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1.1 bertrand 1: *> \brief \b DGGHD3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGGHD3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgghd3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgghd3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgghd3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGGHD3( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
22: * LDQ, Z, LDZ, WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER COMPQ, COMPZ
26: * INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N, LWORK
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
30: * $ Z( LDZ, * ), WORK( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DGGHD3 reduces a pair of real matrices (A,B) to generalized upper
40: *> Hessenberg form using orthogonal transformations, where A is a
41: *> general matrix and B is upper triangular. The form of the
42: *> generalized eigenvalue problem is
43: *> A*x = lambda*B*x,
44: *> and B is typically made upper triangular by computing its QR
45: *> factorization and moving the orthogonal matrix Q to the left side
46: *> of the equation.
47: *>
48: *> This subroutine simultaneously reduces A to a Hessenberg matrix H:
49: *> Q**T*A*Z = H
50: *> and transforms B to another upper triangular matrix T:
51: *> Q**T*B*Z = T
52: *> in order to reduce the problem to its standard form
53: *> H*y = lambda*T*y
54: *> where y = Z**T*x.
55: *>
56: *> The orthogonal matrices Q and Z are determined as products of Givens
57: *> rotations. They may either be formed explicitly, or they may be
58: *> postmultiplied into input matrices Q1 and Z1, so that
59: *>
60: *> Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
61: *>
62: *> Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
63: *>
64: *> If Q1 is the orthogonal matrix from the QR factorization of B in the
65: *> original equation A*x = lambda*B*x, then DGGHD3 reduces the original
66: *> problem to generalized Hessenberg form.
67: *>
68: *> This is a blocked variant of DGGHRD, using matrix-matrix
69: *> multiplications for parts of the computation to enhance performance.
70: *> \endverbatim
71: *
72: * Arguments:
73: * ==========
74: *
75: *> \param[in] COMPQ
76: *> \verbatim
77: *> COMPQ is CHARACTER*1
78: *> = 'N': do not compute Q;
79: *> = 'I': Q is initialized to the unit matrix, and the
80: *> orthogonal matrix Q is returned;
81: *> = 'V': Q must contain an orthogonal matrix Q1 on entry,
82: *> and the product Q1*Q is returned.
83: *> \endverbatim
84: *>
85: *> \param[in] COMPZ
86: *> \verbatim
87: *> COMPZ is CHARACTER*1
88: *> = 'N': do not compute Z;
89: *> = 'I': Z is initialized to the unit matrix, and the
90: *> orthogonal matrix Z is returned;
91: *> = 'V': Z must contain an orthogonal matrix Z1 on entry,
92: *> and the product Z1*Z is returned.
93: *> \endverbatim
94: *>
95: *> \param[in] N
96: *> \verbatim
97: *> N is INTEGER
98: *> The order of the matrices A and B. N >= 0.
99: *> \endverbatim
100: *>
101: *> \param[in] ILO
102: *> \verbatim
103: *> ILO is INTEGER
104: *> \endverbatim
105: *>
106: *> \param[in] IHI
107: *> \verbatim
108: *> IHI is INTEGER
109: *>
110: *> ILO and IHI mark the rows and columns of A which are to be
111: *> reduced. It is assumed that A is already upper triangular
112: *> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
113: *> normally set by a previous call to DGGBAL; otherwise they
114: *> should be set to 1 and N respectively.
115: *> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
116: *> \endverbatim
117: *>
118: *> \param[in,out] A
119: *> \verbatim
120: *> A is DOUBLE PRECISION array, dimension (LDA, N)
121: *> On entry, the N-by-N general matrix to be reduced.
122: *> On exit, the upper triangle and the first subdiagonal of A
123: *> are overwritten with the upper Hessenberg matrix H, and the
124: *> rest is set to zero.
125: *> \endverbatim
126: *>
127: *> \param[in] LDA
128: *> \verbatim
129: *> LDA is INTEGER
130: *> The leading dimension of the array A. LDA >= max(1,N).
131: *> \endverbatim
132: *>
133: *> \param[in,out] B
134: *> \verbatim
135: *> B is DOUBLE PRECISION array, dimension (LDB, N)
136: *> On entry, the N-by-N upper triangular matrix B.
137: *> On exit, the upper triangular matrix T = Q**T B Z. The
138: *> elements below the diagonal are set to zero.
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] Q
148: *> \verbatim
149: *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
150: *> On entry, if COMPQ = 'V', the orthogonal matrix Q1,
151: *> typically from the QR factorization of B.
152: *> On exit, if COMPQ='I', the orthogonal matrix Q, and if
153: *> COMPQ = 'V', the product Q1*Q.
154: *> Not referenced if COMPQ='N'.
155: *> \endverbatim
156: *>
157: *> \param[in] LDQ
158: *> \verbatim
159: *> LDQ is INTEGER
160: *> The leading dimension of the array Q.
161: *> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
162: *> \endverbatim
163: *>
164: *> \param[in,out] Z
165: *> \verbatim
166: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
167: *> On entry, if COMPZ = 'V', the orthogonal matrix Z1.
168: *> On exit, if COMPZ='I', the orthogonal matrix Z, and if
169: *> COMPZ = 'V', the product Z1*Z.
170: *> Not referenced if COMPZ='N'.
171: *> \endverbatim
172: *>
173: *> \param[in] LDZ
174: *> \verbatim
175: *> LDZ is INTEGER
176: *> The leading dimension of the array Z.
177: *> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
178: *> \endverbatim
179: *>
180: *> \param[out] WORK
181: *> \verbatim
182: *> WORK is DOUBLE PRECISION array, dimension (LWORK)
183: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
184: *> \endverbatim
185: *>
186: *> \param[in] LWORK
187: *> \verbatim
188: *> LWORK is INTEGER
189: *> The length of the array WORK. LWORK >= 1.
190: *> For optimum performance LWORK >= 6*N*NB, where NB is the
191: *> optimal blocksize.
192: *>
193: *> If LWORK = -1, then a workspace query is assumed; the routine
194: *> only calculates the optimal size of the WORK array, returns
195: *> this value as the first entry of the WORK array, and no error
196: *> message related to LWORK is issued by XERBLA.
197: *> \endverbatim
198: *>
199: *> \param[out] INFO
200: *> \verbatim
201: *> INFO is INTEGER
202: *> = 0: successful exit.
203: *> < 0: if INFO = -i, the i-th argument had an illegal value.
204: *> \endverbatim
205: *
206: * Authors:
207: * ========
208: *
209: *> \author Univ. of Tennessee
210: *> \author Univ. of California Berkeley
211: *> \author Univ. of Colorado Denver
212: *> \author NAG Ltd.
213: *
214: *> \ingroup doubleOTHERcomputational
215: *
216: *> \par Further Details:
217: * =====================
218: *>
219: *> \verbatim
220: *>
1.7 ! bertrand 221: *> This routine reduces A to Hessenberg form and maintains B in triangular form
1.1 bertrand 222: *> using a blocked variant of Moler and Stewart's original algorithm,
223: *> as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
224: *> (BIT 2008).
225: *> \endverbatim
226: *>
227: * =====================================================================
228: SUBROUTINE DGGHD3( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
229: $ LDQ, Z, LDZ, WORK, LWORK, INFO )
230: *
1.7 ! bertrand 231: * -- LAPACK computational routine --
1.1 bertrand 232: * -- LAPACK is a software package provided by Univ. of Tennessee, --
233: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
234: *
235: IMPLICIT NONE
236: *
237: * .. Scalar Arguments ..
238: CHARACTER COMPQ, COMPZ
239: INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N, LWORK
240: * ..
241: * .. Array Arguments ..
242: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
243: $ Z( LDZ, * ), WORK( * )
244: * ..
245: *
246: * =====================================================================
247: *
248: * .. Parameters ..
249: DOUBLE PRECISION ZERO, ONE
250: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
251: * ..
252: * .. Local Scalars ..
253: LOGICAL BLK22, INITQ, INITZ, LQUERY, WANTQ, WANTZ
254: CHARACTER*1 COMPQ2, COMPZ2
255: INTEGER COLA, I, IERR, J, J0, JCOL, JJ, JROW, K,
256: $ KACC22, LEN, LWKOPT, N2NB, NB, NBLST, NBMIN,
257: $ NH, NNB, NX, PPW, PPWO, PW, TOP, TOPQ
258: DOUBLE PRECISION C, C1, C2, S, S1, S2, TEMP, TEMP1, TEMP2, TEMP3
259: * ..
260: * .. External Functions ..
261: LOGICAL LSAME
262: INTEGER ILAENV
263: EXTERNAL ILAENV, LSAME
264: * ..
265: * .. External Subroutines ..
1.5 bertrand 266: EXTERNAL DGGHRD, DLARTG, DLASET, DORM22, DROT, DGEMM,
267: $ DGEMV, DTRMV, DLACPY, XERBLA
1.1 bertrand 268: * ..
269: * .. Intrinsic Functions ..
270: INTRINSIC DBLE, MAX
271: * ..
272: * .. Executable Statements ..
273: *
274: * Decode and test the input parameters.
275: *
276: INFO = 0
277: NB = ILAENV( 1, 'DGGHD3', ' ', N, ILO, IHI, -1 )
1.2 bertrand 278: LWKOPT = MAX( 6*N*NB, 1 )
1.1 bertrand 279: WORK( 1 ) = DBLE( LWKOPT )
280: INITQ = LSAME( COMPQ, 'I' )
281: WANTQ = INITQ .OR. LSAME( COMPQ, 'V' )
282: INITZ = LSAME( COMPZ, 'I' )
283: WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
284: LQUERY = ( LWORK.EQ.-1 )
285: *
286: IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
287: INFO = -1
288: ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
289: INFO = -2
290: ELSE IF( N.LT.0 ) THEN
291: INFO = -3
292: ELSE IF( ILO.LT.1 ) THEN
293: INFO = -4
294: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
295: INFO = -5
296: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
297: INFO = -7
298: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
299: INFO = -9
300: ELSE IF( ( WANTQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
301: INFO = -11
302: ELSE IF( ( WANTZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
303: INFO = -13
304: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
305: INFO = -15
306: END IF
307: IF( INFO.NE.0 ) THEN
308: CALL XERBLA( 'DGGHD3', -INFO )
309: RETURN
310: ELSE IF( LQUERY ) THEN
311: RETURN
312: END IF
313: *
314: * Initialize Q and Z if desired.
315: *
316: IF( INITQ )
317: $ CALL DLASET( 'All', N, N, ZERO, ONE, Q, LDQ )
318: IF( INITZ )
319: $ CALL DLASET( 'All', N, N, ZERO, ONE, Z, LDZ )
320: *
321: * Zero out lower triangle of B.
322: *
323: IF( N.GT.1 )
324: $ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, B(2, 1), LDB )
325: *
326: * Quick return if possible
327: *
328: NH = IHI - ILO + 1
329: IF( NH.LE.1 ) THEN
330: WORK( 1 ) = ONE
331: RETURN
332: END IF
333: *
334: * Determine the blocksize.
335: *
336: NBMIN = ILAENV( 2, 'DGGHD3', ' ', N, ILO, IHI, -1 )
337: IF( NB.GT.1 .AND. NB.LT.NH ) THEN
338: *
339: * Determine when to use unblocked instead of blocked code.
340: *
341: NX = MAX( NB, ILAENV( 3, 'DGGHD3', ' ', N, ILO, IHI, -1 ) )
342: IF( NX.LT.NH ) THEN
343: *
344: * Determine if workspace is large enough for blocked code.
345: *
346: IF( LWORK.LT.LWKOPT ) THEN
347: *
348: * Not enough workspace to use optimal NB: determine the
349: * minimum value of NB, and reduce NB or force use of
350: * unblocked code.
351: *
352: NBMIN = MAX( 2, ILAENV( 2, 'DGGHD3', ' ', N, ILO, IHI,
353: $ -1 ) )
354: IF( LWORK.GE.6*N*NBMIN ) THEN
355: NB = LWORK / ( 6*N )
356: ELSE
357: NB = 1
358: END IF
359: END IF
360: END IF
361: END IF
362: *
363: IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
364: *
365: * Use unblocked code below
366: *
367: JCOL = ILO
368: *
369: ELSE
370: *
371: * Use blocked code
372: *
373: KACC22 = ILAENV( 16, 'DGGHD3', ' ', N, ILO, IHI, -1 )
374: BLK22 = KACC22.EQ.2
375: DO JCOL = ILO, IHI-2, NB
376: NNB = MIN( NB, IHI-JCOL-1 )
377: *
378: * Initialize small orthogonal factors that will hold the
379: * accumulated Givens rotations in workspace.
380: * N2NB denotes the number of 2*NNB-by-2*NNB factors
381: * NBLST denotes the (possibly smaller) order of the last
382: * factor.
383: *
384: N2NB = ( IHI-JCOL-1 ) / NNB - 1
385: NBLST = IHI - JCOL - N2NB*NNB
386: CALL DLASET( 'All', NBLST, NBLST, ZERO, ONE, WORK, NBLST )
387: PW = NBLST * NBLST + 1
388: DO I = 1, N2NB
389: CALL DLASET( 'All', 2*NNB, 2*NNB, ZERO, ONE,
390: $ WORK( PW ), 2*NNB )
391: PW = PW + 4*NNB*NNB
392: END DO
393: *
394: * Reduce columns JCOL:JCOL+NNB-1 of A to Hessenberg form.
395: *
396: DO J = JCOL, JCOL+NNB-1
397: *
398: * Reduce Jth column of A. Store cosines and sines in Jth
399: * column of A and B, respectively.
400: *
401: DO I = IHI, J+2, -1
402: TEMP = A( I-1, J )
403: CALL DLARTG( TEMP, A( I, J ), C, S, A( I-1, J ) )
404: A( I, J ) = C
405: B( I, J ) = S
406: END DO
407: *
408: * Accumulate Givens rotations into workspace array.
409: *
410: PPW = ( NBLST + 1 )*( NBLST - 2 ) - J + JCOL + 1
411: LEN = 2 + J - JCOL
412: JROW = J + N2NB*NNB + 2
413: DO I = IHI, JROW, -1
414: C = A( I, J )
415: S = B( I, J )
416: DO JJ = PPW, PPW+LEN-1
417: TEMP = WORK( JJ + NBLST )
418: WORK( JJ + NBLST ) = C*TEMP - S*WORK( JJ )
419: WORK( JJ ) = S*TEMP + C*WORK( JJ )
420: END DO
421: LEN = LEN + 1
422: PPW = PPW - NBLST - 1
423: END DO
424: *
425: PPWO = NBLST*NBLST + ( NNB+J-JCOL-1 )*2*NNB + NNB
426: J0 = JROW - NNB
427: DO JROW = J0, J+2, -NNB
428: PPW = PPWO
429: LEN = 2 + J - JCOL
430: DO I = JROW+NNB-1, JROW, -1
431: C = A( I, J )
432: S = B( I, J )
433: DO JJ = PPW, PPW+LEN-1
434: TEMP = WORK( JJ + 2*NNB )
435: WORK( JJ + 2*NNB ) = C*TEMP - S*WORK( JJ )
436: WORK( JJ ) = S*TEMP + C*WORK( JJ )
437: END DO
438: LEN = LEN + 1
439: PPW = PPW - 2*NNB - 1
440: END DO
441: PPWO = PPWO + 4*NNB*NNB
442: END DO
443: *
444: * TOP denotes the number of top rows in A and B that will
445: * not be updated during the next steps.
446: *
447: IF( JCOL.LE.2 ) THEN
448: TOP = 0
449: ELSE
450: TOP = JCOL
451: END IF
452: *
453: * Propagate transformations through B and replace stored
454: * left sines/cosines by right sines/cosines.
455: *
456: DO JJ = N, J+1, -1
457: *
458: * Update JJth column of B.
459: *
460: DO I = MIN( JJ+1, IHI ), J+2, -1
461: C = A( I, J )
462: S = B( I, J )
463: TEMP = B( I, JJ )
464: B( I, JJ ) = C*TEMP - S*B( I-1, JJ )
465: B( I-1, JJ ) = S*TEMP + C*B( I-1, JJ )
466: END DO
467: *
468: * Annihilate B( JJ+1, JJ ).
469: *
470: IF( JJ.LT.IHI ) THEN
471: TEMP = B( JJ+1, JJ+1 )
472: CALL DLARTG( TEMP, B( JJ+1, JJ ), C, S,
473: $ B( JJ+1, JJ+1 ) )
474: B( JJ+1, JJ ) = ZERO
475: CALL DROT( JJ-TOP, B( TOP+1, JJ+1 ), 1,
476: $ B( TOP+1, JJ ), 1, C, S )
477: A( JJ+1, J ) = C
478: B( JJ+1, J ) = -S
479: END IF
480: END DO
481: *
482: * Update A by transformations from right.
483: * Explicit loop unrolling provides better performance
484: * compared to DLASR.
485: * CALL DLASR( 'Right', 'Variable', 'Backward', IHI-TOP,
486: * $ IHI-J, A( J+2, J ), B( J+2, J ),
487: * $ A( TOP+1, J+1 ), LDA )
488: *
489: JJ = MOD( IHI-J-1, 3 )
490: DO I = IHI-J-3, JJ+1, -3
491: C = A( J+1+I, J )
492: S = -B( J+1+I, J )
493: C1 = A( J+2+I, J )
494: S1 = -B( J+2+I, J )
495: C2 = A( J+3+I, J )
496: S2 = -B( J+3+I, J )
497: *
498: DO K = TOP+1, IHI
499: TEMP = A( K, J+I )
500: TEMP1 = A( K, J+I+1 )
501: TEMP2 = A( K, J+I+2 )
502: TEMP3 = A( K, J+I+3 )
503: A( K, J+I+3 ) = C2*TEMP3 + S2*TEMP2
504: TEMP2 = -S2*TEMP3 + C2*TEMP2
505: A( K, J+I+2 ) = C1*TEMP2 + S1*TEMP1
506: TEMP1 = -S1*TEMP2 + C1*TEMP1
507: A( K, J+I+1 ) = C*TEMP1 + S*TEMP
508: A( K, J+I ) = -S*TEMP1 + C*TEMP
509: END DO
510: END DO
511: *
512: IF( JJ.GT.0 ) THEN
513: DO I = JJ, 1, -1
514: CALL DROT( IHI-TOP, A( TOP+1, J+I+1 ), 1,
515: $ A( TOP+1, J+I ), 1, A( J+1+I, J ),
516: $ -B( J+1+I, J ) )
517: END DO
518: END IF
519: *
520: * Update (J+1)th column of A by transformations from left.
521: *
522: IF ( J .LT. JCOL + NNB - 1 ) THEN
523: LEN = 1 + J - JCOL
524: *
525: * Multiply with the trailing accumulated orthogonal
526: * matrix, which takes the form
527: *
528: * [ U11 U12 ]
529: * U = [ ],
530: * [ U21 U22 ]
531: *
532: * where U21 is a LEN-by-LEN matrix and U12 is lower
533: * triangular.
534: *
535: JROW = IHI - NBLST + 1
536: CALL DGEMV( 'Transpose', NBLST, LEN, ONE, WORK,
537: $ NBLST, A( JROW, J+1 ), 1, ZERO,
538: $ WORK( PW ), 1 )
539: PPW = PW + LEN
540: DO I = JROW, JROW+NBLST-LEN-1
541: WORK( PPW ) = A( I, J+1 )
542: PPW = PPW + 1
543: END DO
544: CALL DTRMV( 'Lower', 'Transpose', 'Non-unit',
545: $ NBLST-LEN, WORK( LEN*NBLST + 1 ), NBLST,
546: $ WORK( PW+LEN ), 1 )
547: CALL DGEMV( 'Transpose', LEN, NBLST-LEN, ONE,
548: $ WORK( (LEN+1)*NBLST - LEN + 1 ), NBLST,
549: $ A( JROW+NBLST-LEN, J+1 ), 1, ONE,
550: $ WORK( PW+LEN ), 1 )
551: PPW = PW
552: DO I = JROW, JROW+NBLST-1
553: A( I, J+1 ) = WORK( PPW )
554: PPW = PPW + 1
555: END DO
556: *
557: * Multiply with the other accumulated orthogonal
558: * matrices, which take the form
559: *
560: * [ U11 U12 0 ]
561: * [ ]
562: * U = [ U21 U22 0 ],
563: * [ ]
564: * [ 0 0 I ]
565: *
566: * where I denotes the (NNB-LEN)-by-(NNB-LEN) identity
567: * matrix, U21 is a LEN-by-LEN upper triangular matrix
568: * and U12 is an NNB-by-NNB lower triangular matrix.
569: *
570: PPWO = 1 + NBLST*NBLST
571: J0 = JROW - NNB
572: DO JROW = J0, JCOL+1, -NNB
573: PPW = PW + LEN
574: DO I = JROW, JROW+NNB-1
575: WORK( PPW ) = A( I, J+1 )
576: PPW = PPW + 1
577: END DO
578: PPW = PW
579: DO I = JROW+NNB, JROW+NNB+LEN-1
580: WORK( PPW ) = A( I, J+1 )
581: PPW = PPW + 1
582: END DO
583: CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', LEN,
584: $ WORK( PPWO + NNB ), 2*NNB, WORK( PW ),
585: $ 1 )
586: CALL DTRMV( 'Lower', 'Transpose', 'Non-unit', NNB,
587: $ WORK( PPWO + 2*LEN*NNB ),
588: $ 2*NNB, WORK( PW + LEN ), 1 )
589: CALL DGEMV( 'Transpose', NNB, LEN, ONE,
590: $ WORK( PPWO ), 2*NNB, A( JROW, J+1 ), 1,
591: $ ONE, WORK( PW ), 1 )
592: CALL DGEMV( 'Transpose', LEN, NNB, ONE,
593: $ WORK( PPWO + 2*LEN*NNB + NNB ), 2*NNB,
594: $ A( JROW+NNB, J+1 ), 1, ONE,
595: $ WORK( PW+LEN ), 1 )
596: PPW = PW
597: DO I = JROW, JROW+LEN+NNB-1
598: A( I, J+1 ) = WORK( PPW )
599: PPW = PPW + 1
600: END DO
601: PPWO = PPWO + 4*NNB*NNB
602: END DO
603: END IF
604: END DO
605: *
606: * Apply accumulated orthogonal matrices to A.
607: *
608: COLA = N - JCOL - NNB + 1
609: J = IHI - NBLST + 1
610: CALL DGEMM( 'Transpose', 'No Transpose', NBLST,
611: $ COLA, NBLST, ONE, WORK, NBLST,
612: $ A( J, JCOL+NNB ), LDA, ZERO, WORK( PW ),
613: $ NBLST )
614: CALL DLACPY( 'All', NBLST, COLA, WORK( PW ), NBLST,
615: $ A( J, JCOL+NNB ), LDA )
616: PPWO = NBLST*NBLST + 1
617: J0 = J - NNB
618: DO J = J0, JCOL+1, -NNB
619: IF ( BLK22 ) THEN
620: *
621: * Exploit the structure of
622: *
623: * [ U11 U12 ]
624: * U = [ ]
625: * [ U21 U22 ],
626: *
627: * where all blocks are NNB-by-NNB, U21 is upper
628: * triangular and U12 is lower triangular.
629: *
630: CALL DORM22( 'Left', 'Transpose', 2*NNB, COLA, NNB,
631: $ NNB, WORK( PPWO ), 2*NNB,
632: $ A( J, JCOL+NNB ), LDA, WORK( PW ),
633: $ LWORK-PW+1, IERR )
634: ELSE
635: *
636: * Ignore the structure of U.
637: *
638: CALL DGEMM( 'Transpose', 'No Transpose', 2*NNB,
639: $ COLA, 2*NNB, ONE, WORK( PPWO ), 2*NNB,
640: $ A( J, JCOL+NNB ), LDA, ZERO, WORK( PW ),
641: $ 2*NNB )
642: CALL DLACPY( 'All', 2*NNB, COLA, WORK( PW ), 2*NNB,
643: $ A( J, JCOL+NNB ), LDA )
644: END IF
645: PPWO = PPWO + 4*NNB*NNB
646: END DO
647: *
648: * Apply accumulated orthogonal matrices to Q.
649: *
650: IF( WANTQ ) THEN
651: J = IHI - NBLST + 1
652: IF ( INITQ ) THEN
653: TOPQ = MAX( 2, J - JCOL + 1 )
654: NH = IHI - TOPQ + 1
655: ELSE
656: TOPQ = 1
657: NH = N
658: END IF
659: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
660: $ NBLST, NBLST, ONE, Q( TOPQ, J ), LDQ,
661: $ WORK, NBLST, ZERO, WORK( PW ), NH )
662: CALL DLACPY( 'All', NH, NBLST, WORK( PW ), NH,
663: $ Q( TOPQ, J ), LDQ )
664: PPWO = NBLST*NBLST + 1
665: J0 = J - NNB
666: DO J = J0, JCOL+1, -NNB
667: IF ( INITQ ) THEN
668: TOPQ = MAX( 2, J - JCOL + 1 )
669: NH = IHI - TOPQ + 1
670: END IF
671: IF ( BLK22 ) THEN
672: *
673: * Exploit the structure of U.
674: *
675: CALL DORM22( 'Right', 'No Transpose', NH, 2*NNB,
676: $ NNB, NNB, WORK( PPWO ), 2*NNB,
677: $ Q( TOPQ, J ), LDQ, WORK( PW ),
678: $ LWORK-PW+1, IERR )
679: ELSE
680: *
681: * Ignore the structure of U.
682: *
683: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
684: $ 2*NNB, 2*NNB, ONE, Q( TOPQ, J ), LDQ,
685: $ WORK( PPWO ), 2*NNB, ZERO, WORK( PW ),
686: $ NH )
687: CALL DLACPY( 'All', NH, 2*NNB, WORK( PW ), NH,
688: $ Q( TOPQ, J ), LDQ )
689: END IF
690: PPWO = PPWO + 4*NNB*NNB
691: END DO
692: END IF
693: *
694: * Accumulate right Givens rotations if required.
695: *
696: IF ( WANTZ .OR. TOP.GT.0 ) THEN
697: *
698: * Initialize small orthogonal factors that will hold the
699: * accumulated Givens rotations in workspace.
700: *
701: CALL DLASET( 'All', NBLST, NBLST, ZERO, ONE, WORK,
702: $ NBLST )
703: PW = NBLST * NBLST + 1
704: DO I = 1, N2NB
705: CALL DLASET( 'All', 2*NNB, 2*NNB, ZERO, ONE,
706: $ WORK( PW ), 2*NNB )
707: PW = PW + 4*NNB*NNB
708: END DO
709: *
710: * Accumulate Givens rotations into workspace array.
711: *
712: DO J = JCOL, JCOL+NNB-1
713: PPW = ( NBLST + 1 )*( NBLST - 2 ) - J + JCOL + 1
714: LEN = 2 + J - JCOL
715: JROW = J + N2NB*NNB + 2
716: DO I = IHI, JROW, -1
717: C = A( I, J )
718: A( I, J ) = ZERO
719: S = B( I, J )
720: B( I, J ) = ZERO
721: DO JJ = PPW, PPW+LEN-1
722: TEMP = WORK( JJ + NBLST )
723: WORK( JJ + NBLST ) = C*TEMP - S*WORK( JJ )
724: WORK( JJ ) = S*TEMP + C*WORK( JJ )
725: END DO
726: LEN = LEN + 1
727: PPW = PPW - NBLST - 1
728: END DO
729: *
730: PPWO = NBLST*NBLST + ( NNB+J-JCOL-1 )*2*NNB + NNB
731: J0 = JROW - NNB
732: DO JROW = J0, J+2, -NNB
733: PPW = PPWO
734: LEN = 2 + J - JCOL
735: DO I = JROW+NNB-1, JROW, -1
736: C = A( I, J )
737: A( I, J ) = ZERO
738: S = B( I, J )
739: B( I, J ) = ZERO
740: DO JJ = PPW, PPW+LEN-1
741: TEMP = WORK( JJ + 2*NNB )
742: WORK( JJ + 2*NNB ) = C*TEMP - S*WORK( JJ )
743: WORK( JJ ) = S*TEMP + C*WORK( JJ )
744: END DO
745: LEN = LEN + 1
746: PPW = PPW - 2*NNB - 1
747: END DO
748: PPWO = PPWO + 4*NNB*NNB
749: END DO
750: END DO
751: ELSE
752: *
753: CALL DLASET( 'Lower', IHI - JCOL - 1, NNB, ZERO, ZERO,
754: $ A( JCOL + 2, JCOL ), LDA )
755: CALL DLASET( 'Lower', IHI - JCOL - 1, NNB, ZERO, ZERO,
756: $ B( JCOL + 2, JCOL ), LDB )
757: END IF
758: *
759: * Apply accumulated orthogonal matrices to A and B.
760: *
761: IF ( TOP.GT.0 ) THEN
762: J = IHI - NBLST + 1
763: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
764: $ NBLST, NBLST, ONE, A( 1, J ), LDA,
765: $ WORK, NBLST, ZERO, WORK( PW ), TOP )
766: CALL DLACPY( 'All', TOP, NBLST, WORK( PW ), TOP,
767: $ A( 1, J ), LDA )
768: PPWO = NBLST*NBLST + 1
769: J0 = J - NNB
770: DO J = J0, JCOL+1, -NNB
771: IF ( BLK22 ) THEN
772: *
773: * Exploit the structure of U.
774: *
775: CALL DORM22( 'Right', 'No Transpose', TOP, 2*NNB,
776: $ NNB, NNB, WORK( PPWO ), 2*NNB,
777: $ A( 1, J ), LDA, WORK( PW ),
778: $ LWORK-PW+1, IERR )
779: ELSE
780: *
781: * Ignore the structure of U.
782: *
783: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
784: $ 2*NNB, 2*NNB, ONE, A( 1, J ), LDA,
785: $ WORK( PPWO ), 2*NNB, ZERO,
786: $ WORK( PW ), TOP )
787: CALL DLACPY( 'All', TOP, 2*NNB, WORK( PW ), TOP,
788: $ A( 1, J ), LDA )
789: END IF
790: PPWO = PPWO + 4*NNB*NNB
791: END DO
792: *
793: J = IHI - NBLST + 1
794: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
795: $ NBLST, NBLST, ONE, B( 1, J ), LDB,
796: $ WORK, NBLST, ZERO, WORK( PW ), TOP )
797: CALL DLACPY( 'All', TOP, NBLST, WORK( PW ), TOP,
798: $ B( 1, J ), LDB )
799: PPWO = NBLST*NBLST + 1
800: J0 = J - NNB
801: DO J = J0, JCOL+1, -NNB
802: IF ( BLK22 ) THEN
803: *
804: * Exploit the structure of U.
805: *
806: CALL DORM22( 'Right', 'No Transpose', TOP, 2*NNB,
807: $ NNB, NNB, WORK( PPWO ), 2*NNB,
808: $ B( 1, J ), LDB, WORK( PW ),
809: $ LWORK-PW+1, IERR )
810: ELSE
811: *
812: * Ignore the structure of U.
813: *
814: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
815: $ 2*NNB, 2*NNB, ONE, B( 1, J ), LDB,
816: $ WORK( PPWO ), 2*NNB, ZERO,
817: $ WORK( PW ), TOP )
818: CALL DLACPY( 'All', TOP, 2*NNB, WORK( PW ), TOP,
819: $ B( 1, J ), LDB )
820: END IF
821: PPWO = PPWO + 4*NNB*NNB
822: END DO
823: END IF
824: *
825: * Apply accumulated orthogonal matrices to Z.
826: *
827: IF( WANTZ ) THEN
828: J = IHI - NBLST + 1
829: IF ( INITQ ) THEN
830: TOPQ = MAX( 2, J - JCOL + 1 )
831: NH = IHI - TOPQ + 1
832: ELSE
833: TOPQ = 1
834: NH = N
835: END IF
836: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
837: $ NBLST, NBLST, ONE, Z( TOPQ, J ), LDZ,
838: $ WORK, NBLST, ZERO, WORK( PW ), NH )
839: CALL DLACPY( 'All', NH, NBLST, WORK( PW ), NH,
840: $ Z( TOPQ, J ), LDZ )
841: PPWO = NBLST*NBLST + 1
842: J0 = J - NNB
843: DO J = J0, JCOL+1, -NNB
844: IF ( INITQ ) THEN
845: TOPQ = MAX( 2, J - JCOL + 1 )
846: NH = IHI - TOPQ + 1
847: END IF
848: IF ( BLK22 ) THEN
849: *
850: * Exploit the structure of U.
851: *
852: CALL DORM22( 'Right', 'No Transpose', NH, 2*NNB,
853: $ NNB, NNB, WORK( PPWO ), 2*NNB,
854: $ Z( TOPQ, J ), LDZ, WORK( PW ),
855: $ LWORK-PW+1, IERR )
856: ELSE
857: *
858: * Ignore the structure of U.
859: *
860: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
861: $ 2*NNB, 2*NNB, ONE, Z( TOPQ, J ), LDZ,
862: $ WORK( PPWO ), 2*NNB, ZERO, WORK( PW ),
863: $ NH )
864: CALL DLACPY( 'All', NH, 2*NNB, WORK( PW ), NH,
865: $ Z( TOPQ, J ), LDZ )
866: END IF
867: PPWO = PPWO + 4*NNB*NNB
868: END DO
869: END IF
870: END DO
871: END IF
872: *
873: * Use unblocked code to reduce the rest of the matrix
874: * Avoid re-initialization of modified Q and Z.
875: *
876: COMPQ2 = COMPQ
877: COMPZ2 = COMPZ
878: IF ( JCOL.NE.ILO ) THEN
879: IF ( WANTQ )
880: $ COMPQ2 = 'V'
881: IF ( WANTZ )
882: $ COMPZ2 = 'V'
883: END IF
884: *
885: IF ( JCOL.LT.IHI )
886: $ CALL DGGHRD( COMPQ2, COMPZ2, N, JCOL, IHI, A, LDA, B, LDB, Q,
887: $ LDQ, Z, LDZ, IERR )
888: WORK( 1 ) = DBLE( LWKOPT )
889: *
890: RETURN
891: *
892: * End of DGGHD3
893: *
894: END