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RECURSIVE SUBROUTINE ZUNCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS,
$ SIGNS, M, P, Q, X11, LDX11, X12,
$ LDX12, X21, LDX21, X22, LDX22, THETA,
$ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
$ LDV2T, WORK, LWORK, RWORK, LRWORK,
$ IWORK, INFO )
IMPLICIT NONE
*
* -- LAPACK routine (version 3.3.1) --
*
* -- Contributed by Brian Sutton of the Randolph-Macon College --
* -- November 2010
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* @precisions normal z -> c
*
* .. Scalar Arguments ..
CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS
INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12,
$ LDX21, LDX22, LRWORK, LWORK, M, P, Q
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION THETA( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
$ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ),
$ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22,
$ * )
* ..
*
* Purpose
* =======
*
* ZUNCSD computes the CS decomposition of an M-by-M partitioned
* unitary matrix X:
*
* [ I 0 0 | 0 0 0 ]
* [ 0 C 0 | 0 -S 0 ]
* [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**H
* 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 unitary 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).
*
* Arguments
* =========
*
* JOBU1 (input) CHARACTER
* = 'Y': U1 is computed;
* otherwise: U1 is not computed.
*
* JOBU2 (input) CHARACTER
* = 'Y': U2 is computed;
* otherwise: U2 is not computed.
*
* JOBV1T (input) CHARACTER
* = 'Y': V1T is computed;
* otherwise: V1T is not computed.
*
* JOBV2T (input) CHARACTER
* = 'Y': V2T is computed;
* otherwise: V2T is not computed.
*
* TRANS (input) 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.
*
* SIGNS (input) CHARACTER
* = 'O': The lower-left block is made nonpositive (the
* "other" convention);
* otherwise: The upper-right block is made nonpositive (the
* "default" convention).
*
* M (input) INTEGER
* The number of rows and columns in X.
*
* P (input) INTEGER
* The number of rows in X11 and X12. 0 <= P <= M.
*
* Q (input) INTEGER
* The number of columns in X11 and X21. 0 <= Q <= M.
*
* X (input/workspace) COMPLEX*16 array, dimension (LDX,M)
* On entry, the unitary matrix whose CSD is desired.
*
* LDX (input) INTEGER
* The leading dimension of X. LDX >= MAX(1,M).
*
* THETA (output) 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)) ).
*
* U1 (output) COMPLEX*16 array, dimension (P)
* If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1.
*
* LDU1 (input) INTEGER
* The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
* MAX(1,P).
*
* U2 (output) COMPLEX*16 array, dimension (M-P)
* If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
* matrix U2.
*
* LDU2 (input) INTEGER
* The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
* MAX(1,M-P).
*
* V1T (output) COMPLEX*16 array, dimension (Q)
* If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
* matrix V1**H.
*
* LDV1T (input) INTEGER
* The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
* MAX(1,Q).
*
* V2T (output) COMPLEX*16 array, dimension (M-Q)
* If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary
* matrix V2**H.
*
* LDV2T (input) INTEGER
* The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
* MAX(1,M-Q).
*
* WORK (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) 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.
*
* RWORK (workspace) DOUBLE PRECISION array, dimension MAX(1,LRWORK)
* On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
* If INFO > 0 on exit, RWORK(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.
*
* LRWORK (input) INTEGER
* The dimension of the array RWORK.
*
* If LRWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the RWORK array, returns
* this value as the first entry of the work array, and no error
* message related to LRWORK is issued by XERBLA.
*
* IWORK (workspace) INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: ZBBCSD did not converge. See the description of RWORK
* above for details.
*
* Reference
* =========
*
* [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
* Algorithms, 50(1):33-65, 2009.
*
* ===================================================================
*
* .. Parameters ..
DOUBLE PRECISION REALONE
PARAMETER ( REALONE = 1.0D0 )
COMPLEX*16 NEGONE, ONE, PIOVER2, ZERO
PARAMETER ( NEGONE = (-1.0D0,0.0D0), ONE = (1.0D0,0.0D0),
$ PIOVER2 = 1.57079632679489662D0,
$ ZERO = (0.0D0,0.0D0) )
* ..
* .. 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
INTEGER LRWORKMIN, LRWORKOPT
LOGICAL LRQUERY
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZBBCSD, ZLACPY, ZLAPMR, ZLAPMT, ZLASCL,
$ ZLASET, ZUNBDB, ZUNGLQ, ZUNGQR
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions
INTRINSIC COS, INT, MAX, MIN, SIN
* ..
* .. 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
LRQUERY = LRWORK .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) ) .OR.
$ ( .NOT.COLMAJOR .AND. LDX11 .LT. MAX(1,Q) ) ) THEN
INFO = -11
ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
INFO = -14
ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
INFO = -16
ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
INFO = -18
ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
INFO = -20
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 ZUNCSD( 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, RWORK, LRWORK, 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 ZUNCSD( 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, RWORK, LRWORK, IWORK, INFO )
RETURN
END IF
*
* Compute workspace
*
IF( INFO .EQ. 0 ) THEN
*
* Real workspace
*
IPHI = 2
IB11D = IPHI + MAX( 1, Q - 1 )
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 ZBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 0,
$ 0, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, 0,
$ 0, 0, 0, 0, 0, 0, 0, RWORK, -1, CHILDINFO )
LBBCSDWORKOPT = INT( RWORK(1) )
LBBCSDWORKMIN = LBBCSDWORKOPT
LRWORKOPT = IBBCSD + LBBCSDWORKOPT - 1
LRWORKMIN = IBBCSD + LBBCSDWORKMIN - 1
RWORK(1) = LRWORKOPT
*
* Complex workspace
*
ITAUP1 = 2
ITAUP2 = ITAUP1 + MAX( 1, P )
ITAUQ1 = ITAUP2 + MAX( 1, M - P )
ITAUQ2 = ITAUQ1 + MAX( 1, Q )
IORGQR = ITAUQ2 + MAX( 1, M - Q )
CALL ZUNGQR( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
$ CHILDINFO )
LORGQRWORKOPT = INT( WORK(1) )
LORGQRWORKMIN = MAX( 1, M - Q )
IORGLQ = ITAUQ2 + MAX( 1, M - Q )
CALL ZUNGLQ( M-Q, M-Q, M-Q, 0, MAX(1,M-Q), 0, WORK, -1,
$ CHILDINFO )
LORGLQWORKOPT = INT( WORK(1) )
LORGLQWORKMIN = MAX( 1, M - Q )
IORBDB = ITAUQ2 + MAX( 1, M - Q )
CALL ZUNBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
$ X21, LDX21, X22, LDX22, 0, 0, 0, 0, 0, 0, WORK,
$ -1, CHILDINFO )
LORBDBWORKOPT = INT( WORK(1) )
LORBDBWORKMIN = LORBDBWORKOPT
LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT,
$ IORBDB + LORBDBWORKOPT ) - 1
LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN,
$ IORBDB + LORBDBWORKMIN ) - 1
WORK(1) = MAX(LWORKOPT,LWORKMIN)
*
IF( LWORK .LT. LWORKMIN
$ .AND. .NOT. ( LQUERY .OR. LRQUERY ) ) THEN
INFO = -22
ELSE IF( LRWORK .LT. LRWORKMIN
$ .AND. .NOT. ( LQUERY .OR. LRQUERY ) ) THEN
INFO = -24
ELSE
LORGQRWORK = LWORK - IORGQR + 1
LORGLQWORK = LWORK - IORGLQ + 1
LORBDBWORK = LWORK - IORBDB + 1
LBBCSDWORK = LRWORK - IBBCSD + 1
END IF
END IF
*
* Abort if any illegal arguments
*
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'ZUNCSD', -INFO )
RETURN
ELSE IF( LQUERY .OR. LRQUERY ) THEN
RETURN
END IF
*
* Transform to bidiagonal block form
*
CALL ZUNBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
$ LDX21, X22, LDX22, THETA, RWORK(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 ZLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 )
CALL ZUNGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR),
$ LORGQRWORK, INFO)
END IF
IF( WANTU2 .AND. M-P .GT. 0 ) THEN
CALL ZLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 )
CALL ZUNGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
$ WORK(IORGQR), LORGQRWORK, INFO )
END IF
IF( WANTV1T .AND. Q .GT. 0 ) THEN
CALL ZLACPY( '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 ZUNGLQ( 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 ZLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T )
CALL ZLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22,
$ V2T(P+1,P+1), LDV2T )
CALL ZUNGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2),
$ WORK(IORGLQ), LORGLQWORK, INFO )
END IF
ELSE
IF( WANTU1 .AND. P .GT. 0 ) THEN
CALL ZLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 )
CALL ZUNGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ),
$ LORGLQWORK, INFO)
END IF
IF( WANTU2 .AND. M-P .GT. 0 ) THEN
CALL ZLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 )
CALL ZUNGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2),
$ WORK(IORGLQ), LORGLQWORK, INFO )
END IF
IF( WANTV1T .AND. Q .GT. 0 ) THEN
CALL ZLACPY( '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 ZUNGQR( 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 ZLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T )
CALL ZLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22,
$ V2T(P+1,P+1), LDV2T )
CALL ZUNGQR( 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 ZBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA,
$ RWORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
$ LDV2T, RWORK(IB11D), RWORK(IB11E), RWORK(IB12D),
$ RWORK(IB12E), RWORK(IB21D), RWORK(IB21E),
$ RWORK(IB22D), RWORK(IB22E), RWORK(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 ZLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK )
ELSE
CALL ZLAPMR( .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 ZLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
ELSE
CALL ZLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK )
END IF
END IF
*
RETURN
*
* End ZUNCSD
*
END
|
