
267: * ==================
268: *>
269: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
270: *> Umea University, S-901 87 Umea, Sweden.
271: *
272: *> \par References:
273: * ================
274: *>
275: *> \verbatim
276: *>
277: *> [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
278: *> for Solving the Generalized Sylvester Equation and Estimating the
279: *> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
280: *> Department of Computing Science, Umea University, S-901 87 Umea,
281: *> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
282: *> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
283: *> No 1, 1996.
284: *>
285: *> [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
286: *> Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
287: *> Appl., 15(4):1045-1060, 1994
288: *>
289: *> [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
290: *> Condition Estimators for Solving the Generalized Sylvester
291: *> Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
292: *> July 1989, pp 745-751.
293: *> \endverbatim
294: *>
295: * =====================================================================
296: SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
297: $ LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
298: $ IWORK, INFO )
299: *
300: * -- LAPACK computational routine --
301: * -- LAPACK is a software package provided by Univ. of Tennessee, --
302: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
303: *
304: * .. Scalar Arguments ..
305: CHARACTER TRANS
306: INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
307: $ LWORK, M, N
308: DOUBLE PRECISION DIF, SCALE
309: * ..
310: * .. Array Arguments ..
311: INTEGER IWORK( * )
312: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
313: $ D( LDD, * ), E( LDE, * ), F( LDF, * ),
314: $ WORK( * )
315: * ..
316: *
317: * =====================================================================
318: * Replaced various illegal calls to DCOPY by calls to DLASET.
319: * Sven Hammarling, 1/5/02.
320: *
321: * .. Parameters ..
322: DOUBLE PRECISION ZERO, ONE
323: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
324: * ..
325: * .. Local Scalars ..
326: LOGICAL LQUERY, NOTRAN
327: INTEGER I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K,
328: $ LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q
329: DOUBLE PRECISION DSCALE, DSUM, SCALE2, SCALOC
330: * ..
331: * .. External Functions ..
332: LOGICAL LSAME
333: INTEGER ILAENV
334: EXTERNAL LSAME, ILAENV
335: * ..
336: * .. External Subroutines ..
337: EXTERNAL DGEMM, DLACPY, DLASET, DSCAL, DTGSY2, XERBLA
338: * ..
339: * .. Intrinsic Functions ..
340: INTRINSIC DBLE, MAX, SQRT
341: * ..
342: * .. Executable Statements ..
343: *
344: * Decode and test input parameters
345: *
346: INFO = 0
347: NOTRAN = LSAME( TRANS, 'N' )
348: LQUERY = ( LWORK.EQ.-1 )
349: *
350: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
351: INFO = -1
352: ELSE IF( NOTRAN ) THEN
353: IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) THEN
354: INFO = -2
355: END IF
356: END IF
357: IF( INFO.EQ.0 ) THEN
358: IF( M.LE.0 ) THEN
359: INFO = -3
360: ELSE IF( N.LE.0 ) THEN
361: INFO = -4
362: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
363: INFO = -6
364: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
365: INFO = -8
366: ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
367: INFO = -10
368: ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
369: INFO = -12
370: ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
371: INFO = -14
372: ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
373: INFO = -16
374: END IF
375: END IF
376: *
377: IF( INFO.EQ.0 ) THEN
378: IF( NOTRAN ) THEN
379: IF( IJOB.EQ.1 .OR. IJOB.EQ.2 ) THEN
380: LWMIN = MAX( 1, 2*M*N )
381: ELSE
382: LWMIN = 1
383: END IF
384: ELSE
385: LWMIN = 1
386: END IF
387: WORK( 1 ) = LWMIN
388: *
389: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
390: INFO = -20
391: END IF
392: END IF
393: *
394: IF( INFO.NE.0 ) THEN
395: CALL XERBLA( 'DTGSYL', -INFO )
396: RETURN
397: ELSE IF( LQUERY ) THEN
398: RETURN
399: END IF
400: *
401: * Quick return if possible
402: *
403: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
404: SCALE = 1
405: IF( NOTRAN ) THEN
406: IF( IJOB.NE.0 ) THEN
407: DIF = 0
408: END IF
409: END IF
410: RETURN
411: END IF
412: *
413: * Determine optimal block sizes MB and NB
414: *
415: MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 )
416: NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 )
417: *
418: ISOLVE = 1
419: IFUNC = 0
420: IF( NOTRAN ) THEN
421: IF( IJOB.GE.3 ) THEN
422: IFUNC = IJOB - 2
423: CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
424: CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
425: ELSE IF( IJOB.GE.1 ) THEN
426: ISOLVE = 2
427: END IF
428: END IF
429: *
430: IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) )
431: $ THEN
432: *
433: DO 30 IROUND = 1, ISOLVE
434: *
435: * Use unblocked Level 2 solver
436: *
437: DSCALE = ZERO
438: DSUM = ONE
439: PQ = 0
440: CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC, D,
441: $ LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE,
442: $ IWORK, PQ, INFO )
443: IF( DSCALE.NE.ZERO ) THEN
444: IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
445: DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
446: ELSE
447: DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
448: END IF
449: END IF
450: *
451: IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
452: IF( NOTRAN ) THEN
453: IFUNC = IJOB
454: END IF
455: SCALE2 = SCALE
456: CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
457: CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
458: CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
459: CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
460: ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
461: CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
462: CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
463: SCALE = SCALE2
464: END IF
465: 30 CONTINUE
466: *
467: RETURN
468: END IF
469: *
470: * Determine block structure of A
471: *
472: P = 0
473: I = 1
474: 40 CONTINUE
475: IF( I.GT.M )
476: $ GO TO 50
477: P = P + 1
478: IWORK( P ) = I
479: I = I + MB
480: IF( I.GE.M )
481: $ GO TO 50
482: IF( A( I, I-1 ).NE.ZERO )
483: $ I = I + 1
484: GO TO 40
485: 50 CONTINUE
486: *
487: IWORK( P+1 ) = M + 1
488: IF( IWORK( P ).EQ.IWORK( P+1 ) )
489: $ P = P - 1
490: *
491: * Determine block structure of B
492: *
493: Q = P + 1
494: J = 1
495: 60 CONTINUE
496: IF( J.GT.N )
497: $ GO TO 70
498: Q = Q + 1
499: IWORK( Q ) = J
500: J = J + NB
501: IF( J.GE.N )
502: $ GO TO 70
503: IF( B( J, J-1 ).NE.ZERO )
504: $ J = J + 1
505: GO TO 60
506: 70 CONTINUE
507: *
508: IWORK( Q+1 ) = N + 1
509: IF( IWORK( Q ).EQ.IWORK( Q+1 ) )
510: $ Q = Q - 1
511: *
512: IF( NOTRAN ) THEN
513: *
514: DO 150 IROUND = 1, ISOLVE
515: *
516: * Solve (I, J)-subsystem
517: * A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
518: * D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
519: * for I = P, P - 1,..., 1; J = 1, 2,..., Q
520: *
521: DSCALE = ZERO
522: DSUM = ONE
523: PQ = 0
524: SCALE = ONE
525: DO 130 J = P + 2, Q
526: JS = IWORK( J )
527: JE = IWORK( J+1 ) - 1
528: NB = JE - JS + 1
529: DO 120 I = P, 1, -1
530: IS = IWORK( I )
531: IE = IWORK( I+1 ) - 1
532: MB = IE - IS + 1
533: PPQQ = 0
534: CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
535: $ B( JS, JS ), LDB, C( IS, JS ), LDC,
536: $ D( IS, IS ), LDD, E( JS, JS ), LDE,
537: $ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
538: $ IWORK( Q+2 ), PPQQ, LINFO )
539: IF( LINFO.GT.0 )
540: $ INFO = LINFO
541: *
542: PQ = PQ + PPQQ
543: IF( SCALOC.NE.ONE ) THEN
544: DO 80 K = 1, JS - 1
545: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
546: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
547: 80 CONTINUE
548: DO 90 K = JS, JE
549: CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
550: CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
551: 90 CONTINUE
552: DO 100 K = JS, JE
553: CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
554: CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
555: 100 CONTINUE
556: DO 110 K = JE + 1, N
557: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
558: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
559: 110 CONTINUE
560: SCALE = SCALE*SCALOC
561: END IF
562: *
563: * Substitute R(I, J) and L(I, J) into remaining
564: * equation.
565: *
566: IF( I.GT.1 ) THEN
567: CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
568: $ A( 1, IS ), LDA, C( IS, JS ), LDC, ONE,
569: $ C( 1, JS ), LDC )
570: CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
571: $ D( 1, IS ), LDD, C( IS, JS ), LDC, ONE,
572: $ F( 1, JS ), LDF )
573: END IF
574: IF( J.LT.Q ) THEN
575: CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
576: $ F( IS, JS ), LDF, B( JS, JE+1 ), LDB,
577: $ ONE, C( IS, JE+1 ), LDC )
578: CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
579: $ F( IS, JS ), LDF, E( JS, JE+1 ), LDE,
580: $ ONE, F( IS, JE+1 ), LDF )
581: END IF
582: 120 CONTINUE
583: 130 CONTINUE
584: IF( DSCALE.NE.ZERO ) THEN
585: IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
586: DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
587: ELSE
588: DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
589: END IF
590: END IF
591: IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
592: IF( NOTRAN ) THEN
593: IFUNC = IJOB
594: END IF
595: SCALE2 = SCALE
596: CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
597: CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
598: CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
599: CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
600: ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
601: CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
602: CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
603: SCALE = SCALE2
604: END IF
605: 150 CONTINUE
606: *
607: ELSE
608: *
609: * Solve transposed (I, J)-subsystem
610: * A(I, I)**T * R(I, J) + D(I, I)**T * L(I, J) = C(I, J)
611: * R(I, J) * B(J, J)**T + L(I, J) * E(J, J)**T = -F(I, J)
612: * for I = 1,2,..., P; J = Q, Q-1,..., 1
613: *
614: SCALE = ONE
615: DO 210 I = 1, P
616: IS = IWORK( I )
617: IE = IWORK( I+1 ) - 1
618: MB = IE - IS + 1
619: DO 200 J = Q, P + 2, -1
620: JS = IWORK( J )
621: JE = IWORK( J+1 ) - 1
622: NB = JE - JS + 1
623: CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
624: $ B( JS, JS ), LDB, C( IS, JS ), LDC,
625: $ D( IS, IS ), LDD, E( JS, JS ), LDE,
626: $ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
627: $ IWORK( Q+2 ), PPQQ, LINFO )
628: IF( LINFO.GT.0 )
629: $ INFO = LINFO
630: IF( SCALOC.NE.ONE ) THEN
631: DO 160 K = 1, JS - 1
632: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
633: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
634: 160 CONTINUE
635: DO 170 K = JS, JE
636: CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
637: CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
638: 170 CONTINUE
639: DO 180 K = JS, JE
640: CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
641: CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
642: 180 CONTINUE
643: DO 190 K = JE + 1, N
644: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
645: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
646: 190 CONTINUE
647: SCALE = SCALE*SCALOC
648: END IF
649: *
650: * Substitute R(I, J) and L(I, J) into remaining equation.
651: *
652: IF( J.GT.P+2 ) THEN
653: CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, C( IS, JS ),
654: $ LDC, B( 1, JS ), LDB, ONE, F( IS, 1 ),
655: $ LDF )
656: CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, F( IS, JS ),
657: $ LDF, E( 1, JS ), LDE, ONE, F( IS, 1 ),
658: $ LDF )
659: END IF
660: IF( I.LT.P ) THEN
661: CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
662: $ A( IS, IE+1 ), LDA, C( IS, JS ), LDC, ONE,
663: $ C( IE+1, JS ), LDC )
664: CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
665: $ D( IS, IE+1 ), LDD, F( IS, JS ), LDF, ONE,
666: $ C( IE+1, JS ), LDC )
667: END IF
668: 200 CONTINUE
669: 210 CONTINUE
670: *
671: END IF
672: *
673: WORK( 1 ) = LWMIN
674: *
675: RETURN
676: *
677: * End of DTGSYL
678: *
679: END
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