LAPACK: SRC/zggsvp.f Source File

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00001       SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
00002      $                   TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
00003      $                   IWORK, RWORK, TAU, WORK, INFO )
00004 *
00005 *  -- LAPACK routine (version 3.2) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *     November 2006
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          JOBQ, JOBU, JOBV
00012       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
00013       DOUBLE PRECISION   TOLA, TOLB
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IWORK( * )
00017       DOUBLE PRECISION   RWORK( * )
00018       COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
00019      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  ZGGSVP computes unitary matrices U, V and Q such that
00026 *
00027 *                   N-K-L  K    L
00028 *   U'*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
00029 *                L ( 0     0   A23 )
00030 *            M-K-L ( 0     0    0  )
00031 *
00032 *                   N-K-L  K    L
00033 *          =     K ( 0    A12  A13 )  if M-K-L < 0;
00034 *              M-K ( 0     0   A23 )
00035 *
00036 *                 N-K-L  K    L
00037 *   V'*B*Q =   L ( 0     0   B13 )
00038 *            P-L ( 0     0    0  )
00039 *
00040 *  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
00041 *  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
00042 *  otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
00043 *  numerical rank of the (M+P)-by-N matrix (A',B')'.  Z' denotes the
00044 *  conjugate transpose of Z.
00045 *
00046 *  This decomposition is the preprocessing step for computing the
00047 *  Generalized Singular Value Decomposition (GSVD), see subroutine
00048 *  ZGGSVD.
00049 *
00050 *  Arguments
00051 *  =========
00052 *
00053 *  JOBU    (input) CHARACTER*1
00054 *          = 'U':  Unitary matrix U is computed;
00055 *          = 'N':  U is not computed.
00056 *
00057 *  JOBV    (input) CHARACTER*1
00058 *          = 'V':  Unitary matrix V is computed;
00059 *          = 'N':  V is not computed.
00060 *
00061 *  JOBQ    (input) CHARACTER*1
00062 *          = 'Q':  Unitary matrix Q is computed;
00063 *          = 'N':  Q is not computed.
00064 *
00065 *  M       (input) INTEGER
00066 *          The number of rows of the matrix A.  M >= 0.
00067 *
00068 *  P       (input) INTEGER
00069 *          The number of rows of the matrix B.  P >= 0.
00070 *
00071 *  N       (input) INTEGER
00072 *          The number of columns of the matrices A and B.  N >= 0.
00073 *
00074 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
00075 *          On entry, the M-by-N matrix A.
00076 *          On exit, A contains the triangular (or trapezoidal) matrix
00077 *          described in the Purpose section.
00078 *
00079 *  LDA     (input) INTEGER
00080 *          The leading dimension of the array A. LDA >= max(1,M).
00081 *
00082 *  B       (input/output) COMPLEX*16 array, dimension (LDB,N)
00083 *          On entry, the P-by-N matrix B.
00084 *          On exit, B contains the triangular matrix described in
00085 *          the Purpose section.
00086 *
00087 *  LDB     (input) INTEGER
00088 *          The leading dimension of the array B. LDB >= max(1,P).
00089 *
00090 *  TOLA    (input) DOUBLE PRECISION
00091 *  TOLB    (input) DOUBLE PRECISION
00092 *          TOLA and TOLB are the thresholds to determine the effective
00093 *          numerical rank of matrix B and a subblock of A. Generally,
00094 *          they are set to
00095 *             TOLA = MAX(M,N)*norm(A)*MAZHEPS,
00096 *             TOLB = MAX(P,N)*norm(B)*MAZHEPS.
00097 *          The size of TOLA and TOLB may affect the size of backward
00098 *          errors of the decomposition.
00099 *
00100 *  K       (output) INTEGER
00101 *  L       (output) INTEGER
00102 *          On exit, K and L specify the dimension of the subblocks
00103 *          described in Purpose section.
00104 *          K + L = effective numerical rank of (A',B')'.
00105 *
00106 *  U       (output) COMPLEX*16 array, dimension (LDU,M)
00107 *          If JOBU = 'U', U contains the unitary matrix U.
00108 *          If JOBU = 'N', U is not referenced.
00109 *
00110 *  LDU     (input) INTEGER
00111 *          The leading dimension of the array U. LDU >= max(1,M) if
00112 *          JOBU = 'U'; LDU >= 1 otherwise.
00113 *
00114 *  V       (output) COMPLEX*16 array, dimension (LDV,P)
00115 *          If JOBV = 'V', V contains the unitary matrix V.
00116 *          If JOBV = 'N', V is not referenced.
00117 *
00118 *  LDV     (input) INTEGER
00119 *          The leading dimension of the array V. LDV >= max(1,P) if
00120 *          JOBV = 'V'; LDV >= 1 otherwise.
00121 *
00122 *  Q       (output) COMPLEX*16 array, dimension (LDQ,N)
00123 *          If JOBQ = 'Q', Q contains the unitary matrix Q.
00124 *          If JOBQ = 'N', Q is not referenced.
00125 *
00126 *  LDQ     (input) INTEGER
00127 *          The leading dimension of the array Q. LDQ >= max(1,N) if
00128 *          JOBQ = 'Q'; LDQ >= 1 otherwise.
00129 *
00130 *  IWORK   (workspace) INTEGER array, dimension (N)
00131 *
00132 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
00133 *
00134 *  TAU     (workspace) COMPLEX*16 array, dimension (N)
00135 *
00136 *  WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P))
00137 *
00138 *  INFO    (output) INTEGER
00139 *          = 0:  successful exit
00140 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00141 *
00142 *  Further Details
00143 *  ===============
00144 *
00145 *  The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
00146 *  with column pivoting to detect the effective numerical rank of the
00147 *  a matrix. It may be replaced by a better rank determination strategy.
00148 *
00149 *  =====================================================================
00150 *
00151 *     .. Parameters ..
00152       COMPLEX*16         CZERO, CONE
00153       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
00154      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
00155 *     ..
00156 *     .. Local Scalars ..
00157       LOGICAL            FORWRD, WANTQ, WANTU, WANTV
00158       INTEGER            I, J
00159       COMPLEX*16         T
00160 *     ..
00161 *     .. External Functions ..
00162       LOGICAL            LSAME
00163       EXTERNAL           LSAME
00164 *     ..
00165 *     .. External Subroutines ..
00166       EXTERNAL           XERBLA, ZGEQPF, ZGEQR2, ZGERQ2, ZLACPY, ZLAPMT,
00167      $                   ZLASET, ZUNG2R, ZUNM2R, ZUNMR2
00168 *     ..
00169 *     .. Intrinsic Functions ..
00170       INTRINSIC          ABS, DBLE, DIMAG, MAX, MIN
00171 *     ..
00172 *     .. Statement Functions ..
00173       DOUBLE PRECISION   CABS1
00174 *     ..
00175 *     .. Statement Function definitions ..
00176       CABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
00177 *     ..
00178 *     .. Executable Statements ..
00179 *
00180 *     Test the input parameters
00181 *
00182       WANTU = LSAME( JOBU, 'U' )
00183       WANTV = LSAME( JOBV, 'V' )
00184       WANTQ = LSAME( JOBQ, 'Q' )
00185       FORWRD = .TRUE.
00186 *
00187       INFO = 0
00188       IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
00189          INFO = -1
00190       ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
00191          INFO = -2
00192       ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
00193          INFO = -3
00194       ELSE IF( M.LT.0 ) THEN
00195          INFO = -4
00196       ELSE IF( P.LT.0 ) THEN
00197          INFO = -5
00198       ELSE IF( N.LT.0 ) THEN
00199          INFO = -6
00200       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00201          INFO = -8
00202       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
00203          INFO = -10
00204       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
00205          INFO = -16
00206       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
00207          INFO = -18
00208       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
00209          INFO = -20
00210       END IF
00211       IF( INFO.NE.0 ) THEN
00212          CALL XERBLA( 'ZGGSVP', -INFO )
00213          RETURN
00214       END IF
00215 *
00216 *     QR with column pivoting of B: B*P = V*( S11 S12 )
00217 *                                           (  0   0  )
00218 *
00219       DO 10 I = 1, N
00220          IWORK( I ) = 0
00221    10 CONTINUE
00222       CALL ZGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO )
00223 *
00224 *     Update A := A*P
00225 *
00226       CALL ZLAPMT( FORWRD, M, N, A, LDA, IWORK )
00227 *
00228 *     Determine the effective rank of matrix B.
00229 *
00230       L = 0
00231       DO 20 I = 1, MIN( P, N )
00232          IF( CABS1( B( I, I ) ).GT.TOLB )
00233      $      L = L + 1
00234    20 CONTINUE
00235 *
00236       IF( WANTV ) THEN
00237 *
00238 *        Copy the details of V, and form V.
00239 *
00240          CALL ZLASET( 'Full', P, P, CZERO, CZERO, V, LDV )
00241          IF( P.GT.1 )
00242      $      CALL ZLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
00243      $                   LDV )
00244          CALL ZUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
00245       END IF
00246 *
00247 *     Clean up B
00248 *
00249       DO 40 J = 1, L - 1
00250          DO 30 I = J + 1, L
00251             B( I, J ) = CZERO
00252    30    CONTINUE
00253    40 CONTINUE
00254       IF( P.GT.L )
00255      $   CALL ZLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB )
00256 *
00257       IF( WANTQ ) THEN
00258 *
00259 *        Set Q = I and Update Q := Q*P
00260 *
00261          CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
00262          CALL ZLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
00263       END IF
00264 *
00265       IF( P.GE.L .AND. N.NE.L ) THEN
00266 *
00267 *        RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
00268 *
00269          CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO )
00270 *
00271 *        Update A := A*Z'
00272 *
00273          CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB,
00274      $                TAU, A, LDA, WORK, INFO )
00275          IF( WANTQ ) THEN
00276 *
00277 *           Update Q := Q*Z'
00278 *
00279             CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B,
00280      $                   LDB, TAU, Q, LDQ, WORK, INFO )
00281          END IF
00282 *
00283 *        Clean up B
00284 *
00285          CALL ZLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB )
00286          DO 60 J = N - L + 1, N
00287             DO 50 I = J - N + L + 1, L
00288                B( I, J ) = CZERO
00289    50       CONTINUE
00290    60    CONTINUE
00291 *
00292       END IF
00293 *
00294 *     Let              N-L     L
00295 *                A = ( A11    A12 ) M,
00296 *
00297 *     then the following does the complete QR decomposition of A11:
00298 *
00299 *              A11 = U*(  0  T12 )*P1'
00300 *                      (  0   0  )
00301 *
00302       DO 70 I = 1, N - L
00303          IWORK( I ) = 0
00304    70 CONTINUE
00305       CALL ZGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO )
00306 *
00307 *     Determine the effective rank of A11
00308 *
00309       K = 0
00310       DO 80 I = 1, MIN( M, N-L )
00311          IF( CABS1( A( I, I ) ).GT.TOLA )
00312      $      K = K + 1
00313    80 CONTINUE
00314 *
00315 *     Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N )
00316 *
00317       CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ),
00318      $             A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
00319 *
00320       IF( WANTU ) THEN
00321 *
00322 *        Copy the details of U, and form U
00323 *
00324          CALL ZLASET( 'Full', M, M, CZERO, CZERO, U, LDU )
00325          IF( M.GT.1 )
00326      $      CALL ZLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
00327      $                   LDU )
00328          CALL ZUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
00329       END IF
00330 *
00331       IF( WANTQ ) THEN
00332 *
00333 *        Update Q( 1:N, 1:N-L )  = Q( 1:N, 1:N-L )*P1
00334 *
00335          CALL ZLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
00336       END IF
00337 *
00338 *     Clean up A: set the strictly lower triangular part of
00339 *     A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
00340 *
00341       DO 100 J = 1, K - 1
00342          DO 90 I = J + 1, K
00343             A( I, J ) = CZERO
00344    90    CONTINUE
00345   100 CONTINUE
00346       IF( M.GT.K )
00347      $   CALL ZLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA )
00348 *
00349       IF( N-L.GT.K ) THEN
00350 *
00351 *        RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
00352 *
00353          CALL ZGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
00354 *
00355          IF( WANTQ ) THEN
00356 *
00357 *           Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1'
00358 *
00359             CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A,
00360      $                   LDA, TAU, Q, LDQ, WORK, INFO )
00361          END IF
00362 *
00363 *        Clean up A
00364 *
00365          CALL ZLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA )
00366          DO 120 J = N - L - K + 1, N - L
00367             DO 110 I = J - N + L + K + 1, K
00368                A( I, J ) = CZERO
00369   110       CONTINUE
00370   120    CONTINUE
00371 *
00372       END IF
00373 *
00374       IF( M.GT.K ) THEN
00375 *
00376 *        QR factorization of A( K+1:M,N-L+1:N )
00377 *
00378          CALL ZGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
00379 *
00380          IF( WANTU ) THEN
00381 *
00382 *           Update U(:,K+1:M) := U(:,K+1:M)*U1
00383 *
00384             CALL ZUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
00385      $                   A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
00386      $                   WORK, INFO )
00387          END IF
00388 *
00389 *        Clean up
00390 *
00391          DO 140 J = N - L + 1, N
00392             DO 130 I = J - N + K + L + 1, M
00393                A( I, J ) = CZERO
00394   130       CONTINUE
00395   140    CONTINUE
00396 *
00397       END IF
00398 *
00399       RETURN
00400 *
00401 *     End of ZGGSVP
00402 *
00403       END