Annotation of rpl/lapack/lapack/zggsvd3.f, revision 1.1
1.1 ! bertrand 1: *> \brief <b> ZGGSVD3 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 ZGGSVD3 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggsvd3.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggsvd3.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggsvd3.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
! 22: * LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
! 23: * LWORK, RWORK, IWORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER JOBQ, JOBU, JOBV
! 27: * INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * INTEGER IWORK( * )
! 31: * DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
! 32: * COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 33: * $ U( LDU, * ), V( LDV, * ), WORK( * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> ZGGSVD3 computes the generalized singular value decomposition (GSVD)
! 43: *> of an M-by-N complex matrix A and P-by-N complex matrix B:
! 44: *>
! 45: *> U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
! 46: *>
! 47: *> where U, V and Q are unitary matrices.
! 48: *> Let K+L = the effective numerical rank of the
! 49: *> matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
! 50: *> triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
! 51: *> matrices and of the 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: *> where
! 68: *>
! 69: *> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
! 70: *> S = diag( BETA(K+1), ... , BETA(K+L) ),
! 71: *> C**2 + S**2 = I.
! 72: *>
! 73: *> R is stored in A(1:K+L,N-K-L+1:N) on exit.
! 74: *>
! 75: *> If M-K-L < 0,
! 76: *>
! 77: *> K M-K K+L-M
! 78: *> D1 = K ( I 0 0 )
! 79: *> M-K ( 0 C 0 )
! 80: *>
! 81: *> K M-K K+L-M
! 82: *> D2 = M-K ( 0 S 0 )
! 83: *> K+L-M ( 0 0 I )
! 84: *> P-L ( 0 0 0 )
! 85: *>
! 86: *> N-K-L K M-K K+L-M
! 87: *> ( 0 R ) = K ( 0 R11 R12 R13 )
! 88: *> M-K ( 0 0 R22 R23 )
! 89: *> K+L-M ( 0 0 0 R33 )
! 90: *>
! 91: *> where
! 92: *>
! 93: *> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
! 94: *> S = diag( BETA(K+1), ... , BETA(M) ),
! 95: *> C**2 + S**2 = I.
! 96: *>
! 97: *> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
! 98: *> ( 0 R22 R23 )
! 99: *> in B(M-K+1:L,N+M-K-L+1:N) on exit.
! 100: *>
! 101: *> The routine computes C, S, R, and optionally the unitary
! 102: *> transformation matrices U, V and Q.
! 103: *>
! 104: *> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
! 105: *> A and B implicitly gives the SVD of A*inv(B):
! 106: *> A*inv(B) = U*(D1*inv(D2))*V**H.
! 107: *> If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
! 108: *> equal to the CS decomposition of A and B. Furthermore, the GSVD can
! 109: *> be used to derive the solution of the eigenvalue problem:
! 110: *> A**H*A x = lambda* B**H*B x.
! 111: *> In some literature, the GSVD of A and B is presented in the form
! 112: *> U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
! 113: *> where U and V are orthogonal and X is nonsingular, and D1 and D2 are
! 114: *> ``diagonal''. The former GSVD form can be converted to the latter
! 115: *> form by taking the nonsingular matrix X as
! 116: *>
! 117: *> X = Q*( I 0 )
! 118: *> ( 0 inv(R) )
! 119: *> \endverbatim
! 120: *
! 121: * Arguments:
! 122: * ==========
! 123: *
! 124: *> \param[in] JOBU
! 125: *> \verbatim
! 126: *> JOBU is CHARACTER*1
! 127: *> = 'U': Unitary matrix U is computed;
! 128: *> = 'N': U is not computed.
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[in] JOBV
! 132: *> \verbatim
! 133: *> JOBV is CHARACTER*1
! 134: *> = 'V': Unitary matrix V is computed;
! 135: *> = 'N': V is not computed.
! 136: *> \endverbatim
! 137: *>
! 138: *> \param[in] JOBQ
! 139: *> \verbatim
! 140: *> JOBQ is CHARACTER*1
! 141: *> = 'Q': Unitary matrix Q is computed;
! 142: *> = 'N': Q is not computed.
! 143: *> \endverbatim
! 144: *>
! 145: *> \param[in] M
! 146: *> \verbatim
! 147: *> M is INTEGER
! 148: *> The number of rows of the matrix A. M >= 0.
! 149: *> \endverbatim
! 150: *>
! 151: *> \param[in] N
! 152: *> \verbatim
! 153: *> N is INTEGER
! 154: *> The number of columns of the matrices A and B. N >= 0.
! 155: *> \endverbatim
! 156: *>
! 157: *> \param[in] P
! 158: *> \verbatim
! 159: *> P is INTEGER
! 160: *> The number of rows of the matrix B. P >= 0.
! 161: *> \endverbatim
! 162: *>
! 163: *> \param[out] K
! 164: *> \verbatim
! 165: *> K is INTEGER
! 166: *> \endverbatim
! 167: *>
! 168: *> \param[out] L
! 169: *> \verbatim
! 170: *> L is INTEGER
! 171: *>
! 172: *> On exit, K and L specify the dimension of the subblocks
! 173: *> described in Purpose.
! 174: *> K + L = effective numerical rank of (A**H,B**H)**H.
! 175: *> \endverbatim
! 176: *>
! 177: *> \param[in,out] A
! 178: *> \verbatim
! 179: *> A is COMPLEX*16 array, dimension (LDA,N)
! 180: *> On entry, the M-by-N matrix A.
! 181: *> On exit, A contains the triangular matrix R, or part of R.
! 182: *> See Purpose for details.
! 183: *> \endverbatim
! 184: *>
! 185: *> \param[in] LDA
! 186: *> \verbatim
! 187: *> LDA is INTEGER
! 188: *> The leading dimension of the array A. LDA >= max(1,M).
! 189: *> \endverbatim
! 190: *>
! 191: *> \param[in,out] B
! 192: *> \verbatim
! 193: *> B is COMPLEX*16 array, dimension (LDB,N)
! 194: *> On entry, the P-by-N matrix B.
! 195: *> On exit, B contains part of the triangular matrix R if
! 196: *> M-K-L < 0. See Purpose for details.
! 197: *> \endverbatim
! 198: *>
! 199: *> \param[in] LDB
! 200: *> \verbatim
! 201: *> LDB is INTEGER
! 202: *> The leading dimension of the array B. LDB >= max(1,P).
! 203: *> \endverbatim
! 204: *>
! 205: *> \param[out] ALPHA
! 206: *> \verbatim
! 207: *> ALPHA is DOUBLE PRECISION array, dimension (N)
! 208: *> \endverbatim
! 209: *>
! 210: *> \param[out] BETA
! 211: *> \verbatim
! 212: *> BETA is DOUBLE PRECISION array, dimension (N)
! 213: *>
! 214: *> On exit, ALPHA and BETA contain the generalized singular
! 215: *> value pairs of A and B;
! 216: *> ALPHA(1:K) = 1,
! 217: *> BETA(1:K) = 0,
! 218: *> and if M-K-L >= 0,
! 219: *> ALPHA(K+1:K+L) = C,
! 220: *> BETA(K+1:K+L) = S,
! 221: *> or if M-K-L < 0,
! 222: *> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
! 223: *> BETA(K+1:M) =S, BETA(M+1:K+L) =1
! 224: *> and
! 225: *> ALPHA(K+L+1:N) = 0
! 226: *> BETA(K+L+1:N) = 0
! 227: *> \endverbatim
! 228: *>
! 229: *> \param[out] U
! 230: *> \verbatim
! 231: *> U is COMPLEX*16 array, dimension (LDU,M)
! 232: *> If JOBU = 'U', U contains the M-by-M unitary matrix U.
! 233: *> If JOBU = 'N', U is not referenced.
! 234: *> \endverbatim
! 235: *>
! 236: *> \param[in] LDU
! 237: *> \verbatim
! 238: *> LDU is INTEGER
! 239: *> The leading dimension of the array U. LDU >= max(1,M) if
! 240: *> JOBU = 'U'; LDU >= 1 otherwise.
! 241: *> \endverbatim
! 242: *>
! 243: *> \param[out] V
! 244: *> \verbatim
! 245: *> V is COMPLEX*16 array, dimension (LDV,P)
! 246: *> If JOBV = 'V', V contains the P-by-P unitary matrix V.
! 247: *> If JOBV = 'N', V is not referenced.
! 248: *> \endverbatim
! 249: *>
! 250: *> \param[in] LDV
! 251: *> \verbatim
! 252: *> LDV is INTEGER
! 253: *> The leading dimension of the array V. LDV >= max(1,P) if
! 254: *> JOBV = 'V'; LDV >= 1 otherwise.
! 255: *> \endverbatim
! 256: *>
! 257: *> \param[out] Q
! 258: *> \verbatim
! 259: *> Q is COMPLEX*16 array, dimension (LDQ,N)
! 260: *> If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
! 261: *> If JOBQ = 'N', Q is not referenced.
! 262: *> \endverbatim
! 263: *>
! 264: *> \param[in] LDQ
! 265: *> \verbatim
! 266: *> LDQ is INTEGER
! 267: *> The leading dimension of the array Q. LDQ >= max(1,N) if
! 268: *> JOBQ = 'Q'; LDQ >= 1 otherwise.
! 269: *> \endverbatim
! 270: *>
! 271: *> \param[out] WORK
! 272: *> \verbatim
! 273: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
! 274: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 275: *> \endverbatim
! 276: *>
! 277: *> \param[in] LWORK
! 278: *> \verbatim
! 279: *> LWORK is INTEGER
! 280: *> The dimension of the array WORK.
! 281: *>
! 282: *> If LWORK = -1, then a workspace query is assumed; the routine
! 283: *> only calculates the optimal size of the WORK array, returns
! 284: *> this value as the first entry of the WORK array, and no error
! 285: *> message related to LWORK is issued by XERBLA.
! 286: *> \endverbatim
! 287: *>
! 288: *> \param[out] RWORK
! 289: *> \verbatim
! 290: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
! 291: *> \endverbatim
! 292: *>
! 293: *> \param[out] IWORK
! 294: *> \verbatim
! 295: *> IWORK is INTEGER array, dimension (N)
! 296: *> On exit, IWORK stores the sorting information. More
! 297: *> precisely, the following loop will sort ALPHA
! 298: *> for I = K+1, min(M,K+L)
! 299: *> swap ALPHA(I) and ALPHA(IWORK(I))
! 300: *> endfor
! 301: *> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
! 302: *> \endverbatim
! 303: *>
! 304: *> \param[out] INFO
! 305: *> \verbatim
! 306: *> INFO is INTEGER
! 307: *> = 0: successful exit.
! 308: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 309: *> > 0: if INFO = 1, the Jacobi-type procedure failed to
! 310: *> converge. For further details, see subroutine ZTGSJA.
! 311: *> \endverbatim
! 312: *
! 313: *> \par Internal Parameters:
! 314: * =========================
! 315: *>
! 316: *> \verbatim
! 317: *> TOLA DOUBLE PRECISION
! 318: *> TOLB DOUBLE PRECISION
! 319: *> TOLA and TOLB are the thresholds to determine the effective
! 320: *> rank of (A**H,B**H)**H. Generally, they are set to
! 321: *> TOLA = MAX(M,N)*norm(A)*MACHEPS,
! 322: *> TOLB = MAX(P,N)*norm(B)*MACHEPS.
! 323: *> The size of TOLA and TOLB may affect the size of backward
! 324: *> errors of the decomposition.
! 325: *> \endverbatim
! 326: *
! 327: * Authors:
! 328: * ========
! 329: *
! 330: *> \author Univ. of Tennessee
! 331: *> \author Univ. of California Berkeley
! 332: *> \author Univ. of Colorado Denver
! 333: *> \author NAG Ltd.
! 334: *
! 335: *> \date August 2015
! 336: *
! 337: *> \ingroup complex16OTHERsing
! 338: *
! 339: *> \par Contributors:
! 340: * ==================
! 341: *>
! 342: *> Ming Gu and Huan Ren, Computer Science Division, University of
! 343: *> California at Berkeley, USA
! 344: *>
! 345: *
! 346: *> \par Further Details:
! 347: * =====================
! 348: *>
! 349: *> ZGGSVD3 replaces the deprecated subroutine ZGGSVD.
! 350: *>
! 351: * =====================================================================
! 352: SUBROUTINE ZGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
! 353: $ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
! 354: $ WORK, LWORK, RWORK, IWORK, INFO )
! 355: *
! 356: * -- LAPACK driver routine (version 3.6.0) --
! 357: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 358: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 359: * August 2015
! 360: *
! 361: * .. Scalar Arguments ..
! 362: CHARACTER JOBQ, JOBU, JOBV
! 363: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
! 364: $ LWORK
! 365: * ..
! 366: * .. Array Arguments ..
! 367: INTEGER IWORK( * )
! 368: DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
! 369: COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
! 370: $ U( LDU, * ), V( LDV, * ), WORK( * )
! 371: * ..
! 372: *
! 373: * =====================================================================
! 374: *
! 375: * .. Local Scalars ..
! 376: LOGICAL WANTQ, WANTU, WANTV, LQUERY
! 377: INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
! 378: DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
! 379: * ..
! 380: * .. External Functions ..
! 381: LOGICAL LSAME
! 382: DOUBLE PRECISION DLAMCH, ZLANGE
! 383: EXTERNAL LSAME, DLAMCH, ZLANGE
! 384: * ..
! 385: * .. External Subroutines ..
! 386: EXTERNAL DCOPY, XERBLA, ZGGSVP, ZTGSJA
! 387: * ..
! 388: * .. Intrinsic Functions ..
! 389: INTRINSIC MAX, MIN
! 390: * ..
! 391: * .. Executable Statements ..
! 392: *
! 393: * Decode and test the input parameters
! 394: *
! 395: WANTU = LSAME( JOBU, 'U' )
! 396: WANTV = LSAME( JOBV, 'V' )
! 397: WANTQ = LSAME( JOBQ, 'Q' )
! 398: LQUERY = ( LWORK.EQ.-1 )
! 399: LWKOPT = 1
! 400: *
! 401: * Test the input arguments
! 402: *
! 403: INFO = 0
! 404: IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
! 405: INFO = -1
! 406: ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
! 407: INFO = -2
! 408: ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
! 409: INFO = -3
! 410: ELSE IF( M.LT.0 ) THEN
! 411: INFO = -4
! 412: ELSE IF( N.LT.0 ) THEN
! 413: INFO = -5
! 414: ELSE IF( P.LT.0 ) THEN
! 415: INFO = -6
! 416: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 417: INFO = -10
! 418: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
! 419: INFO = -12
! 420: ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
! 421: INFO = -16
! 422: ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
! 423: INFO = -18
! 424: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
! 425: INFO = -20
! 426: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
! 427: INFO = -24
! 428: END IF
! 429: *
! 430: * Compute workspace
! 431: *
! 432: IF( INFO.EQ.0 ) THEN
! 433: CALL ZGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
! 434: $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
! 435: $ WORK, WORK, -1, INFO )
! 436: LWKOPT = N + INT( WORK( 1 ) )
! 437: LWKOPT = MAX( 2*N, LWKOPT )
! 438: LWKOPT = MAX( 1, LWKOPT )
! 439: WORK( 1 ) = DCMPLX( LWKOPT )
! 440: END IF
! 441: *
! 442: IF( INFO.NE.0 ) THEN
! 443: CALL XERBLA( 'ZGGSVD3', -INFO )
! 444: RETURN
! 445: END IF
! 446: IF( LQUERY ) THEN
! 447: RETURN
! 448: ENDIF
! 449: *
! 450: * Compute the Frobenius norm of matrices A and B
! 451: *
! 452: ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
! 453: BNORM = ZLANGE( '1', P, N, B, LDB, RWORK )
! 454: *
! 455: * Get machine precision and set up threshold for determining
! 456: * the effective numerical rank of the matrices A and B.
! 457: *
! 458: ULP = DLAMCH( 'Precision' )
! 459: UNFL = DLAMCH( 'Safe Minimum' )
! 460: TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
! 461: TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
! 462: *
! 463: CALL ZGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
! 464: $ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK,
! 465: $ WORK, WORK( N+1 ), LWORK-N, INFO )
! 466: *
! 467: * Compute the GSVD of two upper "triangular" matrices
! 468: *
! 469: CALL ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
! 470: $ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
! 471: $ WORK, NCYCLE, INFO )
! 472: *
! 473: * Sort the singular values and store the pivot indices in IWORK
! 474: * Copy ALPHA to RWORK, then sort ALPHA in RWORK
! 475: *
! 476: CALL DCOPY( N, ALPHA, 1, RWORK, 1 )
! 477: IBND = MIN( L, M-K )
! 478: DO 20 I = 1, IBND
! 479: *
! 480: * Scan for largest ALPHA(K+I)
! 481: *
! 482: ISUB = I
! 483: SMAX = RWORK( K+I )
! 484: DO 10 J = I + 1, IBND
! 485: TEMP = RWORK( K+J )
! 486: IF( TEMP.GT.SMAX ) THEN
! 487: ISUB = J
! 488: SMAX = TEMP
! 489: END IF
! 490: 10 CONTINUE
! 491: IF( ISUB.NE.I ) THEN
! 492: RWORK( K+ISUB ) = RWORK( K+I )
! 493: RWORK( K+I ) = SMAX
! 494: IWORK( K+I ) = K + ISUB
! 495: ELSE
! 496: IWORK( K+I ) = K + I
! 497: END IF
! 498: 20 CONTINUE
! 499: *
! 500: WORK( 1 ) = DCMPLX( LWKOPT )
! 501: RETURN
! 502: *
! 503: * End of ZGGSVD3
! 504: *
! 505: END
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