zunbdb4.f

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*> \brief \b ZUNBDB4
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZUNBDB4( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
*                           TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK,
*                           INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LWORK, M, P, Q, LDX11, LDX21
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   PHI(*), THETA(*)
*       COMPLEX*16         PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*),
*      $                   WORK(*), X11(LDX11,*), X21(LDX21,*)
*       ..
*
*
*> \par Purpose:
*  =============
*>
*>\verbatim
*>
*> ZUNBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
*> matrix X with orthonormal columns:
*>
*>                            [ B11 ]
*>      [ X11 ]   [ P1 |    ] [  0  ]
*>      [-----] = [---------] [-----] Q1**T .
*>      [ X21 ]   [    | P2 ] [ B21 ]
*>                            [  0  ]
*>
*> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
*> M-P, or Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB3 handle cases in
*> which M-Q is not the minimum dimension.
*>
*> The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
*> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
*> Householder vectors.
*>
*> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
*> implicitly by angles THETA, PHI.
*>
*>\endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>           The number of rows X11 plus the number of rows in X21.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>           The number of rows in X11. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is INTEGER
*>           The number of columns in X11 and X21. 0 <= Q <= M and
*>           M-Q <= min(P,M-P,Q).
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*>          X11 is COMPLEX*16 array, dimension (LDX11,Q)
*>           On entry, the top block of the matrix X to be reduced. On
*>           exit, the columns of tril(X11) specify reflectors for P1 and
*>           the rows of triu(X11,1) specify reflectors for Q1.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*>          LDX11 is INTEGER
*>           The leading dimension of X11. LDX11 >= P.
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*>          X21 is COMPLEX*16 array, dimension (LDX21,Q)
*>           On entry, the bottom block of the matrix X to be reduced. On
*>           exit, the columns of tril(X21) specify reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*>          LDX21 is INTEGER
*>           The leading dimension of X21. LDX21 >= M-P.
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*>          THETA is DOUBLE PRECISION array, dimension (Q)
*>           The entries of the bidiagonal blocks B11, B21 are defined by
*>           THETA and PHI. See Further Details.
*> \endverbatim
*>
*> \param[out] PHI
*> \verbatim
*>          PHI is DOUBLE PRECISION array, dimension (Q-1)
*>           The entries of the bidiagonal blocks B11, B21 are defined by
*>           THETA and PHI. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP1
*> \verbatim
*>          TAUP1 is COMPLEX*16 array, dimension (M-Q)
*>           The scalar factors of the elementary reflectors that define
*>           P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*>          TAUP2 is COMPLEX*16 array, dimension (M-Q)
*>           The scalar factors of the elementary reflectors that define
*>           P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*>          TAUQ1 is COMPLEX*16 array, dimension (Q)
*>           The scalar factors of the elementary reflectors that define
*>           Q1.
*> \endverbatim
*>
*> \param[out] PHANTOM
*> \verbatim
*>          PHANTOM is COMPLEX*16 array, dimension (M)
*>           The routine computes an M-by-1 column vector Y that is
*>           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
*>           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
*>           Y(P+1:M), respectively.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>           The dimension of the array WORK. LWORK >= M-Q.
*>
*>           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 unbdb4
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The upper-bidiagonal blocks B11, B21 are represented implicitly by
*>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
*>  in each bidiagonal band is a product of a sine or cosine of a THETA
*>  with a sine or cosine of a PHI. See [1] or ZUNCSD for details.
*>
*>  P1, P2, and Q1 are represented as products of elementary reflectors.
*>  See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
*>  and ZUNGLQ.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*>      Algorithms, 50(1):33-65, 2009.
*>
*  =====================================================================
      SUBROUTINE zunbdb4( M, P, Q, X11, LDX11, X21, LDX21, THETA,
     $                    PHI,
     $                    TAUP1, TAUP2, TAUQ1, PHANTOM, 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..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LWORK, M, P, Q, LDX11, LDX21
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   PHI(*), THETA(*)
      COMPLEX*16         PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*),
     $                   WORK(*), X11(LDX11,*), X21(LDX21,*)
*     ..
*
*  ====================================================================
*
*     .. Parameters ..
      COMPLEX*16         NEGONE, ONE, ZERO
      PARAMETER          ( NEGONE = (-1.0d0,0.0d0), one = (1.0d0,0.0d0),
     $                     zero = (0.0d0,0.0d0) )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   C, S
      INTEGER            CHILDINFO, I, ILARF, IORBDB5, J, LLARF,
     $                   LORBDB5, LWORKMIN, LWORKOPT
      LOGICAL            LQUERY
*     ..
*     .. External Subroutines ..
      EXTERNAL           zlarf1f, zlarfgp, zunbdb5, zdrot, zscal,
     $                   zlacgv,
     $                   xerbla
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DZNRM2
      EXTERNAL           DZNRM2
*     ..
*     .. Intrinsic Function ..
      INTRINSIC          atan2, cos, max, sin, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*
      info = 0
      lquery = lwork .EQ. -1
*
      IF( m .LT. 0 ) THEN
         info = -1
      ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
         info = -2
      ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
         info = -3
      ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
         info = -5
      ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
         info = -7
      END IF
*
*     Compute workspace
*
      IF( info .EQ. 0 ) THEN
         ilarf = 2
         llarf = max( q-1, p-1, m-p-1 )
         iorbdb5 = 2
         lorbdb5 = q
         lworkopt = ilarf + llarf - 1
         lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
         lworkmin = lworkopt
         work(1) = lworkopt
         IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
           info = -14
         END IF
      END IF
      IF( info .NE. 0 ) THEN
         CALL xerbla( 'ZUNBDB4', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Reduce columns 1, ..., M-Q of X11 and X21
*
      DO i = 1, m-q
*
         IF( i .EQ. 1 ) THEN
            DO j = 1, m
               phantom(j) = zero
            END DO
            CALL zunbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
     $                    x11, ldx11, x21, ldx21, work(iorbdb5),
     $                    lorbdb5, childinfo )
            CALL zscal( p, negone, phantom(1), 1 )
            CALL zlarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
            CALL zlarfgp( m-p, phantom(p+1), phantom(p+2), 1,
     $                    taup2(1) )
            theta(i) = atan2( dble( phantom(1) ), dble( phantom(p+1) ) )
            c = cos( theta(i) )
            s = sin( theta(i) )
            CALL zlarf1f( 'L', p, q, phantom(1), 1, conjg(taup1(1)),
     $                    x11,
     $                    ldx11, work(ilarf) )
            CALL zlarf1f( 'L', m-p, q, phantom(p+1), 1,
     $                    conjg(taup2(1)),
     $                    x21, ldx21, work(ilarf) )
         ELSE
            CALL zunbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
     $                    x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
     $                    ldx21, work(iorbdb5), lorbdb5, childinfo )
            CALL zscal( p-i+1, negone, x11(i,i-1), 1 )
            CALL zlarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1,
     $                    taup1(i) )
            CALL zlarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
     $                    taup2(i) )
            theta(i) = atan2( dble( x11(i,i-1) ), dble( x21(i,i-1) ) )
            c = cos( theta(i) )
            s = sin( theta(i) )
            CALL zlarf1f( 'L', p-i+1, q-i+1, x11(i,i-1), 1,
     $                    conjg(taup1(i)), x11(i,i), ldx11,
     $                    work(ilarf) )
            CALL zlarf1f( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1,
     $                    conjg(taup2(i)), x21(i,i), ldx21,
     $                    work(ilarf) )
         END IF
*
         CALL zdrot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
         CALL zlacgv( q-i+1, x21(i,i), ldx21 )
         CALL zlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
         c = dble( x21(i,i) )
         CALL zlarf1f( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
     $                 x11(i+1,i), ldx11, work(ilarf) )
         CALL zlarf1f( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
     $                 x21(i+1,i), ldx21, work(ilarf) )
         CALL zlacgv( q-i+1, x21(i,i), ldx21 )
         IF( i .LT. m-q ) THEN
            s = sqrt( dznrm2( p-i, x11(i+1,i), 1 )**2
     $              + dznrm2( m-p-i, x21(i+1,i), 1 )**2 )
            phi(i) = atan2( s, c )
         END IF
*
      END DO
*
*     Reduce the bottom-right portion of X11 to [ I 0 ]
*
      DO i = m - q + 1, p
         CALL zlacgv( q-i+1, x11(i,i), ldx11 )
         CALL zlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
         CALL zlarf1f( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
     $                 x11(i+1,i), ldx11, work(ilarf) )
         CALL zlarf1f( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
     $                 x21(m-q+1,i), ldx21, work(ilarf) )
         CALL zlacgv( q-i+1, x11(i,i), ldx11 )
      END DO
*
*     Reduce the bottom-right portion of X21 to [ 0 I ]
*
      DO i = p + 1, q
         CALL zlacgv( q-i+1, x21(m-q+i-p,i), ldx21 )
         CALL zlarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1),
     $                 ldx21,
     $                 tauq1(i) )
         CALL zlarf1f( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21,
     $                 tauq1(i),
     $                 x21(m-q+i-p+1,i), ldx21, work(ilarf) )
         CALL zlacgv( q-i+1, x21(m-q+i-p,i), ldx21 )
      END DO
*
      RETURN
*
*     End of ZUNBDB4
*
      SUBROUTINE zunbdb4( M, P, Q, X11, LDX11, X21, LDX21, THETA, …
      END