LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cggsvp.f
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00001 *> \brief \b CGGSVP
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CGGSVP + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggsvp.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cggsvp.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggsvp.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
00022 *                          TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
00023 *                          IWORK, RWORK, TAU, WORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBQ, JOBU, JOBV
00027 *       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
00028 *       REAL               TOLA, TOLB
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IWORK( * )
00032 *       REAL               RWORK( * )
00033 *       COMPLEX            A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
00034 *      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> CGGSVP computes unitary matrices U, V and Q such that
00044 *>
00045 *>                    N-K-L  K    L
00046 *>  U**H*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
00047 *>                 L ( 0     0   A23 )
00048 *>             M-K-L ( 0     0    0  )
00049 *>
00050 *>                  N-K-L  K    L
00051 *>         =     K ( 0    A12  A13 )  if M-K-L < 0;
00052 *>             M-K ( 0     0   A23 )
00053 *>
00054 *>                  N-K-L  K    L
00055 *>  V**H*B*Q =   L ( 0     0   B13 )
00056 *>             P-L ( 0     0    0  )
00057 *>
00058 *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
00059 *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
00060 *> otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
00061 *> numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H. 
00062 *>
00063 *> This decomposition is the preprocessing step for computing the
00064 *> Generalized Singular Value Decomposition (GSVD), see subroutine
00065 *> CGGSVD.
00066 *> \endverbatim
00067 *
00068 *  Arguments:
00069 *  ==========
00070 *
00071 *> \param[in] JOBU
00072 *> \verbatim
00073 *>          JOBU is CHARACTER*1
00074 *>          = 'U':  Unitary matrix U is computed;
00075 *>          = 'N':  U is not computed.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] JOBV
00079 *> \verbatim
00080 *>          JOBV is CHARACTER*1
00081 *>          = 'V':  Unitary matrix V is computed;
00082 *>          = 'N':  V is not computed.
00083 *> \endverbatim
00084 *>
00085 *> \param[in] JOBQ
00086 *> \verbatim
00087 *>          JOBQ is CHARACTER*1
00088 *>          = 'Q':  Unitary matrix Q is computed;
00089 *>          = 'N':  Q is not computed.
00090 *> \endverbatim
00091 *>
00092 *> \param[in] M
00093 *> \verbatim
00094 *>          M is INTEGER
00095 *>          The number of rows of the matrix A.  M >= 0.
00096 *> \endverbatim
00097 *>
00098 *> \param[in] P
00099 *> \verbatim
00100 *>          P is INTEGER
00101 *>          The number of rows of the matrix B.  P >= 0.
00102 *> \endverbatim
00103 *>
00104 *> \param[in] N
00105 *> \verbatim
00106 *>          N is INTEGER
00107 *>          The number of columns of the matrices A and B.  N >= 0.
00108 *> \endverbatim
00109 *>
00110 *> \param[in,out] A
00111 *> \verbatim
00112 *>          A is COMPLEX array, dimension (LDA,N)
00113 *>          On entry, the M-by-N matrix A.
00114 *>          On exit, A contains the triangular (or trapezoidal) matrix
00115 *>          described in the Purpose section.
00116 *> \endverbatim
00117 *>
00118 *> \param[in] LDA
00119 *> \verbatim
00120 *>          LDA is INTEGER
00121 *>          The leading dimension of the array A. LDA >= max(1,M).
00122 *> \endverbatim
00123 *>
00124 *> \param[in,out] B
00125 *> \verbatim
00126 *>          B is COMPLEX array, dimension (LDB,N)
00127 *>          On entry, the P-by-N matrix B.
00128 *>          On exit, B contains the triangular matrix described in
00129 *>          the Purpose section.
00130 *> \endverbatim
00131 *>
00132 *> \param[in] LDB
00133 *> \verbatim
00134 *>          LDB is INTEGER
00135 *>          The leading dimension of the array B. LDB >= max(1,P).
00136 *> \endverbatim
00137 *>
00138 *> \param[in] TOLA
00139 *> \verbatim
00140 *>          TOLA is REAL
00141 *> \endverbatim
00142 *>
00143 *> \param[in] TOLB
00144 *> \verbatim
00145 *>          TOLB is REAL
00146 *>
00147 *>          TOLA and TOLB are the thresholds to determine the effective
00148 *>          numerical rank of matrix B and a subblock of A. Generally,
00149 *>          they are set to
00150 *>             TOLA = MAX(M,N)*norm(A)*MACHEPS,
00151 *>             TOLB = MAX(P,N)*norm(B)*MACHEPS.
00152 *>          The size of TOLA and TOLB may affect the size of backward
00153 *>          errors of the decomposition.
00154 *> \endverbatim
00155 *>
00156 *> \param[out] K
00157 *> \verbatim
00158 *>          K is INTEGER
00159 *> \endverbatim
00160 *>
00161 *> \param[out] L
00162 *> \verbatim
00163 *>          L is INTEGER
00164 *>
00165 *>          On exit, K and L specify the dimension of the subblocks
00166 *>          described in Purpose section.
00167 *>          K + L = effective numerical rank of (A**H,B**H)**H.
00168 *> \endverbatim
00169 *>
00170 *> \param[out] U
00171 *> \verbatim
00172 *>          U is COMPLEX array, dimension (LDU,M)
00173 *>          If JOBU = 'U', U contains the unitary matrix U.
00174 *>          If JOBU = 'N', U is not referenced.
00175 *> \endverbatim
00176 *>
00177 *> \param[in] LDU
00178 *> \verbatim
00179 *>          LDU is INTEGER
00180 *>          The leading dimension of the array U. LDU >= max(1,M) if
00181 *>          JOBU = 'U'; LDU >= 1 otherwise.
00182 *> \endverbatim
00183 *>
00184 *> \param[out] V
00185 *> \verbatim
00186 *>          V is COMPLEX array, dimension (LDV,P)
00187 *>          If JOBV = 'V', V contains the unitary matrix V.
00188 *>          If JOBV = 'N', V is not referenced.
00189 *> \endverbatim
00190 *>
00191 *> \param[in] LDV
00192 *> \verbatim
00193 *>          LDV is INTEGER
00194 *>          The leading dimension of the array V. LDV >= max(1,P) if
00195 *>          JOBV = 'V'; LDV >= 1 otherwise.
00196 *> \endverbatim
00197 *>
00198 *> \param[out] Q
00199 *> \verbatim
00200 *>          Q is COMPLEX array, dimension (LDQ,N)
00201 *>          If JOBQ = 'Q', Q contains the unitary matrix Q.
00202 *>          If JOBQ = 'N', Q is not referenced.
00203 *> \endverbatim
00204 *>
00205 *> \param[in] LDQ
00206 *> \verbatim
00207 *>          LDQ is INTEGER
00208 *>          The leading dimension of the array Q. LDQ >= max(1,N) if
00209 *>          JOBQ = 'Q'; LDQ >= 1 otherwise.
00210 *> \endverbatim
00211 *>
00212 *> \param[out] IWORK
00213 *> \verbatim
00214 *>          IWORK is INTEGER array, dimension (N)
00215 *> \endverbatim
00216 *>
00217 *> \param[out] RWORK
00218 *> \verbatim
00219 *>          RWORK is REAL array, dimension (2*N)
00220 *> \endverbatim
00221 *>
00222 *> \param[out] TAU
00223 *> \verbatim
00224 *>          TAU is COMPLEX array, dimension (N)
00225 *> \endverbatim
00226 *>
00227 *> \param[out] WORK
00228 *> \verbatim
00229 *>          WORK is COMPLEX array, dimension (max(3*N,M,P))
00230 *> \endverbatim
00231 *>
00232 *> \param[out] INFO
00233 *> \verbatim
00234 *>          INFO is INTEGER
00235 *>          = 0:  successful exit
00236 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00237 *> \endverbatim
00238 *
00239 *  Authors:
00240 *  ========
00241 *
00242 *> \author Univ. of Tennessee 
00243 *> \author Univ. of California Berkeley 
00244 *> \author Univ. of Colorado Denver 
00245 *> \author NAG Ltd. 
00246 *
00247 *> \date November 2011
00248 *
00249 *> \ingroup complexOTHERcomputational
00250 *
00251 *> \par Further Details:
00252 *  =====================
00253 *>
00254 *>  The subroutine uses LAPACK subroutine CGEQPF for the QR factorization
00255 *>  with column pivoting to detect the effective numerical rank of the
00256 *>  a matrix. It may be replaced by a better rank determination strategy.
00257 *>
00258 *  =====================================================================
00259       SUBROUTINE CGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
00260      $                   TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
00261      $                   IWORK, RWORK, TAU, WORK, INFO )
00262 *
00263 *  -- LAPACK computational routine (version 3.4.0) --
00264 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00265 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00266 *     November 2011
00267 *
00268 *     .. Scalar Arguments ..
00269       CHARACTER          JOBQ, JOBU, JOBV
00270       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
00271       REAL               TOLA, TOLB
00272 *     ..
00273 *     .. Array Arguments ..
00274       INTEGER            IWORK( * )
00275       REAL               RWORK( * )
00276       COMPLEX            A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
00277      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
00278 *     ..
00279 *
00280 *  =====================================================================
00281 *
00282 *     .. Parameters ..
00283       COMPLEX            CZERO, CONE
00284       PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+0 ),
00285      $                   CONE = ( 1.0E+0, 0.0E+0 ) )
00286 *     ..
00287 *     .. Local Scalars ..
00288       LOGICAL            FORWRD, WANTQ, WANTU, WANTV
00289       INTEGER            I, J
00290       COMPLEX            T
00291 *     ..
00292 *     .. External Functions ..
00293       LOGICAL            LSAME
00294       EXTERNAL           LSAME
00295 *     ..
00296 *     .. External Subroutines ..
00297       EXTERNAL           CGEQPF, CGEQR2, CGERQ2, CLACPY, CLAPMT, CLASET,
00298      $                   CUNG2R, CUNM2R, CUNMR2, XERBLA
00299 *     ..
00300 *     .. Intrinsic Functions ..
00301       INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
00302 *     ..
00303 *     .. Statement Functions ..
00304       REAL               CABS1
00305 *     ..
00306 *     .. Statement Function definitions ..
00307       CABS1( T ) = ABS( REAL( T ) ) + ABS( AIMAG( T ) )
00308 *     ..
00309 *     .. Executable Statements ..
00310 *
00311 *     Test the input parameters
00312 *
00313       WANTU = LSAME( JOBU, 'U' )
00314       WANTV = LSAME( JOBV, 'V' )
00315       WANTQ = LSAME( JOBQ, 'Q' )
00316       FORWRD = .TRUE.
00317 *
00318       INFO = 0
00319       IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
00320          INFO = -1
00321       ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
00322          INFO = -2
00323       ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
00324          INFO = -3
00325       ELSE IF( M.LT.0 ) THEN
00326          INFO = -4
00327       ELSE IF( P.LT.0 ) THEN
00328          INFO = -5
00329       ELSE IF( N.LT.0 ) THEN
00330          INFO = -6
00331       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00332          INFO = -8
00333       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
00334          INFO = -10
00335       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
00336          INFO = -16
00337       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
00338          INFO = -18
00339       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
00340          INFO = -20
00341       END IF
00342       IF( INFO.NE.0 ) THEN
00343          CALL XERBLA( 'CGGSVP', -INFO )
00344          RETURN
00345       END IF
00346 *
00347 *     QR with column pivoting of B: B*P = V*( S11 S12 )
00348 *                                           (  0   0  )
00349 *
00350       DO 10 I = 1, N
00351          IWORK( I ) = 0
00352    10 CONTINUE
00353       CALL CGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO )
00354 *
00355 *     Update A := A*P
00356 *
00357       CALL CLAPMT( FORWRD, M, N, A, LDA, IWORK )
00358 *
00359 *     Determine the effective rank of matrix B.
00360 *
00361       L = 0
00362       DO 20 I = 1, MIN( P, N )
00363          IF( CABS1( B( I, I ) ).GT.TOLB )
00364      $      L = L + 1
00365    20 CONTINUE
00366 *
00367       IF( WANTV ) THEN
00368 *
00369 *        Copy the details of V, and form V.
00370 *
00371          CALL CLASET( 'Full', P, P, CZERO, CZERO, V, LDV )
00372          IF( P.GT.1 )
00373      $      CALL CLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
00374      $                   LDV )
00375          CALL CUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
00376       END IF
00377 *
00378 *     Clean up B
00379 *
00380       DO 40 J = 1, L - 1
00381          DO 30 I = J + 1, L
00382             B( I, J ) = CZERO
00383    30    CONTINUE
00384    40 CONTINUE
00385       IF( P.GT.L )
00386      $   CALL CLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB )
00387 *
00388       IF( WANTQ ) THEN
00389 *
00390 *        Set Q = I and Update Q := Q*P
00391 *
00392          CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
00393          CALL CLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
00394       END IF
00395 *
00396       IF( P.GE.L .AND. N.NE.L ) THEN
00397 *
00398 *        RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
00399 *
00400          CALL CGERQ2( L, N, B, LDB, TAU, WORK, INFO )
00401 *
00402 *        Update A := A*Z**H
00403 *
00404          CALL CUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB,
00405      $                TAU, A, LDA, WORK, INFO )
00406          IF( WANTQ ) THEN
00407 *
00408 *           Update Q := Q*Z**H
00409 *
00410             CALL CUNMR2( 'Right', 'Conjugate transpose', N, N, L, B,
00411      $                   LDB, TAU, Q, LDQ, WORK, INFO )
00412          END IF
00413 *
00414 *        Clean up B
00415 *
00416          CALL CLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB )
00417          DO 60 J = N - L + 1, N
00418             DO 50 I = J - N + L + 1, L
00419                B( I, J ) = CZERO
00420    50       CONTINUE
00421    60    CONTINUE
00422 *
00423       END IF
00424 *
00425 *     Let              N-L     L
00426 *                A = ( A11    A12 ) M,
00427 *
00428 *     then the following does the complete QR decomposition of A11:
00429 *
00430 *              A11 = U*(  0  T12 )*P1**H
00431 *                      (  0   0  )
00432 *
00433       DO 70 I = 1, N - L
00434          IWORK( I ) = 0
00435    70 CONTINUE
00436       CALL CGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO )
00437 *
00438 *     Determine the effective rank of A11
00439 *
00440       K = 0
00441       DO 80 I = 1, MIN( M, N-L )
00442          IF( CABS1( A( I, I ) ).GT.TOLA )
00443      $      K = K + 1
00444    80 CONTINUE
00445 *
00446 *     Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
00447 *
00448       CALL CUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ),
00449      $             A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
00450 *
00451       IF( WANTU ) THEN
00452 *
00453 *        Copy the details of U, and form U
00454 *
00455          CALL CLASET( 'Full', M, M, CZERO, CZERO, U, LDU )
00456          IF( M.GT.1 )
00457      $      CALL CLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
00458      $                   LDU )
00459          CALL CUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
00460       END IF
00461 *
00462       IF( WANTQ ) THEN
00463 *
00464 *        Update Q( 1:N, 1:N-L )  = Q( 1:N, 1:N-L )*P1
00465 *
00466          CALL CLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
00467       END IF
00468 *
00469 *     Clean up A: set the strictly lower triangular part of
00470 *     A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
00471 *
00472       DO 100 J = 1, K - 1
00473          DO 90 I = J + 1, K
00474             A( I, J ) = CZERO
00475    90    CONTINUE
00476   100 CONTINUE
00477       IF( M.GT.K )
00478      $   CALL CLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA )
00479 *
00480       IF( N-L.GT.K ) THEN
00481 *
00482 *        RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
00483 *
00484          CALL CGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
00485 *
00486          IF( WANTQ ) THEN
00487 *
00488 *           Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
00489 *
00490             CALL CUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A,
00491      $                   LDA, TAU, Q, LDQ, WORK, INFO )
00492          END IF
00493 *
00494 *        Clean up A
00495 *
00496          CALL CLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA )
00497          DO 120 J = N - L - K + 1, N - L
00498             DO 110 I = J - N + L + K + 1, K
00499                A( I, J ) = CZERO
00500   110       CONTINUE
00501   120    CONTINUE
00502 *
00503       END IF
00504 *
00505       IF( M.GT.K ) THEN
00506 *
00507 *        QR factorization of A( K+1:M,N-L+1:N )
00508 *
00509          CALL CGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
00510 *
00511          IF( WANTU ) THEN
00512 *
00513 *           Update U(:,K+1:M) := U(:,K+1:M)*U1
00514 *
00515             CALL CUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
00516      $                   A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
00517      $                   WORK, INFO )
00518          END IF
00519 *
00520 *        Clean up
00521 *
00522          DO 140 J = N - L + 1, N
00523             DO 130 I = J - N + K + L + 1, M
00524                A( I, J ) = CZERO
00525   130       CONTINUE
00526   140    CONTINUE
00527 *
00528       END IF
00529 *
00530       RETURN
00531 *
00532 *     End of CGGSVP
00533 *
00534       END
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