zggsvp.f

          SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
         $                   TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
         $                   IWORK, RWORK, TAU, WORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBQ, JOBU, JOBV
          INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
          DOUBLE PRECISION   TOLA, TOLB
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   RWORK( * )
          COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
         $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZGGSVP computes unitary matrices U, V and Q such that
    *
    *                   N-K-L  K    L
    *   U'*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
    *                L ( 0     0   A23 )
    *            M-K-L ( 0     0    0  )
    *
    *                   N-K-L  K    L
    *          =     K ( 0    A12  A13 )  if M-K-L < 0;
    *              M-K ( 0     0   A23 )
    *
    *                 N-K-L  K    L
    *   V'*B*Q =   L ( 0     0   B13 )
    *            P-L ( 0     0    0  )
    *
    *  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
    *  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
    *  otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
    *  numerical rank of the (M+P)-by-N matrix (A',B')'.  Z' denotes the
    *  conjugate transpose of Z.
    *
    *  This decomposition is the preprocessing step for computing the
    *  Generalized Singular Value Decomposition (GSVD), see subroutine
    *  ZGGSVD.
    *
    *  Arguments
    *  =========
    *
    *  JOBU    (input) CHARACTER*1
    *          = 'U':  Unitary matrix U is computed;
    *          = 'N':  U is not computed.
    *
    *  JOBV    (input) CHARACTER*1
    *          = 'V':  Unitary matrix V is computed;
    *          = 'N':  V is not computed.
    *
    *  JOBQ    (input) CHARACTER*1
    *          = 'Q':  Unitary matrix Q is computed;
    *          = 'N':  Q is not computed.
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A.  M >= 0.
    *
    *  P       (input) INTEGER
    *          The number of rows of the matrix B.  P >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrices A and B.  N >= 0.
    *
    *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
    *          On entry, the M-by-N matrix A.
    *          On exit, A contains the triangular (or trapezoidal) matrix
    *          described in the Purpose section.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,M).
    *
    *  B       (input/output) COMPLEX*16 array, dimension (LDB,N)
    *          On entry, the P-by-N matrix B.
    *          On exit, B contains the triangular matrix described in
    *          the Purpose section.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,P).
    *
    *  TOLA    (input) DOUBLE PRECISION
    *  TOLB    (input) DOUBLE PRECISION
    *          TOLA and TOLB are the thresholds to determine the effective
    *          numerical rank of matrix B and a subblock of A. Generally,
    *          they are set to
    *             TOLA = MAX(M,N)*norm(A)*MAZHEPS,
    *             TOLB = MAX(P,N)*norm(B)*MAZHEPS.
    *          The size of TOLA and TOLB may affect the size of backward
    *          errors of the decomposition.
    *
    *  K       (output) INTEGER
    *  L       (output) INTEGER
    *          On exit, K and L specify the dimension of the subblocks
    *          described in Purpose section.
    *          K + L = effective numerical rank of (A',B')'.
    *
    *  U       (output) COMPLEX*16 array, dimension (LDU,M)
    *          If JOBU = 'U', U contains the unitary matrix U.
    *          If JOBU = 'N', U is not referenced.
    *
    *  LDU     (input) INTEGER
    *          The leading dimension of the array U. LDU >= max(1,M) if
    *          JOBU = 'U'; LDU >= 1 otherwise.
    *
    *  V       (output) COMPLEX*16 array, dimension (LDV,M)
    *          If JOBV = 'V', V contains the unitary matrix V.
    *          If JOBV = 'N', V is not referenced.
    *
    *  LDV     (input) INTEGER
    *          The leading dimension of the array V. LDV >= max(1,P) if
    *          JOBV = 'V'; LDV >= 1 otherwise.
    *
    *  Q       (output) COMPLEX*16 array, dimension (LDQ,N)
    *          If JOBQ = 'Q', Q contains the unitary matrix Q.
    *          If JOBQ = 'N', Q is not referenced.
    *
    *  LDQ     (input) INTEGER
    *          The leading dimension of the array Q. LDQ >= max(1,N) if
    *          JOBQ = 'Q'; LDQ >= 1 otherwise.
    *
    *  IWORK   (workspace) INTEGER array, dimension (N)
    *
    *  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
    *
    *  TAU     (workspace) COMPLEX*16 array, dimension (N)
    *
    *  WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P))
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *
    *  Further Details
    *  ===============
    *
    *  The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
    *  with column pivoting to detect the effective numerical rank of the
    *  a matrix. It may be replaced by a better rank determination strategy.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          COMPLEX*16         CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
         $                   CONE = ( 1.0D+0, 0.0D+0 ) )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            FORWRD, WANTQ, WANTU, WANTV
          INTEGER            I, J
          COMPLEX*16         T
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA, ZGEQPF, ZGEQR2, ZGERQ2, ZLACPY, ZLAPMT,
         $                   ZLASET, ZUNG2R, ZUNM2R, ZUNMR2
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, DIMAG, MAX, MIN
    *     ..
    *     .. Statement Functions ..
          DOUBLE PRECISION   CABS1
    *     ..
    *     .. Statement Function definitions ..
          CABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          WANTU = LSAME( JOBU, 'U' )
          WANTV = LSAME( JOBV, 'V' )
          WANTQ = LSAME( JOBQ, 'Q' )
          FORWRD = .TRUE.
    *
          INFO = 0
          IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
             INFO = -1
          ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
             INFO = -2
          ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
             INFO = -3
          ELSE IF( M.LT.0 ) THEN
             INFO = -4
          ELSE IF( P.LT.0 ) THEN
             INFO = -5
          ELSE IF( N.LT.0 ) THEN
             INFO = -6
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -8
          ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
             INFO = -10
          ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
             INFO = -16
          ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
             INFO = -18
          ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
             INFO = -20
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZGGSVP', -INFO )
             RETURN
          END IF
    *
    *     QR with column pivoting of B: B*P = V*( S11 S12 )
    *                                           (  0   0  )
    *
          DO 10 I = 1, N
             IWORK( I ) = 0
       10 CONTINUE
          CALL ZGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO )
    *
    *     Update A := A*P
    *
          CALL ZLAPMT( FORWRD, M, N, A, LDA, IWORK )
    *
    *     Determine the effective rank of matrix B.
    *
          L = 0
          DO 20 I = 1, MIN( P, N )
             IF( CABS1( B( I, I ) ).GT.TOLB )
         $      L = L + 1
       20 CONTINUE
    *
          IF( WANTV ) THEN
    *
    *        Copy the details of V, and form V.
    *
             CALL ZLASET( 'Full', P, P, CZERO, CZERO, V, LDV )
             IF( P.GT.1 )
         $      CALL ZLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
         $                   LDV )
             CALL ZUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
          END IF
    *
    *     Clean up B
    *
          DO 40 J = 1, L - 1
             DO 30 I = J + 1, L
                B( I, J ) = CZERO
       30    CONTINUE
       40 CONTINUE
          IF( P.GT.L )
         $   CALL ZLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB )
    *
          IF( WANTQ ) THEN
    *
    *        Set Q = I and Update Q := Q*P
    *
             CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
             CALL ZLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
          END IF
    *
          IF( P.GE.L .AND. N.NE.L ) THEN
    *
    *        RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
    *
             CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO )
    *
    *        Update A := A*Z'
    *
             CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB,
         $                TAU, A, LDA, WORK, INFO )
             IF( WANTQ ) THEN
    *
    *           Update Q := Q*Z'
    *
                CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B,
         $                   LDB, TAU, Q, LDQ, WORK, INFO )
             END IF
    *
    *        Clean up B
    *
             CALL ZLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB )
             DO 60 J = N - L + 1, N
                DO 50 I = J - N + L + 1, L
                   B( I, J ) = CZERO
       50       CONTINUE
       60    CONTINUE
    *
          END IF
    *
    *     Let              N-L     L
    *                A = ( A11    A12 ) M,
    *
    *     then the following does the complete QR decomposition of A11:
    *
    *              A11 = U*(  0  T12 )*P1'
    *                      (  0   0  )
    *
          DO 70 I = 1, N - L
             IWORK( I ) = 0
       70 CONTINUE
          CALL ZGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO )
    *
    *     Determine the effective rank of A11
    *
          K = 0
          DO 80 I = 1, MIN( M, N-L )
             IF( CABS1( A( I, I ) ).GT.TOLA )
         $      K = K + 1
       80 CONTINUE
    *
    *     Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N )
    *
          CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ),
         $             A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
    *
          IF( WANTU ) THEN
    *
    *        Copy the details of U, and form U
    *
             CALL ZLASET( 'Full', M, M, CZERO, CZERO, U, LDU )
             IF( M.GT.1 )
         $      CALL ZLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
         $                   LDU )
             CALL ZUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
          END IF
    *
          IF( WANTQ ) THEN
    *
    *        Update Q( 1:N, 1:N-L )  = Q( 1:N, 1:N-L )*P1
    *
             CALL ZLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
          END IF
    *
    *     Clean up A: set the strictly lower triangular part of
    *     A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
    *
          DO 100 J = 1, K - 1
             DO 90 I = J + 1, K
                A( I, J ) = CZERO
       90    CONTINUE
      100 CONTINUE
          IF( M.GT.K )
         $   CALL ZLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA )
    *
          IF( N-L.GT.K ) THEN
    *
    *        RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
    *
             CALL ZGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
    *
             IF( WANTQ ) THEN
    *
    *           Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1'
    *
                CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A,
         $                   LDA, TAU, Q, LDQ, WORK, INFO )
             END IF
    *
    *        Clean up A
    *
             CALL ZLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA )
             DO 120 J = N - L - K + 1, N - L
                DO 110 I = J - N + L + K + 1, K
                   A( I, J ) = CZERO
      110       CONTINUE
      120    CONTINUE
    *
          END IF
    *
          IF( M.GT.K ) THEN
    *
    *        QR factorization of A( K+1:M,N-L+1:N )
    *
             CALL ZGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
    *
             IF( WANTU ) THEN
    *
    *           Update U(:,K+1:M) := U(:,K+1:M)*U1
    *
                CALL ZUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
         $                   A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
         $                   WORK, INFO )
             END IF
    *
    *        Clean up
    *
             DO 140 J = N - L + 1, N
                DO 130 I = J - N + K + L + 1, M
                   A( I, J ) = CZERO
      130       CONTINUE
      140    CONTINUE
    *
          END IF
    *
          RETURN
    *
    *     End of ZGGSVP
    *
          END