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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 |