dorcsd()

recursive subroutine dorcsd	(	character	jobu1,
		character	jobu2,
		character	jobv1t,
		character	jobv2t,
		character	trans,
		character	signs,
		integer	m,
		integer	p,
		integer	q,
		double precision, dimension( ldx11, * )	x11,
		integer	ldx11,
		double precision, dimension( ldx12, * )	x12,
		integer	ldx12,
		double precision, dimension( ldx21, * )	x21,
		integer	ldx21,
		double precision, dimension( ldx22, * )	x22,
		integer	ldx22,
		double precision, dimension( * )	theta,
		double precision, dimension( ldu1, * )	u1,
		integer	ldu1,
		double precision, dimension( ldu2, * )	u2,
		integer	ldu2,
		double precision, dimension( ldv1t, * )	v1t,
		integer	ldv1t,
		double precision, dimension( ldv2t, * )	v2t,
		integer	ldv2t,
		double precision, dimension( * )	work,
		integer	lwork,
		integer, dimension( * )	iwork,
		integer	info )

DORCSD

Download DORCSD + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> DORCSD computes the CS decomposition of an M-by-M partitioned
!> orthogonal matrix X:
!>
!>                                 [  I  0  0 |  0  0  0 ]
!>                                 [  0  C  0 |  0 -S  0 ]
!>     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
!> X = [-----------] = [---------] [---------------------] [---------]   .
!>     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
!>                                 [  0  S  0 |  0  C  0 ]
!>                                 [  0  0  I |  0  0  0 ]
!>
!> X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
!> (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
!> R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
!> which R = MIN(P,M-P,Q,M-Q).
!> 

Parameters

[in]	JOBU1	!> JOBU1 is CHARACTER !> = 'Y': U1 is computed; !> otherwise: U1 is not computed. !>
[in]	JOBU2	!> JOBU2 is CHARACTER !> = 'Y': U2 is computed; !> otherwise: U2 is not computed. !>
[in]	JOBV1T	!> JOBV1T is CHARACTER !> = 'Y': V1T is computed; !> otherwise: V1T is not computed. !>
[in]	JOBV2T	!> JOBV2T is CHARACTER !> = 'Y': V2T is computed; !> otherwise: V2T is not computed. !>
[in]	TRANS	!> TRANS is CHARACTER !> = 'T': X, U1, U2, V1T, and V2T are stored in row-major !> order; !> otherwise: X, U1, U2, V1T, and V2T are stored in column- !> major order. !>
[in]	SIGNS	!> SIGNS is CHARACTER !> = 'O': The lower-left block is made nonpositive (the !> convention); !> otherwise: The upper-right block is made nonpositive (the !> convention). !>
[in]	M	!> M is INTEGER !> The number of rows and columns in X. !>
[in]	P	!> P is INTEGER !> The number of rows in X11 and X12. 0 <= P <= M. !>
[in]	Q	!> Q is INTEGER !> The number of columns in X11 and X21. 0 <= Q <= M. !>
[in,out]	X11	!> X11 is DOUBLE PRECISION array, dimension (LDX11,Q) !> On entry, part of the orthogonal matrix whose CSD is desired. !>
[in]	LDX11	!> LDX11 is INTEGER !> The leading dimension of X11. LDX11 >= MAX(1,P). !>
[in,out]	X12	!> X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q) !> On entry, part of the orthogonal matrix whose CSD is desired. !>
[in]	LDX12	!> LDX12 is INTEGER !> The leading dimension of X12. LDX12 >= MAX(1,P). !>
[in,out]	X21	!> X21 is DOUBLE PRECISION array, dimension (LDX21,Q) !> On entry, part of the orthogonal matrix whose CSD is desired. !>
[in]	LDX21	!> LDX21 is INTEGER !> The leading dimension of X11. LDX21 >= MAX(1,M-P). !>
[in,out]	X22	!> X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q) !> On entry, part of the orthogonal matrix whose CSD is desired. !>
[in]	LDX22	!> LDX22 is INTEGER !> The leading dimension of X11. LDX22 >= MAX(1,M-P). !>
[out]	THETA	!> THETA is DOUBLE PRECISION array, dimension (R), in which R = !> MIN(P,M-P,Q,M-Q). !> C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and !> S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). !>
[out]	U1	!> U1 is DOUBLE PRECISION array, dimension (LDU1,P) !> If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1. !>
[in]	LDU1	!> LDU1 is INTEGER !> The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= !> MAX(1,P). !>
[out]	U2	!> U2 is DOUBLE PRECISION array, dimension (LDU2,M-P) !> If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal !> matrix U2. !>
[in]	LDU2	!> LDU2 is INTEGER !> The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= !> MAX(1,M-P). !>
[out]	V1T	!> V1T is DOUBLE PRECISION array, dimension (LDV1T,Q) !> If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal !> matrix V1**T. !>
[in]	LDV1T	!> LDV1T is INTEGER !> The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= !> MAX(1,Q). !>
[out]	V2T	!> V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q) !> If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal !> matrix V2**T. !>
[in]	LDV2T	!> LDV2T is INTEGER !> The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= !> MAX(1,M-Q). !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> If INFO > 0 on exit, WORK(2:R) contains the values PHI(1), !> ..., PHI(R-1) that, together with THETA(1), ..., THETA(R), !> define the matrix in intermediate bidiagonal-block form !> remaining after nonconvergence. INFO specifies the number !> of nonzero PHI's. !>
[in]	LWORK	!> 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. !>
[out]	IWORK	!> IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q)) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: DBBCSD did not converge. See the description of WORK !> above for details. !>

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 293 of file dorcsd.f.

*
*  -- 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 ..
      CHARACTER          JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
      INTEGER            INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
     $                   LDX21, LDX22, LWORK, M, P, Q
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   THETA( * )
      DOUBLE PRECISION   U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
     $                   V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
     $                   X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
     $                   * )
*     ..
*
*  ===================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      parameter( one = 1.0d0,
     $                     zero = 0.0d0 )
*     ..
*     .. Local Scalars ..
      CHARACTER          TRANST, SIGNST
      INTEGER            CHILDINFO, I, IB11D, IB11E, IB12D, IB12E,
     $                   IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB,
     $                   IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1,
     $                   ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN,
     $                   LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN,
     $                   LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN,
     $                   LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN,
     $                   LORGQRWORKOPT, LWORKMIN, LWORKOPT
      LOGICAL            COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2,
     $                   WANTV1T, WANTV2T
*     ..
*     .. External Subroutines ..
      EXTERNAL           dbbcsd, dlacpy, dlapmr, dlapmt,
     $                   dorbdb, dorglq, dorgqr, xerbla
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. Intrinsic Functions
      INTRINSIC          int, max, min
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*
      info = 0
      wantu1 = lsame( jobu1, 'Y' )
      wantu2 = lsame( jobu2, 'Y' )
      wantv1t = lsame( jobv1t, 'Y' )
      wantv2t = lsame( jobv2t, 'Y' )
      colmajor = .NOT. lsame( trans, 'T' )
      defaultsigns = .NOT. lsame( signs, 'O' )
      lquery = lwork .EQ. -1
      IF( m .LT. 0 ) THEN
         info = -7
      ELSE IF( p .LT. 0 .OR. p .GT. m ) THEN
         info = -8
      ELSE IF( q .LT. 0 .OR. q .GT. m ) THEN
         info = -9
      ELSE IF ( colmajor .AND.  ldx11 .LT. max( 1, p ) ) THEN
        info = -11
      ELSE IF (.NOT. colmajor .AND. ldx11 .LT. max( 1, q ) ) THEN
        info = -11
      ELSE IF (colmajor .AND. ldx12 .LT. max( 1, p ) ) THEN
        info = -13
      ELSE IF (.NOT. colmajor .AND. ldx12 .LT. max( 1, m-q ) ) THEN
        info = -13
      ELSE IF (colmajor .AND. ldx21 .LT. max( 1, m-p ) ) THEN
        info = -15
      ELSE IF (.NOT. colmajor .AND. ldx21 .LT. max( 1, q ) ) THEN
        info = -15
      ELSE IF (colmajor .AND. ldx22 .LT. max( 1, m-p ) ) THEN
        info = -17
      ELSE IF (.NOT. colmajor .AND. ldx22 .LT. max( 1, m-q ) ) THEN
        info = -17
      ELSE IF( wantu1 .AND. ldu1 .LT. p ) THEN
         info = -20
      ELSE IF( wantu2 .AND. ldu2 .LT. m-p ) THEN
         info = -22
      ELSE IF( wantv1t .AND. ldv1t .LT. q ) THEN
         info = -24
      ELSE IF( wantv2t .AND. ldv2t .LT. m-q ) THEN
         info = -26
      END IF
*
*     Work with transpose if convenient
*
      IF( info .EQ. 0 .AND. min( p, m-p ) .LT. min( q, m-q ) ) THEN
         IF( colmajor ) THEN
            transt = 'T'
         ELSE
            transt = 'N'
         END IF
         IF( defaultsigns ) THEN
            signst = 'O'
         ELSE
            signst = 'D'
         END IF
         CALL dorcsd( jobv1t, jobv2t, jobu1, jobu2, transt, signst,
     $                m,
     $                q, p, x11, ldx11, x21, ldx21, x12, ldx12, x22,
     $                ldx22, theta, v1t, ldv1t, v2t, ldv2t, u1, ldu1,
     $                u2, ldu2, work, lwork, iwork, info )
         RETURN
      END IF
*
*     Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
*     convenient
*
      IF( info .EQ. 0 .AND. m-q .LT. q ) THEN
         IF( defaultsigns ) THEN
            signst = 'O'
         ELSE
            signst = 'D'
         END IF
         CALL dorcsd( jobu2, jobu1, jobv2t, jobv1t, trans, signst, m,
     $                m-p, m-q, x22, ldx22, x21, ldx21, x12, ldx12, x11,
     $                ldx11, theta, u2, ldu2, u1, ldu1, v2t, ldv2t, v1t,
     $                ldv1t, work, lwork, iwork, info )
         RETURN
      END IF
*
*     Compute workspace
*
      IF( info .EQ. 0 ) THEN
*
         iphi = 2
         itaup1 = iphi + max( 1, q - 1 )
         itaup2 = itaup1 + max( 1, p )
         itauq1 = itaup2 + max( 1, m - p )
         itauq2 = itauq1 + max( 1, q )
         iorgqr = itauq2 + max( 1, m - q )
         CALL dorgqr( m-q, m-q, m-q, u1, max(1,m-q), u1, work, -1,
     $                childinfo )
         lorgqrworkopt = int( work(1) )
         lorgqrworkmin = max( 1, m - q )
         iorglq = itauq2 + max( 1, m - q )
         CALL dorglq( m-q, m-q, m-q, u1, max(1,m-q), u1, work, -1,
     $                childinfo )
         lorglqworkopt = int( work(1) )
         lorglqworkmin = max( 1, m - q )
         iorbdb = itauq2 + max( 1, m - q )
         CALL dorbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12,
     $                x21, ldx21, x22, ldx22, theta, v1t, u1, u2, v1t,
     $                v2t, work, -1, childinfo )
         lorbdbworkopt = int( work(1) )
         lorbdbworkmin = lorbdbworkopt
         ib11d = itauq2 + max( 1, m - q )
         ib11e = ib11d + max( 1, q )
         ib12d = ib11e + max( 1, q - 1 )
         ib12e = ib12d + max( 1, q )
         ib21d = ib12e + max( 1, q - 1 )
         ib21e = ib21d + max( 1, q )
         ib22d = ib21e + max( 1, q - 1 )
         ib22e = ib22d + max( 1, q )
         ibbcsd = ib22e + max( 1, q - 1 )
         CALL dbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q,
     $                theta, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t,
     $                ldv2t, u1, u1, u1, u1, u1, u1, u1, u1, work, -1,
     $                childinfo )
         lbbcsdworkopt = int( work(1) )
         lbbcsdworkmin = lbbcsdworkopt
         lworkopt = max( iorgqr + lorgqrworkopt, iorglq + lorglqworkopt,
     $              iorbdb + lorbdbworkopt, ibbcsd + lbbcsdworkopt ) - 1
         lworkmin = max( iorgqr + lorgqrworkmin, iorglq + lorglqworkmin,
     $              iorbdb + lorbdbworkopt, ibbcsd + lbbcsdworkmin ) - 1
         work(1) = max(lworkopt,lworkmin)
*
         IF( lwork .LT. lworkmin .AND. .NOT. lquery ) THEN
            info = -22
         ELSE
            lorgqrwork = lwork - iorgqr + 1
            lorglqwork = lwork - iorglq + 1
            lorbdbwork = lwork - iorbdb + 1
            lbbcsdwork = lwork - ibbcsd + 1
         END IF
      END IF
*
*     Abort if any illegal arguments
*
      IF( info .NE. 0 ) THEN
         CALL xerbla( 'DORCSD', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Transform to bidiagonal block form
*
      CALL dorbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12,
     $             x21,
     $             ldx21, x22, ldx22, theta, work(iphi), work(itaup1),
     $             work(itaup2), work(itauq1), work(itauq2),
     $             work(iorbdb), lorbdbwork, childinfo )
*
*     Accumulate Householder reflectors
*
      IF( colmajor ) THEN
         IF( wantu1 .AND. p .GT. 0 ) THEN
            CALL dlacpy( 'L', p, q, x11, ldx11, u1, ldu1 )
            CALL dorgqr( p, p, q, u1, ldu1, work(itaup1),
     $                   work(iorgqr),
     $                   lorgqrwork, info)
         END IF
         IF( wantu2 .AND. m-p .GT. 0 ) THEN
            CALL dlacpy( 'L', m-p, q, x21, ldx21, u2, ldu2 )
            CALL dorgqr( m-p, m-p, q, u2, ldu2, work(itaup2),
     $                   work(iorgqr), lorgqrwork, info )
         END IF
         IF( wantv1t .AND. q .GT. 0 ) THEN
            CALL dlacpy( 'U', q-1, q-1, x11(1,2), ldx11, v1t(2,2),
     $                   ldv1t )
            v1t(1, 1) = one
            DO j = 2, q
               v1t(1,j) = zero
               v1t(j,1) = zero
            END DO
            CALL dorglq( q-1, q-1, q-1, v1t(2,2), ldv1t,
     $                   work(itauq1),
     $                   work(iorglq), lorglqwork, info )
         END IF
         IF( wantv2t .AND. m-q .GT. 0 ) THEN
            CALL dlacpy( 'U', p, m-q, x12, ldx12, v2t, ldv2t )
            IF (m-p .GT. q) Then
               CALL dlacpy( 'U', m-p-q, m-p-q, x22(q+1,p+1), ldx22,
     $                      v2t(p+1,p+1), ldv2t )
            END IF
            IF (m .GT. q) THEN
               CALL dorglq( m-q, m-q, m-q, v2t, ldv2t, work(itauq2),
     $                      work(iorglq), lorglqwork, info )
            END IF
         END IF
      ELSE
         IF( wantu1 .AND. p .GT. 0 ) THEN
            CALL dlacpy( 'U', q, p, x11, ldx11, u1, ldu1 )
            CALL dorglq( p, p, q, u1, ldu1, work(itaup1),
     $                   work(iorglq),
     $                   lorglqwork, info)
         END IF
         IF( wantu2 .AND. m-p .GT. 0 ) THEN
            CALL dlacpy( 'U', q, m-p, x21, ldx21, u2, ldu2 )
            CALL dorglq( m-p, m-p, q, u2, ldu2, work(itaup2),
     $                   work(iorglq), lorglqwork, info )
         END IF
         IF( wantv1t .AND. q .GT. 0 ) THEN
            CALL dlacpy( 'L', q-1, q-1, x11(2,1), ldx11, v1t(2,2),
     $                   ldv1t )
            v1t(1, 1) = one
            DO j = 2, q
               v1t(1,j) = zero
               v1t(j,1) = zero
            END DO
            CALL dorgqr( q-1, q-1, q-1, v1t(2,2), ldv1t,
     $                   work(itauq1),
     $                   work(iorgqr), lorgqrwork, info )
         END IF
         IF( wantv2t .AND. m-q .GT. 0 ) THEN
            CALL dlacpy( 'L', m-q, p, x12, ldx12, v2t, ldv2t )
            CALL dlacpy( 'L', m-p-q, m-p-q, x22(p+1,q+1), ldx22,
     $                   v2t(p+1,p+1), ldv2t )
            CALL dorgqr( m-q, m-q, m-q, v2t, ldv2t, work(itauq2),
     $                   work(iorgqr), lorgqrwork, info )
         END IF
      END IF
*
*     Compute the CSD of the matrix in bidiagonal-block form
*
      CALL dbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q,
     $             theta,
     $             work(iphi), u1, ldu1, u2, ldu2, v1t, ldv1t, v2t,
     $             ldv2t, work(ib11d), work(ib11e), work(ib12d),
     $             work(ib12e), work(ib21d), work(ib21e), work(ib22d),
     $             work(ib22e), work(ibbcsd), lbbcsdwork, info )
*
*     Permute rows and columns to place identity submatrices in top-
*     left corner of (1,1)-block and/or bottom-right corner of (1,2)-
*     block and/or bottom-right corner of (2,1)-block and/or top-left
*     corner of (2,2)-block
*
      IF( q .GT. 0 .AND. wantu2 ) THEN
         DO i = 1, q
            iwork(i) = m - p - q + i
         END DO
         DO i = q + 1, m - p
            iwork(i) = i - q
         END DO
         IF( colmajor ) THEN
            CALL dlapmt( .false., m-p, m-p, u2, ldu2, iwork )
         ELSE
            CALL dlapmr( .false., m-p, m-p, u2, ldu2, iwork )
         END IF
      END IF
      IF( m .GT. 0 .AND. wantv2t ) THEN
         DO i = 1, p
            iwork(i) = m - p - q + i
         END DO
         DO i = p + 1, m - q
            iwork(i) = i - p
         END DO
         IF( .NOT. colmajor ) THEN
            CALL dlapmt( .false., m-q, m-q, v2t, ldv2t, iwork )
         ELSE
            CALL dlapmr( .false., m-q, m-q, v2t, ldv2t, iwork )
         END IF
      END IF
*
      RETURN
*
*     End DORCSD
*

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