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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: *> \date January 2015
215: *
216: *> \ingroup doubleOTHERcomputational
217: *
218: *> \par Further Details:
219: * =====================
220: *>
221: *> \verbatim
222: *>
223: *> This routine reduces A to Hessenberg form and maintains B in
224: *> using a blocked variant of Moler and Stewart's original algorithm,
225: *> as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
226: *> (BIT 2008).
227: *> \endverbatim
228: *>
229: * =====================================================================
230: SUBROUTINE DGGHD3( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
231: $ LDQ, Z, LDZ, WORK, LWORK, INFO )
232: *
1.2 ! bertrand 233: * -- LAPACK computational routine (version 3.6.1) --
1.1 bertrand 234: * -- LAPACK is a software package provided by Univ. of Tennessee, --
235: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
236: * January 2015
237: *
238: IMPLICIT NONE
239: *
240: * .. Scalar Arguments ..
241: CHARACTER COMPQ, COMPZ
242: INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N, LWORK
243: * ..
244: * .. Array Arguments ..
245: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
246: $ Z( LDZ, * ), WORK( * )
247: * ..
248: *
249: * =====================================================================
250: *
251: * .. Parameters ..
252: DOUBLE PRECISION ZERO, ONE
253: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
254: * ..
255: * .. Local Scalars ..
256: LOGICAL BLK22, INITQ, INITZ, LQUERY, WANTQ, WANTZ
257: CHARACTER*1 COMPQ2, COMPZ2
258: INTEGER COLA, I, IERR, J, J0, JCOL, JJ, JROW, K,
259: $ KACC22, LEN, LWKOPT, N2NB, NB, NBLST, NBMIN,
260: $ NH, NNB, NX, PPW, PPWO, PW, TOP, TOPQ
261: DOUBLE PRECISION C, C1, C2, S, S1, S2, TEMP, TEMP1, TEMP2, TEMP3
262: * ..
263: * .. External Functions ..
264: LOGICAL LSAME
265: INTEGER ILAENV
266: EXTERNAL ILAENV, LSAME
267: * ..
268: * .. External Subroutines ..
269: EXTERNAL DGGHRD, DLARTG, DLASET, DORM22, DROT, XERBLA
270: * ..
271: * .. Intrinsic Functions ..
272: INTRINSIC DBLE, MAX
273: * ..
274: * .. Executable Statements ..
275: *
276: * Decode and test the input parameters.
277: *
278: INFO = 0
279: NB = ILAENV( 1, 'DGGHD3', ' ', N, ILO, IHI, -1 )
1.2 ! bertrand 280: LWKOPT = MAX( 6*N*NB, 1 )
1.1 bertrand 281: WORK( 1 ) = DBLE( LWKOPT )
282: INITQ = LSAME( COMPQ, 'I' )
283: WANTQ = INITQ .OR. LSAME( COMPQ, 'V' )
284: INITZ = LSAME( COMPZ, 'I' )
285: WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
286: LQUERY = ( LWORK.EQ.-1 )
287: *
288: IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
289: INFO = -1
290: ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
291: INFO = -2
292: ELSE IF( N.LT.0 ) THEN
293: INFO = -3
294: ELSE IF( ILO.LT.1 ) THEN
295: INFO = -4
296: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
297: INFO = -5
298: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
299: INFO = -7
300: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
301: INFO = -9
302: ELSE IF( ( WANTQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN
303: INFO = -11
304: ELSE IF( ( WANTZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
305: INFO = -13
306: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
307: INFO = -15
308: END IF
309: IF( INFO.NE.0 ) THEN
310: CALL XERBLA( 'DGGHD3', -INFO )
311: RETURN
312: ELSE IF( LQUERY ) THEN
313: RETURN
314: END IF
315: *
316: * Initialize Q and Z if desired.
317: *
318: IF( INITQ )
319: $ CALL DLASET( 'All', N, N, ZERO, ONE, Q, LDQ )
320: IF( INITZ )
321: $ CALL DLASET( 'All', N, N, ZERO, ONE, Z, LDZ )
322: *
323: * Zero out lower triangle of B.
324: *
325: IF( N.GT.1 )
326: $ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, B(2, 1), LDB )
327: *
328: * Quick return if possible
329: *
330: NH = IHI - ILO + 1
331: IF( NH.LE.1 ) THEN
332: WORK( 1 ) = ONE
333: RETURN
334: END IF
335: *
336: * Determine the blocksize.
337: *
338: NBMIN = ILAENV( 2, 'DGGHD3', ' ', N, ILO, IHI, -1 )
339: IF( NB.GT.1 .AND. NB.LT.NH ) THEN
340: *
341: * Determine when to use unblocked instead of blocked code.
342: *
343: NX = MAX( NB, ILAENV( 3, 'DGGHD3', ' ', N, ILO, IHI, -1 ) )
344: IF( NX.LT.NH ) THEN
345: *
346: * Determine if workspace is large enough for blocked code.
347: *
348: IF( LWORK.LT.LWKOPT ) THEN
349: *
350: * Not enough workspace to use optimal NB: determine the
351: * minimum value of NB, and reduce NB or force use of
352: * unblocked code.
353: *
354: NBMIN = MAX( 2, ILAENV( 2, 'DGGHD3', ' ', N, ILO, IHI,
355: $ -1 ) )
356: IF( LWORK.GE.6*N*NBMIN ) THEN
357: NB = LWORK / ( 6*N )
358: ELSE
359: NB = 1
360: END IF
361: END IF
362: END IF
363: END IF
364: *
365: IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
366: *
367: * Use unblocked code below
368: *
369: JCOL = ILO
370: *
371: ELSE
372: *
373: * Use blocked code
374: *
375: KACC22 = ILAENV( 16, 'DGGHD3', ' ', N, ILO, IHI, -1 )
376: BLK22 = KACC22.EQ.2
377: DO JCOL = ILO, IHI-2, NB
378: NNB = MIN( NB, IHI-JCOL-1 )
379: *
380: * Initialize small orthogonal factors that will hold the
381: * accumulated Givens rotations in workspace.
382: * N2NB denotes the number of 2*NNB-by-2*NNB factors
383: * NBLST denotes the (possibly smaller) order of the last
384: * factor.
385: *
386: N2NB = ( IHI-JCOL-1 ) / NNB - 1
387: NBLST = IHI - JCOL - N2NB*NNB
388: CALL DLASET( 'All', NBLST, NBLST, ZERO, ONE, WORK, NBLST )
389: PW = NBLST * NBLST + 1
390: DO I = 1, N2NB
391: CALL DLASET( 'All', 2*NNB, 2*NNB, ZERO, ONE,
392: $ WORK( PW ), 2*NNB )
393: PW = PW + 4*NNB*NNB
394: END DO
395: *
396: * Reduce columns JCOL:JCOL+NNB-1 of A to Hessenberg form.
397: *
398: DO J = JCOL, JCOL+NNB-1
399: *
400: * Reduce Jth column of A. Store cosines and sines in Jth
401: * column of A and B, respectively.
402: *
403: DO I = IHI, J+2, -1
404: TEMP = A( I-1, J )
405: CALL DLARTG( TEMP, A( I, J ), C, S, A( I-1, J ) )
406: A( I, J ) = C
407: B( I, J ) = S
408: END DO
409: *
410: * Accumulate Givens rotations into workspace array.
411: *
412: PPW = ( NBLST + 1 )*( NBLST - 2 ) - J + JCOL + 1
413: LEN = 2 + J - JCOL
414: JROW = J + N2NB*NNB + 2
415: DO I = IHI, JROW, -1
416: C = A( I, J )
417: S = B( I, J )
418: DO JJ = PPW, PPW+LEN-1
419: TEMP = WORK( JJ + NBLST )
420: WORK( JJ + NBLST ) = C*TEMP - S*WORK( JJ )
421: WORK( JJ ) = S*TEMP + C*WORK( JJ )
422: END DO
423: LEN = LEN + 1
424: PPW = PPW - NBLST - 1
425: END DO
426: *
427: PPWO = NBLST*NBLST + ( NNB+J-JCOL-1 )*2*NNB + NNB
428: J0 = JROW - NNB
429: DO JROW = J0, J+2, -NNB
430: PPW = PPWO
431: LEN = 2 + J - JCOL
432: DO I = JROW+NNB-1, JROW, -1
433: C = A( I, J )
434: S = B( I, J )
435: DO JJ = PPW, PPW+LEN-1
436: TEMP = WORK( JJ + 2*NNB )
437: WORK( JJ + 2*NNB ) = C*TEMP - S*WORK( JJ )
438: WORK( JJ ) = S*TEMP + C*WORK( JJ )
439: END DO
440: LEN = LEN + 1
441: PPW = PPW - 2*NNB - 1
442: END DO
443: PPWO = PPWO + 4*NNB*NNB
444: END DO
445: *
446: * TOP denotes the number of top rows in A and B that will
447: * not be updated during the next steps.
448: *
449: IF( JCOL.LE.2 ) THEN
450: TOP = 0
451: ELSE
452: TOP = JCOL
453: END IF
454: *
455: * Propagate transformations through B and replace stored
456: * left sines/cosines by right sines/cosines.
457: *
458: DO JJ = N, J+1, -1
459: *
460: * Update JJth column of B.
461: *
462: DO I = MIN( JJ+1, IHI ), J+2, -1
463: C = A( I, J )
464: S = B( I, J )
465: TEMP = B( I, JJ )
466: B( I, JJ ) = C*TEMP - S*B( I-1, JJ )
467: B( I-1, JJ ) = S*TEMP + C*B( I-1, JJ )
468: END DO
469: *
470: * Annihilate B( JJ+1, JJ ).
471: *
472: IF( JJ.LT.IHI ) THEN
473: TEMP = B( JJ+1, JJ+1 )
474: CALL DLARTG( TEMP, B( JJ+1, JJ ), C, S,
475: $ B( JJ+1, JJ+1 ) )
476: B( JJ+1, JJ ) = ZERO
477: CALL DROT( JJ-TOP, B( TOP+1, JJ+1 ), 1,
478: $ B( TOP+1, JJ ), 1, C, S )
479: A( JJ+1, J ) = C
480: B( JJ+1, J ) = -S
481: END IF
482: END DO
483: *
484: * Update A by transformations from right.
485: * Explicit loop unrolling provides better performance
486: * compared to DLASR.
487: * CALL DLASR( 'Right', 'Variable', 'Backward', IHI-TOP,
488: * $ IHI-J, A( J+2, J ), B( J+2, J ),
489: * $ A( TOP+1, J+1 ), LDA )
490: *
491: JJ = MOD( IHI-J-1, 3 )
492: DO I = IHI-J-3, JJ+1, -3
493: C = A( J+1+I, J )
494: S = -B( J+1+I, J )
495: C1 = A( J+2+I, J )
496: S1 = -B( J+2+I, J )
497: C2 = A( J+3+I, J )
498: S2 = -B( J+3+I, J )
499: *
500: DO K = TOP+1, IHI
501: TEMP = A( K, J+I )
502: TEMP1 = A( K, J+I+1 )
503: TEMP2 = A( K, J+I+2 )
504: TEMP3 = A( K, J+I+3 )
505: A( K, J+I+3 ) = C2*TEMP3 + S2*TEMP2
506: TEMP2 = -S2*TEMP3 + C2*TEMP2
507: A( K, J+I+2 ) = C1*TEMP2 + S1*TEMP1
508: TEMP1 = -S1*TEMP2 + C1*TEMP1
509: A( K, J+I+1 ) = C*TEMP1 + S*TEMP
510: A( K, J+I ) = -S*TEMP1 + C*TEMP
511: END DO
512: END DO
513: *
514: IF( JJ.GT.0 ) THEN
515: DO I = JJ, 1, -1
516: CALL DROT( IHI-TOP, A( TOP+1, J+I+1 ), 1,
517: $ A( TOP+1, J+I ), 1, A( J+1+I, J ),
518: $ -B( J+1+I, J ) )
519: END DO
520: END IF
521: *
522: * Update (J+1)th column of A by transformations from left.
523: *
524: IF ( J .LT. JCOL + NNB - 1 ) THEN
525: LEN = 1 + J - JCOL
526: *
527: * Multiply with the trailing accumulated orthogonal
528: * matrix, which takes the form
529: *
530: * [ U11 U12 ]
531: * U = [ ],
532: * [ U21 U22 ]
533: *
534: * where U21 is a LEN-by-LEN matrix and U12 is lower
535: * triangular.
536: *
537: JROW = IHI - NBLST + 1
538: CALL DGEMV( 'Transpose', NBLST, LEN, ONE, WORK,
539: $ NBLST, A( JROW, J+1 ), 1, ZERO,
540: $ WORK( PW ), 1 )
541: PPW = PW + LEN
542: DO I = JROW, JROW+NBLST-LEN-1
543: WORK( PPW ) = A( I, J+1 )
544: PPW = PPW + 1
545: END DO
546: CALL DTRMV( 'Lower', 'Transpose', 'Non-unit',
547: $ NBLST-LEN, WORK( LEN*NBLST + 1 ), NBLST,
548: $ WORK( PW+LEN ), 1 )
549: CALL DGEMV( 'Transpose', LEN, NBLST-LEN, ONE,
550: $ WORK( (LEN+1)*NBLST - LEN + 1 ), NBLST,
551: $ A( JROW+NBLST-LEN, J+1 ), 1, ONE,
552: $ WORK( PW+LEN ), 1 )
553: PPW = PW
554: DO I = JROW, JROW+NBLST-1
555: A( I, J+1 ) = WORK( PPW )
556: PPW = PPW + 1
557: END DO
558: *
559: * Multiply with the other accumulated orthogonal
560: * matrices, which take the form
561: *
562: * [ U11 U12 0 ]
563: * [ ]
564: * U = [ U21 U22 0 ],
565: * [ ]
566: * [ 0 0 I ]
567: *
568: * where I denotes the (NNB-LEN)-by-(NNB-LEN) identity
569: * matrix, U21 is a LEN-by-LEN upper triangular matrix
570: * and U12 is an NNB-by-NNB lower triangular matrix.
571: *
572: PPWO = 1 + NBLST*NBLST
573: J0 = JROW - NNB
574: DO JROW = J0, JCOL+1, -NNB
575: PPW = PW + LEN
576: DO I = JROW, JROW+NNB-1
577: WORK( PPW ) = A( I, J+1 )
578: PPW = PPW + 1
579: END DO
580: PPW = PW
581: DO I = JROW+NNB, JROW+NNB+LEN-1
582: WORK( PPW ) = A( I, J+1 )
583: PPW = PPW + 1
584: END DO
585: CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', LEN,
586: $ WORK( PPWO + NNB ), 2*NNB, WORK( PW ),
587: $ 1 )
588: CALL DTRMV( 'Lower', 'Transpose', 'Non-unit', NNB,
589: $ WORK( PPWO + 2*LEN*NNB ),
590: $ 2*NNB, WORK( PW + LEN ), 1 )
591: CALL DGEMV( 'Transpose', NNB, LEN, ONE,
592: $ WORK( PPWO ), 2*NNB, A( JROW, J+1 ), 1,
593: $ ONE, WORK( PW ), 1 )
594: CALL DGEMV( 'Transpose', LEN, NNB, ONE,
595: $ WORK( PPWO + 2*LEN*NNB + NNB ), 2*NNB,
596: $ A( JROW+NNB, J+1 ), 1, ONE,
597: $ WORK( PW+LEN ), 1 )
598: PPW = PW
599: DO I = JROW, JROW+LEN+NNB-1
600: A( I, J+1 ) = WORK( PPW )
601: PPW = PPW + 1
602: END DO
603: PPWO = PPWO + 4*NNB*NNB
604: END DO
605: END IF
606: END DO
607: *
608: * Apply accumulated orthogonal matrices to A.
609: *
610: COLA = N - JCOL - NNB + 1
611: J = IHI - NBLST + 1
612: CALL DGEMM( 'Transpose', 'No Transpose', NBLST,
613: $ COLA, NBLST, ONE, WORK, NBLST,
614: $ A( J, JCOL+NNB ), LDA, ZERO, WORK( PW ),
615: $ NBLST )
616: CALL DLACPY( 'All', NBLST, COLA, WORK( PW ), NBLST,
617: $ A( J, JCOL+NNB ), LDA )
618: PPWO = NBLST*NBLST + 1
619: J0 = J - NNB
620: DO J = J0, JCOL+1, -NNB
621: IF ( BLK22 ) THEN
622: *
623: * Exploit the structure of
624: *
625: * [ U11 U12 ]
626: * U = [ ]
627: * [ U21 U22 ],
628: *
629: * where all blocks are NNB-by-NNB, U21 is upper
630: * triangular and U12 is lower triangular.
631: *
632: CALL DORM22( 'Left', 'Transpose', 2*NNB, COLA, NNB,
633: $ NNB, WORK( PPWO ), 2*NNB,
634: $ A( J, JCOL+NNB ), LDA, WORK( PW ),
635: $ LWORK-PW+1, IERR )
636: ELSE
637: *
638: * Ignore the structure of U.
639: *
640: CALL DGEMM( 'Transpose', 'No Transpose', 2*NNB,
641: $ COLA, 2*NNB, ONE, WORK( PPWO ), 2*NNB,
642: $ A( J, JCOL+NNB ), LDA, ZERO, WORK( PW ),
643: $ 2*NNB )
644: CALL DLACPY( 'All', 2*NNB, COLA, WORK( PW ), 2*NNB,
645: $ A( J, JCOL+NNB ), LDA )
646: END IF
647: PPWO = PPWO + 4*NNB*NNB
648: END DO
649: *
650: * Apply accumulated orthogonal matrices to Q.
651: *
652: IF( WANTQ ) THEN
653: J = IHI - NBLST + 1
654: IF ( INITQ ) THEN
655: TOPQ = MAX( 2, J - JCOL + 1 )
656: NH = IHI - TOPQ + 1
657: ELSE
658: TOPQ = 1
659: NH = N
660: END IF
661: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
662: $ NBLST, NBLST, ONE, Q( TOPQ, J ), LDQ,
663: $ WORK, NBLST, ZERO, WORK( PW ), NH )
664: CALL DLACPY( 'All', NH, NBLST, WORK( PW ), NH,
665: $ Q( TOPQ, J ), LDQ )
666: PPWO = NBLST*NBLST + 1
667: J0 = J - NNB
668: DO J = J0, JCOL+1, -NNB
669: IF ( INITQ ) THEN
670: TOPQ = MAX( 2, J - JCOL + 1 )
671: NH = IHI - TOPQ + 1
672: END IF
673: IF ( BLK22 ) THEN
674: *
675: * Exploit the structure of U.
676: *
677: CALL DORM22( 'Right', 'No Transpose', NH, 2*NNB,
678: $ NNB, NNB, WORK( PPWO ), 2*NNB,
679: $ Q( TOPQ, J ), LDQ, WORK( PW ),
680: $ LWORK-PW+1, IERR )
681: ELSE
682: *
683: * Ignore the structure of U.
684: *
685: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
686: $ 2*NNB, 2*NNB, ONE, Q( TOPQ, J ), LDQ,
687: $ WORK( PPWO ), 2*NNB, ZERO, WORK( PW ),
688: $ NH )
689: CALL DLACPY( 'All', NH, 2*NNB, WORK( PW ), NH,
690: $ Q( TOPQ, J ), LDQ )
691: END IF
692: PPWO = PPWO + 4*NNB*NNB
693: END DO
694: END IF
695: *
696: * Accumulate right Givens rotations if required.
697: *
698: IF ( WANTZ .OR. TOP.GT.0 ) THEN
699: *
700: * Initialize small orthogonal factors that will hold the
701: * accumulated Givens rotations in workspace.
702: *
703: CALL DLASET( 'All', NBLST, NBLST, ZERO, ONE, WORK,
704: $ NBLST )
705: PW = NBLST * NBLST + 1
706: DO I = 1, N2NB
707: CALL DLASET( 'All', 2*NNB, 2*NNB, ZERO, ONE,
708: $ WORK( PW ), 2*NNB )
709: PW = PW + 4*NNB*NNB
710: END DO
711: *
712: * Accumulate Givens rotations into workspace array.
713: *
714: DO J = JCOL, JCOL+NNB-1
715: PPW = ( NBLST + 1 )*( NBLST - 2 ) - J + JCOL + 1
716: LEN = 2 + J - JCOL
717: JROW = J + N2NB*NNB + 2
718: DO I = IHI, JROW, -1
719: C = A( I, J )
720: A( I, J ) = ZERO
721: S = B( I, J )
722: B( I, J ) = ZERO
723: DO JJ = PPW, PPW+LEN-1
724: TEMP = WORK( JJ + NBLST )
725: WORK( JJ + NBLST ) = C*TEMP - S*WORK( JJ )
726: WORK( JJ ) = S*TEMP + C*WORK( JJ )
727: END DO
728: LEN = LEN + 1
729: PPW = PPW - NBLST - 1
730: END DO
731: *
732: PPWO = NBLST*NBLST + ( NNB+J-JCOL-1 )*2*NNB + NNB
733: J0 = JROW - NNB
734: DO JROW = J0, J+2, -NNB
735: PPW = PPWO
736: LEN = 2 + J - JCOL
737: DO I = JROW+NNB-1, JROW, -1
738: C = A( I, J )
739: A( I, J ) = ZERO
740: S = B( I, J )
741: B( I, J ) = ZERO
742: DO JJ = PPW, PPW+LEN-1
743: TEMP = WORK( JJ + 2*NNB )
744: WORK( JJ + 2*NNB ) = C*TEMP - S*WORK( JJ )
745: WORK( JJ ) = S*TEMP + C*WORK( JJ )
746: END DO
747: LEN = LEN + 1
748: PPW = PPW - 2*NNB - 1
749: END DO
750: PPWO = PPWO + 4*NNB*NNB
751: END DO
752: END DO
753: ELSE
754: *
755: CALL DLASET( 'Lower', IHI - JCOL - 1, NNB, ZERO, ZERO,
756: $ A( JCOL + 2, JCOL ), LDA )
757: CALL DLASET( 'Lower', IHI - JCOL - 1, NNB, ZERO, ZERO,
758: $ B( JCOL + 2, JCOL ), LDB )
759: END IF
760: *
761: * Apply accumulated orthogonal matrices to A and B.
762: *
763: IF ( TOP.GT.0 ) THEN
764: J = IHI - NBLST + 1
765: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
766: $ NBLST, NBLST, ONE, A( 1, J ), LDA,
767: $ WORK, NBLST, ZERO, WORK( PW ), TOP )
768: CALL DLACPY( 'All', TOP, NBLST, WORK( PW ), TOP,
769: $ A( 1, J ), LDA )
770: PPWO = NBLST*NBLST + 1
771: J0 = J - NNB
772: DO J = J0, JCOL+1, -NNB
773: IF ( BLK22 ) THEN
774: *
775: * Exploit the structure of U.
776: *
777: CALL DORM22( 'Right', 'No Transpose', TOP, 2*NNB,
778: $ NNB, NNB, WORK( PPWO ), 2*NNB,
779: $ A( 1, J ), LDA, WORK( PW ),
780: $ LWORK-PW+1, IERR )
781: ELSE
782: *
783: * Ignore the structure of U.
784: *
785: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
786: $ 2*NNB, 2*NNB, ONE, A( 1, J ), LDA,
787: $ WORK( PPWO ), 2*NNB, ZERO,
788: $ WORK( PW ), TOP )
789: CALL DLACPY( 'All', TOP, 2*NNB, WORK( PW ), TOP,
790: $ A( 1, J ), LDA )
791: END IF
792: PPWO = PPWO + 4*NNB*NNB
793: END DO
794: *
795: J = IHI - NBLST + 1
796: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
797: $ NBLST, NBLST, ONE, B( 1, J ), LDB,
798: $ WORK, NBLST, ZERO, WORK( PW ), TOP )
799: CALL DLACPY( 'All', TOP, NBLST, WORK( PW ), TOP,
800: $ B( 1, J ), LDB )
801: PPWO = NBLST*NBLST + 1
802: J0 = J - NNB
803: DO J = J0, JCOL+1, -NNB
804: IF ( BLK22 ) THEN
805: *
806: * Exploit the structure of U.
807: *
808: CALL DORM22( 'Right', 'No Transpose', TOP, 2*NNB,
809: $ NNB, NNB, WORK( PPWO ), 2*NNB,
810: $ B( 1, J ), LDB, WORK( PW ),
811: $ LWORK-PW+1, IERR )
812: ELSE
813: *
814: * Ignore the structure of U.
815: *
816: CALL DGEMM( 'No Transpose', 'No Transpose', TOP,
817: $ 2*NNB, 2*NNB, ONE, B( 1, J ), LDB,
818: $ WORK( PPWO ), 2*NNB, ZERO,
819: $ WORK( PW ), TOP )
820: CALL DLACPY( 'All', TOP, 2*NNB, WORK( PW ), TOP,
821: $ B( 1, J ), LDB )
822: END IF
823: PPWO = PPWO + 4*NNB*NNB
824: END DO
825: END IF
826: *
827: * Apply accumulated orthogonal matrices to Z.
828: *
829: IF( WANTZ ) THEN
830: J = IHI - NBLST + 1
831: IF ( INITQ ) THEN
832: TOPQ = MAX( 2, J - JCOL + 1 )
833: NH = IHI - TOPQ + 1
834: ELSE
835: TOPQ = 1
836: NH = N
837: END IF
838: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
839: $ NBLST, NBLST, ONE, Z( TOPQ, J ), LDZ,
840: $ WORK, NBLST, ZERO, WORK( PW ), NH )
841: CALL DLACPY( 'All', NH, NBLST, WORK( PW ), NH,
842: $ Z( TOPQ, J ), LDZ )
843: PPWO = NBLST*NBLST + 1
844: J0 = J - NNB
845: DO J = J0, JCOL+1, -NNB
846: IF ( INITQ ) THEN
847: TOPQ = MAX( 2, J - JCOL + 1 )
848: NH = IHI - TOPQ + 1
849: END IF
850: IF ( BLK22 ) THEN
851: *
852: * Exploit the structure of U.
853: *
854: CALL DORM22( 'Right', 'No Transpose', NH, 2*NNB,
855: $ NNB, NNB, WORK( PPWO ), 2*NNB,
856: $ Z( TOPQ, J ), LDZ, WORK( PW ),
857: $ LWORK-PW+1, IERR )
858: ELSE
859: *
860: * Ignore the structure of U.
861: *
862: CALL DGEMM( 'No Transpose', 'No Transpose', NH,
863: $ 2*NNB, 2*NNB, ONE, Z( TOPQ, J ), LDZ,
864: $ WORK( PPWO ), 2*NNB, ZERO, WORK( PW ),
865: $ NH )
866: CALL DLACPY( 'All', NH, 2*NNB, WORK( PW ), NH,
867: $ Z( TOPQ, J ), LDZ )
868: END IF
869: PPWO = PPWO + 4*NNB*NNB
870: END DO
871: END IF
872: END DO
873: END IF
874: *
875: * Use unblocked code to reduce the rest of the matrix
876: * Avoid re-initialization of modified Q and Z.
877: *
878: COMPQ2 = COMPQ
879: COMPZ2 = COMPZ
880: IF ( JCOL.NE.ILO ) THEN
881: IF ( WANTQ )
882: $ COMPQ2 = 'V'
883: IF ( WANTZ )
884: $ COMPZ2 = 'V'
885: END IF
886: *
887: IF ( JCOL.LT.IHI )
888: $ CALL DGGHRD( COMPQ2, COMPZ2, N, JCOL, IHI, A, LDA, B, LDB, Q,
889: $ LDQ, Z, LDZ, IERR )
890: WORK( 1 ) = DBLE( LWKOPT )
891: *
892: RETURN
893: *
894: * End of DGGHD3
895: *
896: END