dorbdb3.f

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*> \brief \b DORBDB3
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DORBDB3( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
*                           TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LWORK, M, P, Q, LDX11, LDX21
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   PHI(*), THETA(*)
*       DOUBLE PRECISION   TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
*      $                   X11(LDX11,*), X21(LDX21,*)
*       ..
*
*
*> \par Purpose:
*  =============
*>
*>\verbatim
*>
*> DORBDB3 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-P must be no larger than P,
*> Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in
*> which M-P is not the minimum dimension.
*>
*> The orthogonal 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-P)-by-(M-P) 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. M-P <= min(P,Q,M-Q).
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is INTEGER
*>           The number of columns in X11 and X21. 0 <= Q <= M.
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*>          X11 is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (P)
*>           The scalar factors of the elementary reflectors that define
*>           P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*>          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
*>           The scalar factors of the elementary reflectors that define
*>           P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*>          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
*>           The scalar factors of the elementary reflectors that define
*>           Q1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION 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 unbdb3
*
*> \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 DORCSD for details.
*>
*>  P1, P2, and Q1 are represented as products of elementary reflectors.
*>  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
*>  and DORGLQ.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*>      Algorithms, 50(1):33-65, 2009.
*>
*  =====================================================================
      SUBROUTINE dorbdb3( M, P, Q, X11, LDX11, X21, LDX21, THETA,
     $                    PHI,
     $                    TAUP1, TAUP2, TAUQ1, 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(*)
      DOUBLE PRECISION   TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
     $                   x11(ldx11,*), x21(ldx21,*)
*     ..
*
*  ====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0d0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   C, S
      INTEGER            CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
     $                   lworkmin, lworkopt
      LOGICAL            LQUERY
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlarf1f, dlarfgp, dorbdb5, drot,
     $                   xerbla
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DNRM2
      EXTERNAL           DNRM2
*     ..
*     .. 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( 2*p .LT. m .OR. p .GT. m ) THEN
         info = -2
      ELSE IF( q .LT. m-p .OR. m-q .LT. m-p ) 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( p, m-p-1, q-1 )
         iorbdb5 = 2
         lorbdb5 = q-1
         lworkopt = max( ilarf+llarf-1, 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( 'DORBDB3', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Reduce rows 1, ..., M-P of X11 and X21
*
      DO i = 1, m-p
*
         IF( i .GT. 1 ) THEN
            CALL drot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c,
     $                 s )
         END IF
*
         CALL dlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
         s = x21(i,i)
         CALL dlarf1f( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
     $               x11(i,i), ldx11, work(ilarf) )
         CALL dlarf1f( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
     $               x21(i+1,i), ldx21, work(ilarf) )
         c = sqrt( dnrm2( p-i+1, x11(i,i), 1 )**2
     $           + dnrm2( m-p-i, x21(i+1,i), 1 )**2 )
         theta(i) = atan2( s, c )
*
         CALL dorbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
     $                 x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
     $                 work(iorbdb5), lorbdb5, childinfo )
         CALL dlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
         IF( i .LT. m-p ) THEN
            CALL dlarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1,
     $                    taup2(i) )
            phi(i) = atan2( x21(i+1,i), x11(i,i) )
            c = cos( phi(i) )
            s = sin( phi(i) )
            CALL dlarf1f( 'L', m-p-i, q-i, x21(i+1,i), 1, taup2(i),
     $                  x21(i+1,i+1), ldx21, work(ilarf) )
         END IF
         CALL dlarf1f( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,
     $               i+1),
     $               ldx11, work(ilarf) )
*
      END DO
*
*     Reduce the bottom-right portion of X11 to the identity matrix
*
      DO i = m-p + 1, q
         CALL dlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
         CALL dlarf1f( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,
     $               i+1),
     $               ldx11, work(ilarf) )
      END DO
*
      RETURN
*
*     End of DORBDB3
*
      SUBROUTINE dorbdb3( M, P, Q, X11, LDX11, X21, LDX21, THETA, …
      END