Annotation of rpl/lapack/lapack/dggsvp.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DGGSVP
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DGGSVP + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvp.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvp.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvp.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
! 22: * TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
! 23: * IWORK, 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 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 33: * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> DGGSVP computes orthogonal matrices U, V and Q such that
! 43: *>
! 44: *> N-K-L K L
! 45: *> U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
! 46: *> L ( 0 0 A23 )
! 47: *> M-K-L ( 0 0 0 )
! 48: *>
! 49: *> N-K-L K L
! 50: *> = K ( 0 A12 A13 ) if M-K-L < 0;
! 51: *> M-K ( 0 0 A23 )
! 52: *>
! 53: *> N-K-L K L
! 54: *> V**T*B*Q = L ( 0 0 B13 )
! 55: *> P-L ( 0 0 0 )
! 56: *>
! 57: *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
! 58: *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
! 59: *> otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
! 60: *> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
! 61: *>
! 62: *> This decomposition is the preprocessing step for computing the
! 63: *> Generalized Singular Value Decomposition (GSVD), see subroutine
! 64: *> DGGSVD.
! 65: *> \endverbatim
! 66: *
! 67: * Arguments:
! 68: * ==========
! 69: *
! 70: *> \param[in] JOBU
! 71: *> \verbatim
! 72: *> JOBU is CHARACTER*1
! 73: *> = 'U': Orthogonal matrix U is computed;
! 74: *> = 'N': U is not computed.
! 75: *> \endverbatim
! 76: *>
! 77: *> \param[in] JOBV
! 78: *> \verbatim
! 79: *> JOBV is CHARACTER*1
! 80: *> = 'V': Orthogonal matrix V is computed;
! 81: *> = 'N': V is not computed.
! 82: *> \endverbatim
! 83: *>
! 84: *> \param[in] JOBQ
! 85: *> \verbatim
! 86: *> JOBQ is CHARACTER*1
! 87: *> = 'Q': Orthogonal matrix Q is computed;
! 88: *> = 'N': Q is not computed.
! 89: *> \endverbatim
! 90: *>
! 91: *> \param[in] M
! 92: *> \verbatim
! 93: *> M is INTEGER
! 94: *> The number of rows of the matrix A. M >= 0.
! 95: *> \endverbatim
! 96: *>
! 97: *> \param[in] P
! 98: *> \verbatim
! 99: *> P is INTEGER
! 100: *> The number of rows of the matrix B. P >= 0.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[in] N
! 104: *> \verbatim
! 105: *> N is INTEGER
! 106: *> The number of columns of the matrices A and B. N >= 0.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[in,out] A
! 110: *> \verbatim
! 111: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 112: *> On entry, the M-by-N matrix A.
! 113: *> On exit, A contains the triangular (or trapezoidal) matrix
! 114: *> described in the Purpose section.
! 115: *> \endverbatim
! 116: *>
! 117: *> \param[in] LDA
! 118: *> \verbatim
! 119: *> LDA is INTEGER
! 120: *> The leading dimension of the array A. LDA >= max(1,M).
! 121: *> \endverbatim
! 122: *>
! 123: *> \param[in,out] B
! 124: *> \verbatim
! 125: *> B is DOUBLE PRECISION array, dimension (LDB,N)
! 126: *> On entry, the P-by-N matrix B.
! 127: *> On exit, B contains the triangular matrix described in
! 128: *> the Purpose section.
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[in] LDB
! 132: *> \verbatim
! 133: *> LDB is INTEGER
! 134: *> The leading dimension of the array B. LDB >= max(1,P).
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[in] TOLA
! 138: *> \verbatim
! 139: *> TOLA is DOUBLE PRECISION
! 140: *> \endverbatim
! 141: *>
! 142: *> \param[in] TOLB
! 143: *> \verbatim
! 144: *> TOLB is DOUBLE PRECISION
! 145: *>
! 146: *> TOLA and TOLB are the thresholds to determine the effective
! 147: *> numerical rank of matrix B and a subblock of A. Generally,
! 148: *> they are set to
! 149: *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
! 150: *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
! 151: *> The size of TOLA and TOLB may affect the size of backward
! 152: *> errors of the decomposition.
! 153: *> \endverbatim
! 154: *>
! 155: *> \param[out] K
! 156: *> \verbatim
! 157: *> K is INTEGER
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[out] L
! 161: *> \verbatim
! 162: *> L is INTEGER
! 163: *>
! 164: *> On exit, K and L specify the dimension of the subblocks
! 165: *> described in Purpose section.
! 166: *> K + L = effective numerical rank of (A**T,B**T)**T.
! 167: *> \endverbatim
! 168: *>
! 169: *> \param[out] U
! 170: *> \verbatim
! 171: *> U is DOUBLE PRECISION array, dimension (LDU,M)
! 172: *> If JOBU = 'U', U contains the orthogonal matrix U.
! 173: *> If JOBU = 'N', U is not referenced.
! 174: *> \endverbatim
! 175: *>
! 176: *> \param[in] LDU
! 177: *> \verbatim
! 178: *> LDU is INTEGER
! 179: *> The leading dimension of the array U. LDU >= max(1,M) if
! 180: *> JOBU = 'U'; LDU >= 1 otherwise.
! 181: *> \endverbatim
! 182: *>
! 183: *> \param[out] V
! 184: *> \verbatim
! 185: *> V is DOUBLE PRECISION array, dimension (LDV,P)
! 186: *> If JOBV = 'V', V contains the orthogonal matrix V.
! 187: *> If JOBV = 'N', V is not referenced.
! 188: *> \endverbatim
! 189: *>
! 190: *> \param[in] LDV
! 191: *> \verbatim
! 192: *> LDV is INTEGER
! 193: *> The leading dimension of the array V. LDV >= max(1,P) if
! 194: *> JOBV = 'V'; LDV >= 1 otherwise.
! 195: *> \endverbatim
! 196: *>
! 197: *> \param[out] Q
! 198: *> \verbatim
! 199: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
! 200: *> If JOBQ = 'Q', Q contains the orthogonal matrix Q.
! 201: *> If JOBQ = 'N', Q is not referenced.
! 202: *> \endverbatim
! 203: *>
! 204: *> \param[in] LDQ
! 205: *> \verbatim
! 206: *> LDQ is INTEGER
! 207: *> The leading dimension of the array Q. LDQ >= max(1,N) if
! 208: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
! 209: *> \endverbatim
! 210: *>
! 211: *> \param[out] IWORK
! 212: *> \verbatim
! 213: *> IWORK is INTEGER array, dimension (N)
! 214: *> \endverbatim
! 215: *>
! 216: *> \param[out] TAU
! 217: *> \verbatim
! 218: *> TAU is DOUBLE PRECISION array, dimension (N)
! 219: *> \endverbatim
! 220: *>
! 221: *> \param[out] WORK
! 222: *> \verbatim
! 223: *> WORK is DOUBLE PRECISION array, dimension (max(3*N,M,P))
! 224: *> \endverbatim
! 225: *>
! 226: *> \param[out] INFO
! 227: *> \verbatim
! 228: *> INFO is INTEGER
! 229: *> = 0: successful exit
! 230: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 231: *> \endverbatim
! 232: *
! 233: * Authors:
! 234: * ========
! 235: *
! 236: *> \author Univ. of Tennessee
! 237: *> \author Univ. of California Berkeley
! 238: *> \author Univ. of Colorado Denver
! 239: *> \author NAG Ltd.
! 240: *
! 241: *> \date November 2011
! 242: *
! 243: *> \ingroup doubleOTHERcomputational
! 244: *
! 245: *> \par Further Details:
! 246: * =====================
! 247: *>
! 248: *> The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
! 249: *> with column pivoting to detect the effective numerical rank of the
! 250: *> a matrix. It may be replaced by a better rank determination strategy.
! 251: *>
! 252: * =====================================================================
1.1 bertrand 253: SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
254: $ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
255: $ IWORK, TAU, WORK, INFO )
256: *
1.9 ! bertrand 257: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 258: * -- LAPACK is a software package provided by Univ. of Tennessee, --
259: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 260: * November 2011
1.1 bertrand 261: *
262: * .. Scalar Arguments ..
263: CHARACTER JOBQ, JOBU, JOBV
264: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
265: DOUBLE PRECISION TOLA, TOLB
266: * ..
267: * .. Array Arguments ..
268: INTEGER IWORK( * )
269: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
270: $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
271: * ..
272: *
273: * =====================================================================
274: *
275: * .. Parameters ..
276: DOUBLE PRECISION ZERO, ONE
277: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
278: * ..
279: * .. Local Scalars ..
280: LOGICAL FORWRD, WANTQ, WANTU, WANTV
281: INTEGER I, J
282: * ..
283: * .. External Functions ..
284: LOGICAL LSAME
285: EXTERNAL LSAME
286: * ..
287: * .. External Subroutines ..
288: EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET,
289: $ DORG2R, DORM2R, DORMR2, XERBLA
290: * ..
291: * .. Intrinsic Functions ..
292: INTRINSIC ABS, MAX, MIN
293: * ..
294: * .. Executable Statements ..
295: *
296: * Test the input parameters
297: *
298: WANTU = LSAME( JOBU, 'U' )
299: WANTV = LSAME( JOBV, 'V' )
300: WANTQ = LSAME( JOBQ, 'Q' )
301: FORWRD = .TRUE.
302: *
303: INFO = 0
304: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
305: INFO = -1
306: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
307: INFO = -2
308: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
309: INFO = -3
310: ELSE IF( M.LT.0 ) THEN
311: INFO = -4
312: ELSE IF( P.LT.0 ) THEN
313: INFO = -5
314: ELSE IF( N.LT.0 ) THEN
315: INFO = -6
316: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
317: INFO = -8
318: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
319: INFO = -10
320: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
321: INFO = -16
322: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
323: INFO = -18
324: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
325: INFO = -20
326: END IF
327: IF( INFO.NE.0 ) THEN
328: CALL XERBLA( 'DGGSVP', -INFO )
329: RETURN
330: END IF
331: *
332: * QR with column pivoting of B: B*P = V*( S11 S12 )
333: * ( 0 0 )
334: *
335: DO 10 I = 1, N
336: IWORK( I ) = 0
337: 10 CONTINUE
338: CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
339: *
340: * Update A := A*P
341: *
342: CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK )
343: *
344: * Determine the effective rank of matrix B.
345: *
346: L = 0
347: DO 20 I = 1, MIN( P, N )
348: IF( ABS( B( I, I ) ).GT.TOLB )
349: $ L = L + 1
350: 20 CONTINUE
351: *
352: IF( WANTV ) THEN
353: *
354: * Copy the details of V, and form V.
355: *
356: CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
357: IF( P.GT.1 )
358: $ CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
359: $ LDV )
360: CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
361: END IF
362: *
363: * Clean up B
364: *
365: DO 40 J = 1, L - 1
366: DO 30 I = J + 1, L
367: B( I, J ) = ZERO
368: 30 CONTINUE
369: 40 CONTINUE
370: IF( P.GT.L )
371: $ CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
372: *
373: IF( WANTQ ) THEN
374: *
375: * Set Q = I and Update Q := Q*P
376: *
377: CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
378: CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
379: END IF
380: *
381: IF( P.GE.L .AND. N.NE.L ) THEN
382: *
383: * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
384: *
385: CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO )
386: *
1.8 bertrand 387: * Update A := A*Z**T
1.1 bertrand 388: *
389: CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
390: $ LDA, WORK, INFO )
391: *
392: IF( WANTQ ) THEN
393: *
1.8 bertrand 394: * Update Q := Q*Z**T
1.1 bertrand 395: *
396: CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
397: $ LDQ, WORK, INFO )
398: END IF
399: *
400: * Clean up B
401: *
402: CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
403: DO 60 J = N - L + 1, N
404: DO 50 I = J - N + L + 1, L
405: B( I, J ) = ZERO
406: 50 CONTINUE
407: 60 CONTINUE
408: *
409: END IF
410: *
411: * Let N-L L
412: * A = ( A11 A12 ) M,
413: *
414: * then the following does the complete QR decomposition of A11:
415: *
1.8 bertrand 416: * A11 = U*( 0 T12 )*P1**T
1.1 bertrand 417: * ( 0 0 )
418: *
419: DO 70 I = 1, N - L
420: IWORK( I ) = 0
421: 70 CONTINUE
422: CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
423: *
424: * Determine the effective rank of A11
425: *
426: K = 0
427: DO 80 I = 1, MIN( M, N-L )
428: IF( ABS( A( I, I ) ).GT.TOLA )
429: $ K = K + 1
430: 80 CONTINUE
431: *
1.8 bertrand 432: * Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
1.1 bertrand 433: *
434: CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
435: $ TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
436: *
437: IF( WANTU ) THEN
438: *
439: * Copy the details of U, and form U
440: *
441: CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
442: IF( M.GT.1 )
443: $ CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
444: $ LDU )
445: CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
446: END IF
447: *
448: IF( WANTQ ) THEN
449: *
450: * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
451: *
452: CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
453: END IF
454: *
455: * Clean up A: set the strictly lower triangular part of
456: * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
457: *
458: DO 100 J = 1, K - 1
459: DO 90 I = J + 1, K
460: A( I, J ) = ZERO
461: 90 CONTINUE
462: 100 CONTINUE
463: IF( M.GT.K )
464: $ CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
465: *
466: IF( N-L.GT.K ) THEN
467: *
468: * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
469: *
470: CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
471: *
472: IF( WANTQ ) THEN
473: *
1.8 bertrand 474: * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
1.1 bertrand 475: *
476: CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
477: $ Q, LDQ, WORK, INFO )
478: END IF
479: *
480: * Clean up A
481: *
482: CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
483: DO 120 J = N - L - K + 1, N - L
484: DO 110 I = J - N + L + K + 1, K
485: A( I, J ) = ZERO
486: 110 CONTINUE
487: 120 CONTINUE
488: *
489: END IF
490: *
491: IF( M.GT.K ) THEN
492: *
493: * QR factorization of A( K+1:M,N-L+1:N )
494: *
495: CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
496: *
497: IF( WANTU ) THEN
498: *
499: * Update U(:,K+1:M) := U(:,K+1:M)*U1
500: *
501: CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
502: $ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
503: $ WORK, INFO )
504: END IF
505: *
506: * Clean up
507: *
508: DO 140 J = N - L + 1, N
509: DO 130 I = J - N + K + L + 1, M
510: A( I, J ) = ZERO
511: 130 CONTINUE
512: 140 CONTINUE
513: *
514: END IF
515: *
516: RETURN
517: *
518: * End of DGGSVP
519: *
520: END
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