zggsvp3.f

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*> \brief \b ZGGSVP3
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGGSVP3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
*                           TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
*                           IWORK, RWORK, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBQ, JOBU, JOBV
*       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK
*       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( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGGSVP3 computes unitary matrices U, V and Q such that
*>
*>                    N-K-L  K    L
*>  U**H*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**H*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**H,B**H)**H.
*>
*> This decomposition is the preprocessing step for computing the
*> Generalized Singular Value Decomposition (GSVD), see subroutine
*> ZGGSVD3.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          = 'U':  Unitary matrix U is computed;
*>          = 'N':  U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          = 'V':  Unitary matrix V is computed;
*>          = 'N':  V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*>          JOBQ is CHARACTER*1
*>          = 'Q':  Unitary matrix Q is computed;
*>          = 'N':  Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows of the matrix B.  P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is 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.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in] TOLA
*> \verbatim
*>          TOLA is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] TOLB
*> \verbatim
*>          TOLB is 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.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*> \endverbatim
*>
*> \param[out] L
*> \verbatim
*>          L is INTEGER
*>
*>          On exit, K and L specify the dimension of the subblocks
*>          described in Purpose section.
*>          K + L = effective numerical rank of (A**H,B**H)**H.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is COMPLEX*16 array, dimension (LDU,M)
*>          If JOBU = 'U', U contains the unitary matrix U.
*>          If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U. LDU >= max(1,M) if
*>          JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is COMPLEX*16 array, dimension (LDV,P)
*>          If JOBV = 'V', V contains the unitary matrix V.
*>          If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V. LDV >= max(1,P) if
*>          JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is COMPLEX*16 array, dimension (LDQ,N)
*>          If JOBQ = 'Q', Q contains the unitary matrix Q.
*>          If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= max(1,N) if
*>          JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is COMPLEX*16 array, dimension (N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggsvp3
*
*> \par Further Details:
*  =====================
*
*> \verbatim
*>
*>  The subroutine uses LAPACK subroutine ZGEQP3 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.
*>
*>  ZGGSVP3 replaces the deprecated subroutine ZGGSVP.
*>
*> \endverbatim
*>
* =====================================================================
      SUBROUTINE zggsvp3( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
     $                    TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
     $                    IWORK, RWORK, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
      IMPLICIT NONE
*
*     .. Scalar Arguments ..
      CHARACTER          JOBQ, JOBU, JOBV
      INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
     $                   lwork
      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( * )
*     ..
*
*  =====================================================================
*
*     .. 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, LQUERY
      INTEGER            I, J, LWKOPT
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zgeqp3, zgeqr2, zgerq2, zlacpy, zlapmt,
     $                   zlaset, zung2r, zunm2r, zunmr2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      wantu = lsame( jobu, 'U' )
      wantv = lsame( jobv, 'V' )
      wantq = lsame( jobq, 'Q' )
      forwrd = .true.
      lquery = ( lwork.EQ.-1 )
      lwkopt = 1
*
*     Test the input arguments
*
      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
      ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
         info = -24
      END IF
*
*     Compute workspace
*
      IF( info.EQ.0 ) THEN
         CALL zgeqp3( p, n, b, ldb, iwork, tau, work, -1, rwork, info )
         lwkopt = int( work( 1 ) )
         IF( wantv ) THEN
            lwkopt = max( lwkopt, p )
         END IF
         lwkopt = max( lwkopt, min( n, p ) )
         lwkopt = max( lwkopt, m )
         IF( wantq ) THEN
            lwkopt = max( lwkopt, n )
         END IF
         CALL zgeqp3( m, n, a, lda, iwork, tau, work, -1, rwork, info )
         lwkopt = max( lwkopt, int( work( 1 ) ) )
         lwkopt = max( 1, lwkopt )
         work( 1 ) = dcmplx( lwkopt )
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZGGSVP3', -info )
         RETURN
      END IF
      IF( lquery ) THEN
         RETURN
      ENDIF
*
*     QR with column pivoting of B: B*P = V*( S11 S12 )
*                                           (  0   0  )
*
      DO 10 i = 1, n
         iwork( i ) = 0
   10 CONTINUE
      CALL zgeqp3( p, n, b, ldb, iwork, tau, work, lwork, 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( abs( 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**H
*
         CALL zunmr2( 'Right', 'Conjugate transpose', m, n, l, b, ldb,
     $                tau, a, lda, work, info )
         IF( wantq ) THEN
*
*           Update Q := Q*Z**H
*
            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**H
*                      (  0   0  )
*
      DO 70 i = 1, n - l
         iwork( i ) = 0
   70 CONTINUE
      CALL zgeqp3( m, n-l, a, lda, iwork, tau, work, lwork, rwork,
     $             info )
*
*     Determine the effective rank of A11
*
      k = 0
      DO 80 i = 1, min( m, n-l )
         IF( abs( a( i, i ) ).GT.tola )
     $      k = k + 1
   80 CONTINUE
*
*     Update A12 := U**H*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**H
*
            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
*
      work( 1 ) = dcmplx( lwkopt )
      RETURN
*
*     End of ZGGSVP3
*
      END