Annotation of rpl/lapack/lapack/zggsvp.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZGGSVP
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZGGSVP + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggsvp.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggsvp.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggsvp.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
! 22: * TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
! 23: * IWORK, RWORK, TAU, WORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER JOBQ, JOBU, JOBV
! 27: * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
! 28: * DOUBLE PRECISION TOLA, TOLB
! 29: * ..
! 30: * .. Array Arguments ..
! 31: * INTEGER IWORK( * )
! 32: * DOUBLE PRECISION RWORK( * )
! 33: * COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 34: * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
! 35: * ..
! 36: *
! 37: *
! 38: *> \par Purpose:
! 39: * =============
! 40: *>
! 41: *> \verbatim
! 42: *>
! 43: *> ZGGSVP computes unitary matrices U, V and Q such that
! 44: *>
! 45: *> N-K-L K L
! 46: *> U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
! 47: *> L ( 0 0 A23 )
! 48: *> M-K-L ( 0 0 0 )
! 49: *>
! 50: *> N-K-L K L
! 51: *> = K ( 0 A12 A13 ) if M-K-L < 0;
! 52: *> M-K ( 0 0 A23 )
! 53: *>
! 54: *> N-K-L K L
! 55: *> V**H*B*Q = L ( 0 0 B13 )
! 56: *> P-L ( 0 0 0 )
! 57: *>
! 58: *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
! 59: *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
! 60: *> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
! 61: *> numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H.
! 62: *>
! 63: *> This decomposition is the preprocessing step for computing the
! 64: *> Generalized Singular Value Decomposition (GSVD), see subroutine
! 65: *> ZGGSVD.
! 66: *> \endverbatim
! 67: *
! 68: * Arguments:
! 69: * ==========
! 70: *
! 71: *> \param[in] JOBU
! 72: *> \verbatim
! 73: *> JOBU is CHARACTER*1
! 74: *> = 'U': Unitary matrix U is computed;
! 75: *> = 'N': U is not computed.
! 76: *> \endverbatim
! 77: *>
! 78: *> \param[in] JOBV
! 79: *> \verbatim
! 80: *> JOBV is CHARACTER*1
! 81: *> = 'V': Unitary matrix V is computed;
! 82: *> = 'N': V is not computed.
! 83: *> \endverbatim
! 84: *>
! 85: *> \param[in] JOBQ
! 86: *> \verbatim
! 87: *> JOBQ is CHARACTER*1
! 88: *> = 'Q': Unitary matrix Q is computed;
! 89: *> = 'N': Q is not computed.
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[in] M
! 93: *> \verbatim
! 94: *> M is INTEGER
! 95: *> The number of rows of the matrix A. M >= 0.
! 96: *> \endverbatim
! 97: *>
! 98: *> \param[in] P
! 99: *> \verbatim
! 100: *> P is INTEGER
! 101: *> The number of rows of the matrix B. P >= 0.
! 102: *> \endverbatim
! 103: *>
! 104: *> \param[in] N
! 105: *> \verbatim
! 106: *> N is INTEGER
! 107: *> The number of columns of the matrices A and B. N >= 0.
! 108: *> \endverbatim
! 109: *>
! 110: *> \param[in,out] A
! 111: *> \verbatim
! 112: *> A is COMPLEX*16 array, dimension (LDA,N)
! 113: *> On entry, the M-by-N matrix A.
! 114: *> On exit, A contains the triangular (or trapezoidal) matrix
! 115: *> described in the Purpose section.
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in] LDA
! 119: *> \verbatim
! 120: *> LDA is INTEGER
! 121: *> The leading dimension of the array A. LDA >= max(1,M).
! 122: *> \endverbatim
! 123: *>
! 124: *> \param[in,out] B
! 125: *> \verbatim
! 126: *> B is COMPLEX*16 array, dimension (LDB,N)
! 127: *> On entry, the P-by-N matrix B.
! 128: *> On exit, B contains the triangular matrix described in
! 129: *> the Purpose section.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[in] LDB
! 133: *> \verbatim
! 134: *> LDB is INTEGER
! 135: *> The leading dimension of the array B. LDB >= max(1,P).
! 136: *> \endverbatim
! 137: *>
! 138: *> \param[in] TOLA
! 139: *> \verbatim
! 140: *> TOLA is DOUBLE PRECISION
! 141: *> \endverbatim
! 142: *>
! 143: *> \param[in] TOLB
! 144: *> \verbatim
! 145: *> TOLB is DOUBLE PRECISION
! 146: *>
! 147: *> TOLA and TOLB are the thresholds to determine the effective
! 148: *> numerical rank of matrix B and a subblock of A. Generally,
! 149: *> they are set to
! 150: *> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
! 151: *> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
! 152: *> The size of TOLA and TOLB may affect the size of backward
! 153: *> errors of the decomposition.
! 154: *> \endverbatim
! 155: *>
! 156: *> \param[out] K
! 157: *> \verbatim
! 158: *> K is INTEGER
! 159: *> \endverbatim
! 160: *>
! 161: *> \param[out] L
! 162: *> \verbatim
! 163: *> L is INTEGER
! 164: *>
! 165: *> On exit, K and L specify the dimension of the subblocks
! 166: *> described in Purpose section.
! 167: *> K + L = effective numerical rank of (A**H,B**H)**H.
! 168: *> \endverbatim
! 169: *>
! 170: *> \param[out] U
! 171: *> \verbatim
! 172: *> U is COMPLEX*16 array, dimension (LDU,M)
! 173: *> If JOBU = 'U', U contains the unitary matrix U.
! 174: *> If JOBU = 'N', U is not referenced.
! 175: *> \endverbatim
! 176: *>
! 177: *> \param[in] LDU
! 178: *> \verbatim
! 179: *> LDU is INTEGER
! 180: *> The leading dimension of the array U. LDU >= max(1,M) if
! 181: *> JOBU = 'U'; LDU >= 1 otherwise.
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[out] V
! 185: *> \verbatim
! 186: *> V is COMPLEX*16 array, dimension (LDV,P)
! 187: *> If JOBV = 'V', V contains the unitary matrix V.
! 188: *> If JOBV = 'N', V is not referenced.
! 189: *> \endverbatim
! 190: *>
! 191: *> \param[in] LDV
! 192: *> \verbatim
! 193: *> LDV is INTEGER
! 194: *> The leading dimension of the array V. LDV >= max(1,P) if
! 195: *> JOBV = 'V'; LDV >= 1 otherwise.
! 196: *> \endverbatim
! 197: *>
! 198: *> \param[out] Q
! 199: *> \verbatim
! 200: *> Q is COMPLEX*16 array, dimension (LDQ,N)
! 201: *> If JOBQ = 'Q', Q contains the unitary matrix Q.
! 202: *> If JOBQ = 'N', Q is not referenced.
! 203: *> \endverbatim
! 204: *>
! 205: *> \param[in] LDQ
! 206: *> \verbatim
! 207: *> LDQ is INTEGER
! 208: *> The leading dimension of the array Q. LDQ >= max(1,N) if
! 209: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
! 210: *> \endverbatim
! 211: *>
! 212: *> \param[out] IWORK
! 213: *> \verbatim
! 214: *> IWORK is INTEGER array, dimension (N)
! 215: *> \endverbatim
! 216: *>
! 217: *> \param[out] RWORK
! 218: *> \verbatim
! 219: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
! 220: *> \endverbatim
! 221: *>
! 222: *> \param[out] TAU
! 223: *> \verbatim
! 224: *> TAU is COMPLEX*16 array, dimension (N)
! 225: *> \endverbatim
! 226: *>
! 227: *> \param[out] WORK
! 228: *> \verbatim
! 229: *> WORK is COMPLEX*16 array, dimension (max(3*N,M,P))
! 230: *> \endverbatim
! 231: *>
! 232: *> \param[out] INFO
! 233: *> \verbatim
! 234: *> INFO is INTEGER
! 235: *> = 0: successful exit
! 236: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 237: *> \endverbatim
! 238: *
! 239: * Authors:
! 240: * ========
! 241: *
! 242: *> \author Univ. of Tennessee
! 243: *> \author Univ. of California Berkeley
! 244: *> \author Univ. of Colorado Denver
! 245: *> \author NAG Ltd.
! 246: *
! 247: *> \date November 2011
! 248: *
! 249: *> \ingroup complex16OTHERcomputational
! 250: *
! 251: *> \par Further Details:
! 252: * =====================
! 253: *>
! 254: *> \verbatim
! 255: *>
! 256: *> The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
! 257: *> with column pivoting to detect the effective numerical rank of the
! 258: *> a matrix. It may be replaced by a better rank determination strategy.
! 259: *> \endverbatim
! 260: *>
! 261: * =====================================================================
1.1 bertrand 262: SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
263: $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
264: $ IWORK, RWORK, TAU, WORK, INFO )
265: *
1.9 ! bertrand 266: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 267: * -- LAPACK is a software package provided by Univ. of Tennessee, --
268: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 269: * November 2011
1.1 bertrand 270: *
271: * .. Scalar Arguments ..
272: CHARACTER JOBQ, JOBU, JOBV
273: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
274: DOUBLE PRECISION TOLA, TOLB
275: * ..
276: * .. Array Arguments ..
277: INTEGER IWORK( * )
278: DOUBLE PRECISION RWORK( * )
279: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
280: $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
281: * ..
282: *
283: * =====================================================================
284: *
285: * .. Parameters ..
286: COMPLEX*16 CZERO, CONE
287: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
288: $ CONE = ( 1.0D+0, 0.0D+0 ) )
289: * ..
290: * .. Local Scalars ..
291: LOGICAL FORWRD, WANTQ, WANTU, WANTV
292: INTEGER I, J
293: COMPLEX*16 T
294: * ..
295: * .. External Functions ..
296: LOGICAL LSAME
297: EXTERNAL LSAME
298: * ..
299: * .. External Subroutines ..
300: EXTERNAL XERBLA, ZGEQPF, ZGEQR2, ZGERQ2, ZLACPY, ZLAPMT,
301: $ ZLASET, ZUNG2R, ZUNM2R, ZUNMR2
302: * ..
303: * .. Intrinsic Functions ..
304: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
305: * ..
306: * .. Statement Functions ..
307: DOUBLE PRECISION CABS1
308: * ..
309: * .. Statement Function definitions ..
310: CABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
311: * ..
312: * .. Executable Statements ..
313: *
314: * Test the input parameters
315: *
316: WANTU = LSAME( JOBU, 'U' )
317: WANTV = LSAME( JOBV, 'V' )
318: WANTQ = LSAME( JOBQ, 'Q' )
319: FORWRD = .TRUE.
320: *
321: INFO = 0
322: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
323: INFO = -1
324: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
325: INFO = -2
326: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
327: INFO = -3
328: ELSE IF( M.LT.0 ) THEN
329: INFO = -4
330: ELSE IF( P.LT.0 ) THEN
331: INFO = -5
332: ELSE IF( N.LT.0 ) THEN
333: INFO = -6
334: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
335: INFO = -8
336: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
337: INFO = -10
338: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
339: INFO = -16
340: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
341: INFO = -18
342: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
343: INFO = -20
344: END IF
345: IF( INFO.NE.0 ) THEN
346: CALL XERBLA( 'ZGGSVP', -INFO )
347: RETURN
348: END IF
349: *
350: * QR with column pivoting of B: B*P = V*( S11 S12 )
351: * ( 0 0 )
352: *
353: DO 10 I = 1, N
354: IWORK( I ) = 0
355: 10 CONTINUE
356: CALL ZGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO )
357: *
358: * Update A := A*P
359: *
360: CALL ZLAPMT( FORWRD, M, N, A, LDA, IWORK )
361: *
362: * Determine the effective rank of matrix B.
363: *
364: L = 0
365: DO 20 I = 1, MIN( P, N )
366: IF( CABS1( B( I, I ) ).GT.TOLB )
367: $ L = L + 1
368: 20 CONTINUE
369: *
370: IF( WANTV ) THEN
371: *
372: * Copy the details of V, and form V.
373: *
374: CALL ZLASET( 'Full', P, P, CZERO, CZERO, V, LDV )
375: IF( P.GT.1 )
376: $ CALL ZLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
377: $ LDV )
378: CALL ZUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
379: END IF
380: *
381: * Clean up B
382: *
383: DO 40 J = 1, L - 1
384: DO 30 I = J + 1, L
385: B( I, J ) = CZERO
386: 30 CONTINUE
387: 40 CONTINUE
388: IF( P.GT.L )
389: $ CALL ZLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB )
390: *
391: IF( WANTQ ) THEN
392: *
393: * Set Q = I and Update Q := Q*P
394: *
395: CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
396: CALL ZLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
397: END IF
398: *
399: IF( P.GE.L .AND. N.NE.L ) THEN
400: *
401: * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
402: *
403: CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO )
404: *
1.8 bertrand 405: * Update A := A*Z**H
1.1 bertrand 406: *
407: CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB,
408: $ TAU, A, LDA, WORK, INFO )
409: IF( WANTQ ) THEN
410: *
1.8 bertrand 411: * Update Q := Q*Z**H
1.1 bertrand 412: *
413: CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B,
414: $ LDB, TAU, Q, LDQ, WORK, INFO )
415: END IF
416: *
417: * Clean up B
418: *
419: CALL ZLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB )
420: DO 60 J = N - L + 1, N
421: DO 50 I = J - N + L + 1, L
422: B( I, J ) = CZERO
423: 50 CONTINUE
424: 60 CONTINUE
425: *
426: END IF
427: *
428: * Let N-L L
429: * A = ( A11 A12 ) M,
430: *
431: * then the following does the complete QR decomposition of A11:
432: *
1.8 bertrand 433: * A11 = U*( 0 T12 )*P1**H
1.1 bertrand 434: * ( 0 0 )
435: *
436: DO 70 I = 1, N - L
437: IWORK( I ) = 0
438: 70 CONTINUE
439: CALL ZGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO )
440: *
441: * Determine the effective rank of A11
442: *
443: K = 0
444: DO 80 I = 1, MIN( M, N-L )
445: IF( CABS1( A( I, I ) ).GT.TOLA )
446: $ K = K + 1
447: 80 CONTINUE
448: *
1.8 bertrand 449: * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
1.1 bertrand 450: *
451: CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ),
452: $ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
453: *
454: IF( WANTU ) THEN
455: *
456: * Copy the details of U, and form U
457: *
458: CALL ZLASET( 'Full', M, M, CZERO, CZERO, U, LDU )
459: IF( M.GT.1 )
460: $ CALL ZLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
461: $ LDU )
462: CALL ZUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
463: END IF
464: *
465: IF( WANTQ ) THEN
466: *
467: * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
468: *
469: CALL ZLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
470: END IF
471: *
472: * Clean up A: set the strictly lower triangular part of
473: * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
474: *
475: DO 100 J = 1, K - 1
476: DO 90 I = J + 1, K
477: A( I, J ) = CZERO
478: 90 CONTINUE
479: 100 CONTINUE
480: IF( M.GT.K )
481: $ CALL ZLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA )
482: *
483: IF( N-L.GT.K ) THEN
484: *
485: * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
486: *
487: CALL ZGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
488: *
489: IF( WANTQ ) THEN
490: *
1.8 bertrand 491: * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
1.1 bertrand 492: *
493: CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A,
494: $ LDA, TAU, Q, LDQ, WORK, INFO )
495: END IF
496: *
497: * Clean up A
498: *
499: CALL ZLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA )
500: DO 120 J = N - L - K + 1, N - L
501: DO 110 I = J - N + L + K + 1, K
502: A( I, J ) = CZERO
503: 110 CONTINUE
504: 120 CONTINUE
505: *
506: END IF
507: *
508: IF( M.GT.K ) THEN
509: *
510: * QR factorization of A( K+1:M,N-L+1:N )
511: *
512: CALL ZGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
513: *
514: IF( WANTU ) THEN
515: *
516: * Update U(:,K+1:M) := U(:,K+1:M)*U1
517: *
518: CALL ZUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
519: $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
520: $ WORK, INFO )
521: END IF
522: *
523: * Clean up
524: *
525: DO 140 J = N - L + 1, N
526: DO 130 I = J - N + K + L + 1, M
527: A( I, J ) = CZERO
528: 130 CONTINUE
529: 140 CONTINUE
530: *
531: END IF
532: *
533: RETURN
534: *
535: * End of ZGGSVP
536: *
537: END
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