Annotation of rpl/lapack/lapack/dggsvd.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> DGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b>
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DGGSVD + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvd.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvd.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
! 22: * LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
! 23: * IWORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER JOBQ, JOBU, JOBV
! 27: * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * INTEGER IWORK( * )
! 31: * DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
! 32: * $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
! 33: * $ V( LDV, * ), WORK( * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> DGGSVD computes the generalized singular value decomposition (GSVD)
! 43: *> of an M-by-N real matrix A and P-by-N real matrix B:
! 44: *>
! 45: *> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
! 46: *>
! 47: *> where U, V and Q are orthogonal matrices.
! 48: *> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
! 49: *> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
! 50: *> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
! 51: *> following structures, respectively:
! 52: *>
! 53: *> If M-K-L >= 0,
! 54: *>
! 55: *> K L
! 56: *> D1 = K ( I 0 )
! 57: *> L ( 0 C )
! 58: *> M-K-L ( 0 0 )
! 59: *>
! 60: *> K L
! 61: *> D2 = L ( 0 S )
! 62: *> P-L ( 0 0 )
! 63: *>
! 64: *> N-K-L K L
! 65: *> ( 0 R ) = K ( 0 R11 R12 )
! 66: *> L ( 0 0 R22 )
! 67: *>
! 68: *> where
! 69: *>
! 70: *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
! 71: *> S = diag( BETA(K+1), ... , BETA(K+L) ),
! 72: *> C**2 + S**2 = I.
! 73: *>
! 74: *> R is stored in A(1:K+L,N-K-L+1:N) on exit.
! 75: *>
! 76: *> If M-K-L < 0,
! 77: *>
! 78: *> K M-K K+L-M
! 79: *> D1 = K ( I 0 0 )
! 80: *> M-K ( 0 C 0 )
! 81: *>
! 82: *> K M-K K+L-M
! 83: *> D2 = M-K ( 0 S 0 )
! 84: *> K+L-M ( 0 0 I )
! 85: *> P-L ( 0 0 0 )
! 86: *>
! 87: *> N-K-L K M-K K+L-M
! 88: *> ( 0 R ) = K ( 0 R11 R12 R13 )
! 89: *> M-K ( 0 0 R22 R23 )
! 90: *> K+L-M ( 0 0 0 R33 )
! 91: *>
! 92: *> where
! 93: *>
! 94: *> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
! 95: *> S = diag( BETA(K+1), ... , BETA(M) ),
! 96: *> C**2 + S**2 = I.
! 97: *>
! 98: *> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
! 99: *> ( 0 R22 R23 )
! 100: *> in B(M-K+1:L,N+M-K-L+1:N) on exit.
! 101: *>
! 102: *> The routine computes C, S, R, and optionally the orthogonal
! 103: *> transformation matrices U, V and Q.
! 104: *>
! 105: *> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
! 106: *> A and B implicitly gives the SVD of A*inv(B):
! 107: *> A*inv(B) = U*(D1*inv(D2))*V**T.
! 108: *> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is
! 109: *> also equal to the CS decomposition of A and B. Furthermore, the GSVD
! 110: *> can be used to derive the solution of the eigenvalue problem:
! 111: *> A**T*A x = lambda* B**T*B x.
! 112: *> In some literature, the GSVD of A and B is presented in the form
! 113: *> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
! 114: *> where U and V are orthogonal and X is nonsingular, D1 and D2 are
! 115: *> ``diagonal''. The former GSVD form can be converted to the latter
! 116: *> form by taking the nonsingular matrix X as
! 117: *>
! 118: *> X = Q*( I 0 )
! 119: *> ( 0 inv(R) ).
! 120: *> \endverbatim
! 121: *
! 122: * Arguments:
! 123: * ==========
! 124: *
! 125: *> \param[in] JOBU
! 126: *> \verbatim
! 127: *> JOBU is CHARACTER*1
! 128: *> = 'U': Orthogonal matrix U is computed;
! 129: *> = 'N': U is not computed.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[in] JOBV
! 133: *> \verbatim
! 134: *> JOBV is CHARACTER*1
! 135: *> = 'V': Orthogonal matrix V is computed;
! 136: *> = 'N': V is not computed.
! 137: *> \endverbatim
! 138: *>
! 139: *> \param[in] JOBQ
! 140: *> \verbatim
! 141: *> JOBQ is CHARACTER*1
! 142: *> = 'Q': Orthogonal matrix Q is computed;
! 143: *> = 'N': Q is not computed.
! 144: *> \endverbatim
! 145: *>
! 146: *> \param[in] M
! 147: *> \verbatim
! 148: *> M is INTEGER
! 149: *> The number of rows of the matrix A. M >= 0.
! 150: *> \endverbatim
! 151: *>
! 152: *> \param[in] N
! 153: *> \verbatim
! 154: *> N is INTEGER
! 155: *> The number of columns of the matrices A and B. N >= 0.
! 156: *> \endverbatim
! 157: *>
! 158: *> \param[in] P
! 159: *> \verbatim
! 160: *> P is INTEGER
! 161: *> The number of rows of the matrix B. P >= 0.
! 162: *> \endverbatim
! 163: *>
! 164: *> \param[out] K
! 165: *> \verbatim
! 166: *> K is INTEGER
! 167: *> \endverbatim
! 168: *>
! 169: *> \param[out] L
! 170: *> \verbatim
! 171: *> L is INTEGER
! 172: *>
! 173: *> On exit, K and L specify the dimension of the subblocks
! 174: *> described in Purpose.
! 175: *> K + L = effective numerical rank of (A**T,B**T)**T.
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in,out] A
! 179: *> \verbatim
! 180: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 181: *> On entry, the M-by-N matrix A.
! 182: *> On exit, A contains the triangular matrix R, or part of R.
! 183: *> See Purpose for details.
! 184: *> \endverbatim
! 185: *>
! 186: *> \param[in] LDA
! 187: *> \verbatim
! 188: *> LDA is INTEGER
! 189: *> The leading dimension of the array A. LDA >= max(1,M).
! 190: *> \endverbatim
! 191: *>
! 192: *> \param[in,out] B
! 193: *> \verbatim
! 194: *> B is DOUBLE PRECISION array, dimension (LDB,N)
! 195: *> On entry, the P-by-N matrix B.
! 196: *> On exit, B contains the triangular matrix R if M-K-L < 0.
! 197: *> See Purpose for details.
! 198: *> \endverbatim
! 199: *>
! 200: *> \param[in] LDB
! 201: *> \verbatim
! 202: *> LDB is INTEGER
! 203: *> The leading dimension of the array B. LDB >= max(1,P).
! 204: *> \endverbatim
! 205: *>
! 206: *> \param[out] ALPHA
! 207: *> \verbatim
! 208: *> ALPHA is DOUBLE PRECISION array, dimension (N)
! 209: *> \endverbatim
! 210: *>
! 211: *> \param[out] BETA
! 212: *> \verbatim
! 213: *> BETA is DOUBLE PRECISION array, dimension (N)
! 214: *>
! 215: *> On exit, ALPHA and BETA contain the generalized singular
! 216: *> value pairs of A and B;
! 217: *> ALPHA(1:K) = 1,
! 218: *> BETA(1:K) = 0,
! 219: *> and if M-K-L >= 0,
! 220: *> ALPHA(K+1:K+L) = C,
! 221: *> BETA(K+1:K+L) = S,
! 222: *> or if M-K-L < 0,
! 223: *> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
! 224: *> BETA(K+1:M) =S, BETA(M+1:K+L) =1
! 225: *> and
! 226: *> ALPHA(K+L+1:N) = 0
! 227: *> BETA(K+L+1:N) = 0
! 228: *> \endverbatim
! 229: *>
! 230: *> \param[out] U
! 231: *> \verbatim
! 232: *> U is DOUBLE PRECISION array, dimension (LDU,M)
! 233: *> If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
! 234: *> If JOBU = 'N', U is not referenced.
! 235: *> \endverbatim
! 236: *>
! 237: *> \param[in] LDU
! 238: *> \verbatim
! 239: *> LDU is INTEGER
! 240: *> The leading dimension of the array U. LDU >= max(1,M) if
! 241: *> JOBU = 'U'; LDU >= 1 otherwise.
! 242: *> \endverbatim
! 243: *>
! 244: *> \param[out] V
! 245: *> \verbatim
! 246: *> V is DOUBLE PRECISION array, dimension (LDV,P)
! 247: *> If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
! 248: *> If JOBV = 'N', V is not referenced.
! 249: *> \endverbatim
! 250: *>
! 251: *> \param[in] LDV
! 252: *> \verbatim
! 253: *> LDV is INTEGER
! 254: *> The leading dimension of the array V. LDV >= max(1,P) if
! 255: *> JOBV = 'V'; LDV >= 1 otherwise.
! 256: *> \endverbatim
! 257: *>
! 258: *> \param[out] Q
! 259: *> \verbatim
! 260: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
! 261: *> If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
! 262: *> If JOBQ = 'N', Q is not referenced.
! 263: *> \endverbatim
! 264: *>
! 265: *> \param[in] LDQ
! 266: *> \verbatim
! 267: *> LDQ is INTEGER
! 268: *> The leading dimension of the array Q. LDQ >= max(1,N) if
! 269: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
! 270: *> \endverbatim
! 271: *>
! 272: *> \param[out] WORK
! 273: *> \verbatim
! 274: *> WORK is DOUBLE PRECISION array,
! 275: *> dimension (max(3*N,M,P)+N)
! 276: *> \endverbatim
! 277: *>
! 278: *> \param[out] IWORK
! 279: *> \verbatim
! 280: *> IWORK is INTEGER array, dimension (N)
! 281: *> On exit, IWORK stores the sorting information. More
! 282: *> precisely, the following loop will sort ALPHA
! 283: *> for I = K+1, min(M,K+L)
! 284: *> swap ALPHA(I) and ALPHA(IWORK(I))
! 285: *> endfor
! 286: *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
! 287: *> \endverbatim
! 288: *>
! 289: *> \param[out] INFO
! 290: *> \verbatim
! 291: *> INFO is INTEGER
! 292: *> = 0: successful exit
! 293: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 294: *> > 0: if INFO = 1, the Jacobi-type procedure failed to
! 295: *> converge. For further details, see subroutine DTGSJA.
! 296: *> \endverbatim
! 297: *
! 298: *> \par Internal Parameters:
! 299: * =========================
! 300: *>
! 301: *> \verbatim
! 302: *> TOLA DOUBLE PRECISION
! 303: *> TOLB DOUBLE PRECISION
! 304: *> TOLA and TOLB are the thresholds to determine the effective
! 305: *> rank of (A',B')**T. Generally, they are set to
! 306: *> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
! 307: *> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
! 308: *> The size of TOLA and TOLB may affect the size of backward
! 309: *> errors of the decomposition.
! 310: *> \endverbatim
! 311: *
! 312: * Authors:
! 313: * ========
! 314: *
! 315: *> \author Univ. of Tennessee
! 316: *> \author Univ. of California Berkeley
! 317: *> \author Univ. of Colorado Denver
! 318: *> \author NAG Ltd.
! 319: *
! 320: *> \date November 2011
! 321: *
! 322: *> \ingroup doubleOTHERsing
! 323: *
! 324: *> \par Contributors:
! 325: * ==================
! 326: *>
! 327: *> Ming Gu and Huan Ren, Computer Science Division, University of
! 328: *> California at Berkeley, USA
! 329: *>
! 330: * =====================================================================
1.1 bertrand 331: SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
332: $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
333: $ IWORK, INFO )
334: *
1.9 ! bertrand 335: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 336: * -- LAPACK is a software package provided by Univ. of Tennessee, --
337: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 338: * November 2011
1.1 bertrand 339: *
340: * .. Scalar Arguments ..
341: CHARACTER JOBQ, JOBU, JOBV
342: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
343: * ..
344: * .. Array Arguments ..
345: INTEGER IWORK( * )
346: DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
347: $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
348: $ V( LDV, * ), WORK( * )
349: * ..
350: *
351: * =====================================================================
352: *
353: * .. Local Scalars ..
354: LOGICAL WANTQ, WANTU, WANTV
355: INTEGER I, IBND, ISUB, J, NCYCLE
356: DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
357: * ..
358: * .. External Functions ..
359: LOGICAL LSAME
360: DOUBLE PRECISION DLAMCH, DLANGE
361: EXTERNAL LSAME, DLAMCH, DLANGE
362: * ..
363: * .. External Subroutines ..
364: EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA
365: * ..
366: * .. Intrinsic Functions ..
367: INTRINSIC MAX, MIN
368: * ..
369: * .. Executable Statements ..
370: *
371: * Test the input parameters
372: *
373: WANTU = LSAME( JOBU, 'U' )
374: WANTV = LSAME( JOBV, 'V' )
375: WANTQ = LSAME( JOBQ, 'Q' )
376: *
377: INFO = 0
378: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
379: INFO = -1
380: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
381: INFO = -2
382: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
383: INFO = -3
384: ELSE IF( M.LT.0 ) THEN
385: INFO = -4
386: ELSE IF( N.LT.0 ) THEN
387: INFO = -5
388: ELSE IF( P.LT.0 ) THEN
389: INFO = -6
390: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
391: INFO = -10
392: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
393: INFO = -12
394: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
395: INFO = -16
396: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
397: INFO = -18
398: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
399: INFO = -20
400: END IF
401: IF( INFO.NE.0 ) THEN
402: CALL XERBLA( 'DGGSVD', -INFO )
403: RETURN
404: END IF
405: *
406: * Compute the Frobenius norm of matrices A and B
407: *
408: ANORM = DLANGE( '1', M, N, A, LDA, WORK )
409: BNORM = DLANGE( '1', P, N, B, LDB, WORK )
410: *
411: * Get machine precision and set up threshold for determining
412: * the effective numerical rank of the matrices A and B.
413: *
414: ULP = DLAMCH( 'Precision' )
415: UNFL = DLAMCH( 'Safe Minimum' )
416: TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
417: TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
418: *
419: * Preprocessing
420: *
421: CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
422: $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
423: $ WORK( N+1 ), INFO )
424: *
425: * Compute the GSVD of two upper "triangular" matrices
426: *
427: CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
428: $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
429: $ WORK, NCYCLE, INFO )
430: *
431: * Sort the singular values and store the pivot indices in IWORK
432: * Copy ALPHA to WORK, then sort ALPHA in WORK
433: *
434: CALL DCOPY( N, ALPHA, 1, WORK, 1 )
435: IBND = MIN( L, M-K )
436: DO 20 I = 1, IBND
437: *
438: * Scan for largest ALPHA(K+I)
439: *
440: ISUB = I
441: SMAX = WORK( K+I )
442: DO 10 J = I + 1, IBND
443: TEMP = WORK( K+J )
444: IF( TEMP.GT.SMAX ) THEN
445: ISUB = J
446: SMAX = TEMP
447: END IF
448: 10 CONTINUE
449: IF( ISUB.NE.I ) THEN
450: WORK( K+ISUB ) = WORK( K+I )
451: WORK( K+I ) = SMAX
452: IWORK( K+I ) = K + ISUB
453: ELSE
454: IWORK( K+I ) = K + I
455: END IF
456: 20 CONTINUE
457: *
458: RETURN
459: *
460: * End of DGGSVD
461: *
462: END
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