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: *> \date November 2011
265: *
266: *> \ingroup doubleSYcomputational
267: *
268: *> \par Contributors:
269: * ==================
270: *>
271: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
272: *> Umea University, S-901 87 Umea, Sweden.
273: *
274: *> \par References:
275: * ================
276: *>
277: *> \verbatim
278: *>
279: *> [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
280: *> for Solving the Generalized Sylvester Equation and Estimating the
281: *> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
282: *> Department of Computing Science, Umea University, S-901 87 Umea,
283: *> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
284: *> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
285: *> No 1, 1996.
286: *>
287: *> [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
288: *> Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
289: *> Appl., 15(4):1045-1060, 1994
290: *>
291: *> [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
292: *> Condition Estimators for Solving the Generalized Sylvester
293: *> Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
294: *> July 1989, pp 745-751.
295: *> \endverbatim
296: *>
297: * =====================================================================
298: SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
299: $ LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
300: $ IWORK, INFO )
301: *
302: * -- LAPACK computational routine (version 3.4.0) --
303: * -- LAPACK is a software package provided by Univ. of Tennessee, --
304: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
305: * November 2011
306: *
307: * .. Scalar Arguments ..
308: CHARACTER TRANS
309: INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
310: $ LWORK, M, N
311: DOUBLE PRECISION DIF, SCALE
312: * ..
313: * .. Array Arguments ..
314: INTEGER IWORK( * )
315: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
316: $ D( LDD, * ), E( LDE, * ), F( LDF, * ),
317: $ WORK( * )
318: * ..
319: *
320: * =====================================================================
321: * Replaced various illegal calls to DCOPY by calls to DLASET.
322: * Sven Hammarling, 1/5/02.
323: *
324: * .. Parameters ..
325: DOUBLE PRECISION ZERO, ONE
326: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
327: * ..
328: * .. Local Scalars ..
329: LOGICAL LQUERY, NOTRAN
330: INTEGER I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K,
331: $ LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q
332: DOUBLE PRECISION DSCALE, DSUM, SCALE2, SCALOC
333: * ..
334: * .. External Functions ..
335: LOGICAL LSAME
336: INTEGER ILAENV
337: EXTERNAL LSAME, ILAENV
338: * ..
339: * .. External Subroutines ..
340: EXTERNAL DGEMM, DLACPY, DLASET, DSCAL, DTGSY2, XERBLA
341: * ..
342: * .. Intrinsic Functions ..
343: INTRINSIC DBLE, MAX, SQRT
344: * ..
345: * .. Executable Statements ..
346: *
347: * Decode and test input parameters
348: *
349: INFO = 0
350: NOTRAN = LSAME( TRANS, 'N' )
351: LQUERY = ( LWORK.EQ.-1 )
352: *
353: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
354: INFO = -1
355: ELSE IF( NOTRAN ) THEN
356: IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) THEN
357: INFO = -2
358: END IF
359: END IF
360: IF( INFO.EQ.0 ) THEN
361: IF( M.LE.0 ) THEN
362: INFO = -3
363: ELSE IF( N.LE.0 ) THEN
364: INFO = -4
365: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
366: INFO = -6
367: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
368: INFO = -8
369: ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
370: INFO = -10
371: ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
372: INFO = -12
373: ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
374: INFO = -14
375: ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
376: INFO = -16
377: END IF
378: END IF
379: *
380: IF( INFO.EQ.0 ) THEN
381: IF( NOTRAN ) THEN
382: IF( IJOB.EQ.1 .OR. IJOB.EQ.2 ) THEN
383: LWMIN = MAX( 1, 2*M*N )
384: ELSE
385: LWMIN = 1
386: END IF
387: ELSE
388: LWMIN = 1
389: END IF
390: WORK( 1 ) = LWMIN
391: *
392: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
393: INFO = -20
394: END IF
395: END IF
396: *
397: IF( INFO.NE.0 ) THEN
398: CALL XERBLA( 'DTGSYL', -INFO )
399: RETURN
400: ELSE IF( LQUERY ) THEN
401: RETURN
402: END IF
403: *
404: * Quick return if possible
405: *
406: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
407: SCALE = 1
408: IF( NOTRAN ) THEN
409: IF( IJOB.NE.0 ) THEN
410: DIF = 0
411: END IF
412: END IF
413: RETURN
414: END IF
415: *
416: * Determine optimal block sizes MB and NB
417: *
418: MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 )
419: NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 )
420: *
421: ISOLVE = 1
422: IFUNC = 0
423: IF( NOTRAN ) THEN
424: IF( IJOB.GE.3 ) THEN
425: IFUNC = IJOB - 2
426: CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
427: CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
428: ELSE IF( IJOB.GE.1 ) THEN
429: ISOLVE = 2
430: END IF
431: END IF
432: *
433: IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) )
434: $ THEN
435: *
436: DO 30 IROUND = 1, ISOLVE
437: *
438: * Use unblocked Level 2 solver
439: *
440: DSCALE = ZERO
441: DSUM = ONE
442: PQ = 0
443: CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC, D,
444: $ LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE,
445: $ IWORK, PQ, INFO )
446: IF( DSCALE.NE.ZERO ) THEN
447: IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
448: DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
449: ELSE
450: DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
451: END IF
452: END IF
453: *
454: IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
455: IF( NOTRAN ) THEN
456: IFUNC = IJOB
457: END IF
458: SCALE2 = SCALE
459: CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
460: CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
461: CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
462: CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
463: ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
464: CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
465: CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
466: SCALE = SCALE2
467: END IF
468: 30 CONTINUE
469: *
470: RETURN
471: END IF
472: *
473: * Determine block structure of A
474: *
475: P = 0
476: I = 1
477: 40 CONTINUE
478: IF( I.GT.M )
479: $ GO TO 50
480: P = P + 1
481: IWORK( P ) = I
482: I = I + MB
483: IF( I.GE.M )
484: $ GO TO 50
485: IF( A( I, I-1 ).NE.ZERO )
486: $ I = I + 1
487: GO TO 40
488: 50 CONTINUE
489: *
490: IWORK( P+1 ) = M + 1
491: IF( IWORK( P ).EQ.IWORK( P+1 ) )
492: $ P = P - 1
493: *
494: * Determine block structure of B
495: *
496: Q = P + 1
497: J = 1
498: 60 CONTINUE
499: IF( J.GT.N )
500: $ GO TO 70
501: Q = Q + 1
502: IWORK( Q ) = J
503: J = J + NB
504: IF( J.GE.N )
505: $ GO TO 70
506: IF( B( J, J-1 ).NE.ZERO )
507: $ J = J + 1
508: GO TO 60
509: 70 CONTINUE
510: *
511: IWORK( Q+1 ) = N + 1
512: IF( IWORK( Q ).EQ.IWORK( Q+1 ) )
513: $ Q = Q - 1
514: *
515: IF( NOTRAN ) THEN
516: *
517: DO 150 IROUND = 1, ISOLVE
518: *
519: * Solve (I, J)-subsystem
520: * A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
521: * D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
522: * for I = P, P - 1,..., 1; J = 1, 2,..., Q
523: *
524: DSCALE = ZERO
525: DSUM = ONE
526: PQ = 0
527: SCALE = ONE
528: DO 130 J = P + 2, Q
529: JS = IWORK( J )
530: JE = IWORK( J+1 ) - 1
531: NB = JE - JS + 1
532: DO 120 I = P, 1, -1
533: IS = IWORK( I )
534: IE = IWORK( I+1 ) - 1
535: MB = IE - IS + 1
536: PPQQ = 0
537: CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
538: $ B( JS, JS ), LDB, C( IS, JS ), LDC,
539: $ D( IS, IS ), LDD, E( JS, JS ), LDE,
540: $ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
541: $ IWORK( Q+2 ), PPQQ, LINFO )
542: IF( LINFO.GT.0 )
543: $ INFO = LINFO
544: *
545: PQ = PQ + PPQQ
546: IF( SCALOC.NE.ONE ) THEN
547: DO 80 K = 1, JS - 1
548: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
549: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
550: 80 CONTINUE
551: DO 90 K = JS, JE
552: CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
553: CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
554: 90 CONTINUE
555: DO 100 K = JS, JE
556: CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
557: CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
558: 100 CONTINUE
559: DO 110 K = JE + 1, N
560: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
561: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
562: 110 CONTINUE
563: SCALE = SCALE*SCALOC
564: END IF
565: *
566: * Substitute R(I, J) and L(I, J) into remaining
567: * equation.
568: *
569: IF( I.GT.1 ) THEN
570: CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
571: $ A( 1, IS ), LDA, C( IS, JS ), LDC, ONE,
572: $ C( 1, JS ), LDC )
573: CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
574: $ D( 1, IS ), LDD, C( IS, JS ), LDC, ONE,
575: $ F( 1, JS ), LDF )
576: END IF
577: IF( J.LT.Q ) THEN
578: CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
579: $ F( IS, JS ), LDF, B( JS, JE+1 ), LDB,
580: $ ONE, C( IS, JE+1 ), LDC )
581: CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
582: $ F( IS, JS ), LDF, E( JS, JE+1 ), LDE,
583: $ ONE, F( IS, JE+1 ), LDF )
584: END IF
585: 120 CONTINUE
586: 130 CONTINUE
587: IF( DSCALE.NE.ZERO ) THEN
588: IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
589: DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
590: ELSE
591: DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
592: END IF
593: END IF
594: IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
595: IF( NOTRAN ) THEN
596: IFUNC = IJOB
597: END IF
598: SCALE2 = SCALE
599: CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
600: CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
601: CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
602: CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
603: ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
604: CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
605: CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
606: SCALE = SCALE2
607: END IF
608: 150 CONTINUE
609: *
610: ELSE
611: *
612: * Solve transposed (I, J)-subsystem
613: * A(I, I)**T * R(I, J) + D(I, I)**T * L(I, J) = C(I, J)
614: * R(I, J) * B(J, J)**T + L(I, J) * E(J, J)**T = -F(I, J)
615: * for I = 1,2,..., P; J = Q, Q-1,..., 1
616: *
617: SCALE = ONE
618: DO 210 I = 1, P
619: IS = IWORK( I )
620: IE = IWORK( I+1 ) - 1
621: MB = IE - IS + 1
622: DO 200 J = Q, P + 2, -1
623: JS = IWORK( J )
624: JE = IWORK( J+1 ) - 1
625: NB = JE - JS + 1
626: CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
627: $ B( JS, JS ), LDB, C( IS, JS ), LDC,
628: $ D( IS, IS ), LDD, E( JS, JS ), LDE,
629: $ F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
630: $ IWORK( Q+2 ), PPQQ, LINFO )
631: IF( LINFO.GT.0 )
632: $ INFO = LINFO
633: IF( SCALOC.NE.ONE ) THEN
634: DO 160 K = 1, JS - 1
635: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
636: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
637: 160 CONTINUE
638: DO 170 K = JS, JE
639: CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
640: CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
641: 170 CONTINUE
642: DO 180 K = JS, JE
643: CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
644: CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
645: 180 CONTINUE
646: DO 190 K = JE + 1, N
647: CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
648: CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
649: 190 CONTINUE
650: SCALE = SCALE*SCALOC
651: END IF
652: *
653: * Substitute R(I, J) and L(I, J) into remaining equation.
654: *
655: IF( J.GT.P+2 ) THEN
656: CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, C( IS, JS ),
657: $ LDC, B( 1, JS ), LDB, ONE, F( IS, 1 ),
658: $ LDF )
659: CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, F( IS, JS ),
660: $ LDF, E( 1, JS ), LDE, ONE, F( IS, 1 ),
661: $ LDF )
662: END IF
663: IF( I.LT.P ) THEN
664: CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
665: $ A( IS, IE+1 ), LDA, C( IS, JS ), LDC, ONE,
666: $ C( IE+1, JS ), LDC )
667: CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
668: $ D( IS, IE+1 ), LDD, F( IS, JS ), LDF, ONE,
669: $ C( IE+1, JS ), LDC )
670: END IF
671: 200 CONTINUE
672: 210 CONTINUE
673: *
674: END IF
675: *
676: WORK( 1 ) = LWMIN
677: *
678: RETURN
679: *
680: * End of DTGSYL
681: *
682: END
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