1: *> \brief \b ZGGHD3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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15: *> [TXT]</a>
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGGHD3( 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: * COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
30: * $ Z( LDZ, * ), WORK( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZGGHD3 reduces a pair of complex matrices (A,B) to generalized upper
40: *> Hessenberg form using unitary 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 unitary matrix Q to the left side
46: *> of the equation.
47: *>
48: *> This subroutine simultaneously reduces A to a Hessenberg matrix H:
49: *> Q**H*A*Z = H
50: *> and transforms B to another upper triangular matrix T:
51: *> Q**H*B*Z = T
52: *> in order to reduce the problem to its standard form
53: *> H*y = lambda*T*y
54: *> where y = Z**H*x.
55: *>
56: *> The unitary 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: *> Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
60: *> Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
61: *> If Q1 is the unitary matrix from the QR factorization of B in the
62: *> original equation A*x = lambda*B*x, then ZGGHD3 reduces the original
63: *> problem to generalized Hessenberg form.
64: *>
65: *> This is a blocked variant of CGGHRD, using matrix-matrix
66: *> multiplications for parts of the computation to enhance performance.
67: *> \endverbatim
68: *
69: * Arguments:
70: * ==========
71: *
72: *> \param[in] COMPQ
73: *> \verbatim
74: *> COMPQ is CHARACTER*1
75: *> = 'N': do not compute Q;
76: *> = 'I': Q is initialized to the unit matrix, and the
77: *> unitary matrix Q is returned;
78: *> = 'V': Q must contain a unitary matrix Q1 on entry,
79: *> and the product Q1*Q is returned.
80: *> \endverbatim
81: *>
82: *> \param[in] COMPZ
83: *> \verbatim
84: *> COMPZ is CHARACTER*1
85: *> = 'N': do not compute Z;
86: *> = 'I': Z is initialized to the unit matrix, and the
87: *> unitary matrix Z is returned;
88: *> = 'V': Z must contain a unitary matrix Z1 on entry,
89: *> and the product Z1*Z is returned.
90: *> \endverbatim
91: *>
92: *> \param[in] N
93: *> \verbatim
94: *> N is INTEGER
95: *> The order of the matrices A and B. N >= 0.
96: *> \endverbatim
97: *>
98: *> \param[in] ILO
99: *> \verbatim
100: *> ILO is INTEGER
101: *> \endverbatim
102: *>
103: *> \param[in] IHI
104: *> \verbatim
105: *> IHI is INTEGER
106: *>
107: *> ILO and IHI mark the rows and columns of A which are to be
108: *> reduced. It is assumed that A is already upper triangular
109: *> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
110: *> normally set by a previous call to ZGGBAL; otherwise they
111: *> should be set to 1 and N respectively.
112: *> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
113: *> \endverbatim
114: *>
115: *> \param[in,out] A
116: *> \verbatim
117: *> A is COMPLEX*16 array, dimension (LDA, N)
118: *> On entry, the N-by-N general matrix to be reduced.
119: *> On exit, the upper triangle and the first subdiagonal of A
120: *> are overwritten with the upper Hessenberg matrix H, and the
121: *> rest is set to zero.
122: *> \endverbatim
123: *>
124: *> \param[in] LDA
125: *> \verbatim
126: *> LDA is INTEGER
127: *> The leading dimension of the array A. LDA >= max(1,N).
128: *> \endverbatim
129: *>
130: *> \param[in,out] B
131: *> \verbatim
132: *> B is COMPLEX*16 array, dimension (LDB, N)
133: *> On entry, the N-by-N upper triangular matrix B.
134: *> On exit, the upper triangular matrix T = Q**H B Z. The
135: *> elements below the diagonal are set to zero.
136: *> \endverbatim
137: *>
138: *> \param[in] LDB
139: *> \verbatim
140: *> LDB is INTEGER
141: *> The leading dimension of the array B. LDB >= max(1,N).
142: *> \endverbatim
143: *>
144: *> \param[in,out] Q
145: *> \verbatim
146: *> Q is COMPLEX*16 array, dimension (LDQ, N)
147: *> On entry, if COMPQ = 'V', the unitary matrix Q1, typically
148: *> from the QR factorization of B.
149: *> On exit, if COMPQ='I', the unitary matrix Q, and if
150: *> COMPQ = 'V', the product Q1*Q.
151: *> Not referenced if COMPQ='N'.
152: *> \endverbatim
153: *>
154: *> \param[in] LDQ
155: *> \verbatim
156: *> LDQ is INTEGER
157: *> The leading dimension of the array Q.
158: *> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
159: *> \endverbatim
160: *>
161: *> \param[in,out] Z
162: *> \verbatim
163: *> Z is COMPLEX*16 array, dimension (LDZ, N)
164: *> On entry, if COMPZ = 'V', the unitary matrix Z1.
165: *> On exit, if COMPZ='I', the unitary matrix Z, and if
166: *> COMPZ = 'V', the product Z1*Z.
167: *> Not referenced if COMPZ='N'.
168: *> \endverbatim
169: *>
170: *> \param[in] LDZ
171: *> \verbatim
172: *> LDZ is INTEGER
173: *> The leading dimension of the array Z.
174: *> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
175: *> \endverbatim
176: *>
177: *> \param[out] WORK
178: *> \verbatim
179: *> WORK is COMPLEX*16 array, dimension (LWORK)
180: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
181: *> \endverbatim
182: *>
183: *> \param[in] LWORK
184: *> \verbatim
185: *> LWORK is INTEGER
186: *> The length of the array WORK. LWORK >= 1.
187: *> For optimum performance LWORK >= 6*N*NB, where NB is the
188: *> optimal blocksize.
189: *>
190: *> If LWORK = -1, then a workspace query is assumed; the routine
191: *> only calculates the optimal size of the WORK array, returns
192: *> this value as the first entry of the WORK array, and no error
193: *> message related to LWORK is issued by XERBLA.
194: *> \endverbatim
195: *>
196: *> \param[out] INFO
197: *> \verbatim
198: *> INFO is INTEGER
199: *> = 0: successful exit.
200: *> < 0: if INFO = -i, the i-th argument had an illegal value.
201: *> \endverbatim
202: *
203: * Authors:
204: * ========
205: *
206: *> \author Univ. of Tennessee
207: *> \author Univ. of California Berkeley
208: *> \author Univ. of Colorado Denver
209: *> \author NAG Ltd.
210: *
211: *> \ingroup complex16OTHERcomputational
212: *
213: *> \par Further Details:
214: * =====================
215: *>
216: *> \verbatim
217: *>
218: *> This routine reduces A to Hessenberg form and maintains B in triangular form
219: *> using a blocked variant of Moler and Stewart's original algorithm,
220: *> as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
221: *> (BIT 2008).
222: *> \endverbatim
223: *>
224: * =====================================================================
225: SUBROUTINE ZGGHD3( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
226: $ LDQ, Z, LDZ, WORK, LWORK, INFO )
227: *
228: * -- LAPACK computational routine --
229: * -- LAPACK is a software package provided by Univ. of Tennessee, --
230: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
231: *
232: IMPLICIT NONE
233: *
234: * .. Scalar Arguments ..
235: CHARACTER COMPQ, COMPZ
236: INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N, LWORK
237: * ..
238: * .. Array Arguments ..
239: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
240: $ Z( LDZ, * ), WORK( * )
241: * ..
242: *
243: * =====================================================================
244: *
245: * .. Parameters ..
246: COMPLEX*16 CONE, CZERO
247: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
248: $ CZERO = ( 0.0D+0, 0.0D+0 ) )
249: * ..
250: * .. Local Scalars ..
251: LOGICAL BLK22, INITQ, INITZ, LQUERY, WANTQ, WANTZ
252: CHARACTER*1 COMPQ2, COMPZ2
253: INTEGER COLA, I, IERR, J, J0, JCOL, JJ, JROW, K,
254: $ KACC22, LEN, LWKOPT, N2NB, NB, NBLST, NBMIN,
255: $ NH, NNB, NX, PPW, PPWO, PW, TOP, TOPQ
256: DOUBLE PRECISION C
257: COMPLEX*16 C1, C2, CTEMP, S, S1, S2, TEMP, TEMP1, TEMP2,
258: $ TEMP3
259: * ..
260: * .. External Functions ..
261: LOGICAL LSAME
262: INTEGER ILAENV
263: EXTERNAL ILAENV, LSAME
264: * ..
265: * .. External Subroutines ..
266: EXTERNAL ZGGHRD, ZLARTG, ZLASET, ZUNM22, ZROT, ZGEMM,
267: $ ZGEMV, ZTRMV, ZLACPY, XERBLA
268: * ..
269: * .. Intrinsic Functions ..
270: INTRINSIC DBLE, DCMPLX, DCONJG, MAX
271: * ..
272: * .. Executable Statements ..
273: *
274: * Decode and test the input parameters.
275: *
276: INFO = 0
277: NB = ILAENV( 1, 'ZGGHD3', ' ', N, ILO, IHI, -1 )
278: LWKOPT = MAX( 6*N*NB, 1 )
279: WORK( 1 ) = DCMPLX( 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( 'ZGGHD3', -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 ZLASET( 'All', N, N, CZERO, CONE, Q, LDQ )
318: IF( INITZ )
319: $ CALL ZLASET( 'All', N, N, CZERO, CONE, Z, LDZ )
320: *
321: * Zero out lower triangle of B.
322: *
323: IF( N.GT.1 )
324: $ CALL ZLASET( 'Lower', N-1, N-1, CZERO, CZERO, 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 ) = CONE
331: RETURN
332: END IF
333: *
334: * Determine the blocksize.
335: *
336: NBMIN = ILAENV( 2, 'ZGGHD3', ' ', 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, 'ZGGHD3', ' ', 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, 'ZGGHD3', ' ', 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, 'ZGGHD3', ' ', 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 unitary 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 ZLASET( 'All', NBLST, NBLST, CZERO, CONE, WORK, NBLST )
387: PW = NBLST * NBLST + 1
388: DO I = 1, N2NB
389: CALL ZLASET( 'All', 2*NNB, 2*NNB, CZERO, CONE,
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 ZLARTG( TEMP, A( I, J ), C, S, A( I-1, J ) )
404: A( I, J ) = DCMPLX( 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: CTEMP = A( I, J )
415: S = B( I, J )
416: DO JJ = PPW, PPW+LEN-1
417: TEMP = WORK( JJ + NBLST )
418: WORK( JJ + NBLST ) = CTEMP*TEMP - S*WORK( JJ )
419: WORK( JJ ) = DCONJG( S )*TEMP + CTEMP*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: CTEMP = 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 ) = CTEMP*TEMP - S*WORK( JJ )
436: WORK( JJ ) = DCONJG( S )*TEMP + CTEMP*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: CTEMP = A( I, J )
462: S = B( I, J )
463: TEMP = B( I, JJ )
464: B( I, JJ ) = CTEMP*TEMP - DCONJG( S )*B( I-1, JJ )
465: B( I-1, JJ ) = S*TEMP + CTEMP*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 ZLARTG( TEMP, B( JJ+1, JJ ), C, S,
473: $ B( JJ+1, JJ+1 ) )
474: B( JJ+1, JJ ) = CZERO
475: CALL ZROT( JJ-TOP, B( TOP+1, JJ+1 ), 1,
476: $ B( TOP+1, JJ ), 1, C, S )
477: A( JJ+1, J ) = DCMPLX( C )
478: B( JJ+1, J ) = -DCONJG( S )
479: END IF
480: END DO
481: *
482: * Update A by transformations from right.
483: *
484: JJ = MOD( IHI-J-1, 3 )
485: DO I = IHI-J-3, JJ+1, -3
486: CTEMP = A( J+1+I, J )
487: S = -B( J+1+I, J )
488: C1 = A( J+2+I, J )
489: S1 = -B( J+2+I, J )
490: C2 = A( J+3+I, J )
491: S2 = -B( J+3+I, J )
492: *
493: DO K = TOP+1, IHI
494: TEMP = A( K, J+I )
495: TEMP1 = A( K, J+I+1 )
496: TEMP2 = A( K, J+I+2 )
497: TEMP3 = A( K, J+I+3 )
498: A( K, J+I+3 ) = C2*TEMP3 + DCONJG( S2 )*TEMP2
499: TEMP2 = -S2*TEMP3 + C2*TEMP2
500: A( K, J+I+2 ) = C1*TEMP2 + DCONJG( S1 )*TEMP1
501: TEMP1 = -S1*TEMP2 + C1*TEMP1
502: A( K, J+I+1 ) = CTEMP*TEMP1 + DCONJG( S )*TEMP
503: A( K, J+I ) = -S*TEMP1 + CTEMP*TEMP
504: END DO
505: END DO
506: *
507: IF( JJ.GT.0 ) THEN
508: DO I = JJ, 1, -1
509: C = DBLE( A( J+1+I, J ) )
510: CALL ZROT( IHI-TOP, A( TOP+1, J+I+1 ), 1,
511: $ A( TOP+1, J+I ), 1, C,
512: $ -DCONJG( B( J+1+I, J ) ) )
513: END DO
514: END IF
515: *
516: * Update (J+1)th column of A by transformations from left.
517: *
518: IF ( J .LT. JCOL + NNB - 1 ) THEN
519: LEN = 1 + J - JCOL
520: *
521: * Multiply with the trailing accumulated unitary
522: * matrix, which takes the form
523: *
524: * [ U11 U12 ]
525: * U = [ ],
526: * [ U21 U22 ]
527: *
528: * where U21 is a LEN-by-LEN matrix and U12 is lower
529: * triangular.
530: *
531: JROW = IHI - NBLST + 1
532: CALL ZGEMV( 'Conjugate', NBLST, LEN, CONE, WORK,
533: $ NBLST, A( JROW, J+1 ), 1, CZERO,
534: $ WORK( PW ), 1 )
535: PPW = PW + LEN
536: DO I = JROW, JROW+NBLST-LEN-1
537: WORK( PPW ) = A( I, J+1 )
538: PPW = PPW + 1
539: END DO
540: CALL ZTRMV( 'Lower', 'Conjugate', 'Non-unit',
541: $ NBLST-LEN, WORK( LEN*NBLST + 1 ), NBLST,
542: $ WORK( PW+LEN ), 1 )
543: CALL ZGEMV( 'Conjugate', LEN, NBLST-LEN, CONE,
544: $ WORK( (LEN+1)*NBLST - LEN + 1 ), NBLST,
545: $ A( JROW+NBLST-LEN, J+1 ), 1, CONE,
546: $ WORK( PW+LEN ), 1 )
547: PPW = PW
548: DO I = JROW, JROW+NBLST-1
549: A( I, J+1 ) = WORK( PPW )
550: PPW = PPW + 1
551: END DO
552: *
553: * Multiply with the other accumulated unitary
554: * matrices, which take the form
555: *
556: * [ U11 U12 0 ]
557: * [ ]
558: * U = [ U21 U22 0 ],
559: * [ ]
560: * [ 0 0 I ]
561: *
562: * where I denotes the (NNB-LEN)-by-(NNB-LEN) identity
563: * matrix, U21 is a LEN-by-LEN upper triangular matrix
564: * and U12 is an NNB-by-NNB lower triangular matrix.
565: *
566: PPWO = 1 + NBLST*NBLST
567: J0 = JROW - NNB
568: DO JROW = J0, JCOL+1, -NNB
569: PPW = PW + LEN
570: DO I = JROW, JROW+NNB-1
571: WORK( PPW ) = A( I, J+1 )
572: PPW = PPW + 1
573: END DO
574: PPW = PW
575: DO I = JROW+NNB, JROW+NNB+LEN-1
576: WORK( PPW ) = A( I, J+1 )
577: PPW = PPW + 1
578: END DO
579: CALL ZTRMV( 'Upper', 'Conjugate', 'Non-unit', LEN,
580: $ WORK( PPWO + NNB ), 2*NNB, WORK( PW ),
581: $ 1 )
582: CALL ZTRMV( 'Lower', 'Conjugate', 'Non-unit', NNB,
583: $ WORK( PPWO + 2*LEN*NNB ),
584: $ 2*NNB, WORK( PW + LEN ), 1 )
585: CALL ZGEMV( 'Conjugate', NNB, LEN, CONE,
586: $ WORK( PPWO ), 2*NNB, A( JROW, J+1 ), 1,
587: $ CONE, WORK( PW ), 1 )
588: CALL ZGEMV( 'Conjugate', LEN, NNB, CONE,
589: $ WORK( PPWO + 2*LEN*NNB + NNB ), 2*NNB,
590: $ A( JROW+NNB, J+1 ), 1, CONE,
591: $ WORK( PW+LEN ), 1 )
592: PPW = PW
593: DO I = JROW, JROW+LEN+NNB-1
594: A( I, J+1 ) = WORK( PPW )
595: PPW = PPW + 1
596: END DO
597: PPWO = PPWO + 4*NNB*NNB
598: END DO
599: END IF
600: END DO
601: *
602: * Apply accumulated unitary matrices to A.
603: *
604: COLA = N - JCOL - NNB + 1
605: J = IHI - NBLST + 1
606: CALL ZGEMM( 'Conjugate', 'No Transpose', NBLST,
607: $ COLA, NBLST, CONE, WORK, NBLST,
608: $ A( J, JCOL+NNB ), LDA, CZERO, WORK( PW ),
609: $ NBLST )
610: CALL ZLACPY( 'All', NBLST, COLA, WORK( PW ), NBLST,
611: $ A( J, JCOL+NNB ), LDA )
612: PPWO = NBLST*NBLST + 1
613: J0 = J - NNB
614: DO J = J0, JCOL+1, -NNB
615: IF ( BLK22 ) THEN
616: *
617: * Exploit the structure of
618: *
619: * [ U11 U12 ]
620: * U = [ ]
621: * [ U21 U22 ],
622: *
623: * where all blocks are NNB-by-NNB, U21 is upper
624: * triangular and U12 is lower triangular.
625: *
626: CALL ZUNM22( 'Left', 'Conjugate', 2*NNB, COLA, NNB,
627: $ NNB, WORK( PPWO ), 2*NNB,
628: $ A( J, JCOL+NNB ), LDA, WORK( PW ),
629: $ LWORK-PW+1, IERR )
630: ELSE
631: *
632: * Ignore the structure of U.
633: *
634: CALL ZGEMM( 'Conjugate', 'No Transpose', 2*NNB,
635: $ COLA, 2*NNB, CONE, WORK( PPWO ), 2*NNB,
636: $ A( J, JCOL+NNB ), LDA, CZERO, WORK( PW ),
637: $ 2*NNB )
638: CALL ZLACPY( 'All', 2*NNB, COLA, WORK( PW ), 2*NNB,
639: $ A( J, JCOL+NNB ), LDA )
640: END IF
641: PPWO = PPWO + 4*NNB*NNB
642: END DO
643: *
644: * Apply accumulated unitary matrices to Q.
645: *
646: IF( WANTQ ) THEN
647: J = IHI - NBLST + 1
648: IF ( INITQ ) THEN
649: TOPQ = MAX( 2, J - JCOL + 1 )
650: NH = IHI - TOPQ + 1
651: ELSE
652: TOPQ = 1
653: NH = N
654: END IF
655: CALL ZGEMM( 'No Transpose', 'No Transpose', NH,
656: $ NBLST, NBLST, CONE, Q( TOPQ, J ), LDQ,
657: $ WORK, NBLST, CZERO, WORK( PW ), NH )
658: CALL ZLACPY( 'All', NH, NBLST, WORK( PW ), NH,
659: $ Q( TOPQ, J ), LDQ )
660: PPWO = NBLST*NBLST + 1
661: J0 = J - NNB
662: DO J = J0, JCOL+1, -NNB
663: IF ( INITQ ) THEN
664: TOPQ = MAX( 2, J - JCOL + 1 )
665: NH = IHI - TOPQ + 1
666: END IF
667: IF ( BLK22 ) THEN
668: *
669: * Exploit the structure of U.
670: *
671: CALL ZUNM22( 'Right', 'No Transpose', NH, 2*NNB,
672: $ NNB, NNB, WORK( PPWO ), 2*NNB,
673: $ Q( TOPQ, J ), LDQ, WORK( PW ),
674: $ LWORK-PW+1, IERR )
675: ELSE
676: *
677: * Ignore the structure of U.
678: *
679: CALL ZGEMM( 'No Transpose', 'No Transpose', NH,
680: $ 2*NNB, 2*NNB, CONE, Q( TOPQ, J ), LDQ,
681: $ WORK( PPWO ), 2*NNB, CZERO, WORK( PW ),
682: $ NH )
683: CALL ZLACPY( 'All', NH, 2*NNB, WORK( PW ), NH,
684: $ Q( TOPQ, J ), LDQ )
685: END IF
686: PPWO = PPWO + 4*NNB*NNB
687: END DO
688: END IF
689: *
690: * Accumulate right Givens rotations if required.
691: *
692: IF ( WANTZ .OR. TOP.GT.0 ) THEN
693: *
694: * Initialize small unitary factors that will hold the
695: * accumulated Givens rotations in workspace.
696: *
697: CALL ZLASET( 'All', NBLST, NBLST, CZERO, CONE, WORK,
698: $ NBLST )
699: PW = NBLST * NBLST + 1
700: DO I = 1, N2NB
701: CALL ZLASET( 'All', 2*NNB, 2*NNB, CZERO, CONE,
702: $ WORK( PW ), 2*NNB )
703: PW = PW + 4*NNB*NNB
704: END DO
705: *
706: * Accumulate Givens rotations into workspace array.
707: *
708: DO J = JCOL, JCOL+NNB-1
709: PPW = ( NBLST + 1 )*( NBLST - 2 ) - J + JCOL + 1
710: LEN = 2 + J - JCOL
711: JROW = J + N2NB*NNB + 2
712: DO I = IHI, JROW, -1
713: CTEMP = A( I, J )
714: A( I, J ) = CZERO
715: S = B( I, J )
716: B( I, J ) = CZERO
717: DO JJ = PPW, PPW+LEN-1
718: TEMP = WORK( JJ + NBLST )
719: WORK( JJ + NBLST ) = CTEMP*TEMP -
720: $ DCONJG( S )*WORK( JJ )
721: WORK( JJ ) = S*TEMP + CTEMP*WORK( JJ )
722: END DO
723: LEN = LEN + 1
724: PPW = PPW - NBLST - 1
725: END DO
726: *
727: PPWO = NBLST*NBLST + ( NNB+J-JCOL-1 )*2*NNB + NNB
728: J0 = JROW - NNB
729: DO JROW = J0, J+2, -NNB
730: PPW = PPWO
731: LEN = 2 + J - JCOL
732: DO I = JROW+NNB-1, JROW, -1
733: CTEMP = A( I, J )
734: A( I, J ) = CZERO
735: S = B( I, J )
736: B( I, J ) = CZERO
737: DO JJ = PPW, PPW+LEN-1
738: TEMP = WORK( JJ + 2*NNB )
739: WORK( JJ + 2*NNB ) = CTEMP*TEMP -
740: $ DCONJG( S )*WORK( JJ )
741: WORK( JJ ) = S*TEMP + CTEMP*WORK( JJ )
742: END DO
743: LEN = LEN + 1
744: PPW = PPW - 2*NNB - 1
745: END DO
746: PPWO = PPWO + 4*NNB*NNB
747: END DO
748: END DO
749: ELSE
750: *
751: CALL ZLASET( 'Lower', IHI - JCOL - 1, NNB, CZERO, CZERO,
752: $ A( JCOL + 2, JCOL ), LDA )
753: CALL ZLASET( 'Lower', IHI - JCOL - 1, NNB, CZERO, CZERO,
754: $ B( JCOL + 2, JCOL ), LDB )
755: END IF
756: *
757: * Apply accumulated unitary matrices to A and B.
758: *
759: IF ( TOP.GT.0 ) THEN
760: J = IHI - NBLST + 1
761: CALL ZGEMM( 'No Transpose', 'No Transpose', TOP,
762: $ NBLST, NBLST, CONE, A( 1, J ), LDA,
763: $ WORK, NBLST, CZERO, WORK( PW ), TOP )
764: CALL ZLACPY( 'All', TOP, NBLST, WORK( PW ), TOP,
765: $ A( 1, J ), LDA )
766: PPWO = NBLST*NBLST + 1
767: J0 = J - NNB
768: DO J = J0, JCOL+1, -NNB
769: IF ( BLK22 ) THEN
770: *
771: * Exploit the structure of U.
772: *
773: CALL ZUNM22( 'Right', 'No Transpose', TOP, 2*NNB,
774: $ NNB, NNB, WORK( PPWO ), 2*NNB,
775: $ A( 1, J ), LDA, WORK( PW ),
776: $ LWORK-PW+1, IERR )
777: ELSE
778: *
779: * Ignore the structure of U.
780: *
781: CALL ZGEMM( 'No Transpose', 'No Transpose', TOP,
782: $ 2*NNB, 2*NNB, CONE, A( 1, J ), LDA,
783: $ WORK( PPWO ), 2*NNB, CZERO,
784: $ WORK( PW ), TOP )
785: CALL ZLACPY( 'All', TOP, 2*NNB, WORK( PW ), TOP,
786: $ A( 1, J ), LDA )
787: END IF
788: PPWO = PPWO + 4*NNB*NNB
789: END DO
790: *
791: J = IHI - NBLST + 1
792: CALL ZGEMM( 'No Transpose', 'No Transpose', TOP,
793: $ NBLST, NBLST, CONE, B( 1, J ), LDB,
794: $ WORK, NBLST, CZERO, WORK( PW ), TOP )
795: CALL ZLACPY( 'All', TOP, NBLST, WORK( PW ), TOP,
796: $ B( 1, J ), LDB )
797: PPWO = NBLST*NBLST + 1
798: J0 = J - NNB
799: DO J = J0, JCOL+1, -NNB
800: IF ( BLK22 ) THEN
801: *
802: * Exploit the structure of U.
803: *
804: CALL ZUNM22( 'Right', 'No Transpose', TOP, 2*NNB,
805: $ NNB, NNB, WORK( PPWO ), 2*NNB,
806: $ B( 1, J ), LDB, WORK( PW ),
807: $ LWORK-PW+1, IERR )
808: ELSE
809: *
810: * Ignore the structure of U.
811: *
812: CALL ZGEMM( 'No Transpose', 'No Transpose', TOP,
813: $ 2*NNB, 2*NNB, CONE, B( 1, J ), LDB,
814: $ WORK( PPWO ), 2*NNB, CZERO,
815: $ WORK( PW ), TOP )
816: CALL ZLACPY( 'All', TOP, 2*NNB, WORK( PW ), TOP,
817: $ B( 1, J ), LDB )
818: END IF
819: PPWO = PPWO + 4*NNB*NNB
820: END DO
821: END IF
822: *
823: * Apply accumulated unitary matrices to Z.
824: *
825: IF( WANTZ ) THEN
826: J = IHI - NBLST + 1
827: IF ( INITQ ) THEN
828: TOPQ = MAX( 2, J - JCOL + 1 )
829: NH = IHI - TOPQ + 1
830: ELSE
831: TOPQ = 1
832: NH = N
833: END IF
834: CALL ZGEMM( 'No Transpose', 'No Transpose', NH,
835: $ NBLST, NBLST, CONE, Z( TOPQ, J ), LDZ,
836: $ WORK, NBLST, CZERO, WORK( PW ), NH )
837: CALL ZLACPY( 'All', NH, NBLST, WORK( PW ), NH,
838: $ Z( TOPQ, J ), LDZ )
839: PPWO = NBLST*NBLST + 1
840: J0 = J - NNB
841: DO J = J0, JCOL+1, -NNB
842: IF ( INITQ ) THEN
843: TOPQ = MAX( 2, J - JCOL + 1 )
844: NH = IHI - TOPQ + 1
845: END IF
846: IF ( BLK22 ) THEN
847: *
848: * Exploit the structure of U.
849: *
850: CALL ZUNM22( 'Right', 'No Transpose', NH, 2*NNB,
851: $ NNB, NNB, WORK( PPWO ), 2*NNB,
852: $ Z( TOPQ, J ), LDZ, WORK( PW ),
853: $ LWORK-PW+1, IERR )
854: ELSE
855: *
856: * Ignore the structure of U.
857: *
858: CALL ZGEMM( 'No Transpose', 'No Transpose', NH,
859: $ 2*NNB, 2*NNB, CONE, Z( TOPQ, J ), LDZ,
860: $ WORK( PPWO ), 2*NNB, CZERO, WORK( PW ),
861: $ NH )
862: CALL ZLACPY( 'All', NH, 2*NNB, WORK( PW ), NH,
863: $ Z( TOPQ, J ), LDZ )
864: END IF
865: PPWO = PPWO + 4*NNB*NNB
866: END DO
867: END IF
868: END DO
869: END IF
870: *
871: * Use unblocked code to reduce the rest of the matrix
872: * Avoid re-initialization of modified Q and Z.
873: *
874: COMPQ2 = COMPQ
875: COMPZ2 = COMPZ
876: IF ( JCOL.NE.ILO ) THEN
877: IF ( WANTQ )
878: $ COMPQ2 = 'V'
879: IF ( WANTZ )
880: $ COMPZ2 = 'V'
881: END IF
882: *
883: IF ( JCOL.LT.IHI )
884: $ CALL ZGGHRD( COMPQ2, COMPZ2, N, JCOL, IHI, A, LDA, B, LDB, Q,
885: $ LDQ, Z, LDZ, IERR )
886: WORK( 1 ) = DCMPLX( LWKOPT )
887: *
888: RETURN
889: *
890: * End of ZGGHD3
891: *
892: END
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