SORCSD(3)      LAPACK routine of NEC Numeric Library Collection      SORCSD(3)



NAME
       SORCSD

SYNOPSIS
       RECURSIVE SUBROUTINE SORCSD (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,
           IWORK, INFO)



PURPOSE
            SORCSD 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).




ARGUMENTS
           JOBU1     (input)
                     JOBU1 is CHARACTER
                     = 'Y':      U1 is computed;
                     otherwise:  U1 is not computed.

           JOBU2     (input)
                     JOBU2 is CHARACTER
                     = 'Y':      U2 is computed;
                     otherwise:  U2 is not computed.

           JOBV1T    (input)
                     JOBV1T is CHARACTER
                     = 'Y':      V1T is computed;
                     otherwise:  V1T is not computed.

           JOBV2T    (input)
                     JOBV2T is CHARACTER
                     = 'Y':      V2T is computed;
                     otherwise:  V2T is not computed.

           TRANS     (input)
                     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.

           SIGNS     (input)
                     SIGNS is 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)
                     M is INTEGER
                     The number of rows and columns in X.

           P         (input)
                     P is INTEGER
                     The number of rows in X11 and X12. 0 <= P <= M.

           Q         (input)
                     Q is INTEGER
                     The number of columns in X11 and X21. 0 <= Q <= M.

           X11       (input/output)
                     X11 is REAL array, dimension (LDX11,Q)
                     On entry, part of the orthogonal matrix whose CSD is desired.

           LDX11     (input)
                     LDX11 is INTEGER
                     The leading dimension of X11. LDX11 >= MAX(1,P).

           X12       (input/output)
                     X12 is REAL array, dimension (LDX12,M-Q)
                     On entry, part of the orthogonal matrix whose CSD is desired.

           LDX12     (input)
                     LDX12 is INTEGER
                     The leading dimension of X12. LDX12 >= MAX(1,P).

           X21       (input/output)
                     X21 is REAL array, dimension (LDX21,Q)
                     On entry, part of the orthogonal matrix whose CSD is desired.

           LDX21     (input)
                     LDX21 is INTEGER
                     The leading dimension of X11. LDX21 >= MAX(1,M-P).

           X22       (input/output)
                     X22 is REAL array, dimension (LDX22,M-Q)
                     On entry, part of the orthogonal matrix whose CSD is desired.

           LDX22     (input)
                     LDX22 is INTEGER
                     The leading dimension of X11. LDX22 >= MAX(1,M-P).

           THETA     (output)
                     THETA is REAL 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)
                     U1 is REAL array, dimension (P)
                     If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.

           LDU1      (input)
                     LDU1 is INTEGER
                     The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
                     MAX(1,P).

           U2        (output)
                     U2 is REAL array, dimension (M-P)
                     If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
                     matrix U2.

           LDU2      (input)
                     LDU2 is INTEGER
                     The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
                     MAX(1,M-P).

           V1T       (output)
                     V1T is REAL array, dimension (Q)
                     If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
                     matrix V1**T.

           LDV1T     (input)
                     LDV1T is INTEGER
                     The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
                     MAX(1,Q).

           V2T       (output)
                     V2T is REAL array, dimension (M-Q)
                     If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
                     matrix V2**T.

           LDV2T     (input)
                     LDV2T is INTEGER
                     The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
                     MAX(1,M-Q).

           WORK      (output)
                     WORK is REAL 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.

           LWORK     (input)
                     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.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  SBBCSD did not converge. See the description of WORK
                           above for details.



LAPACK routine                  31 October 2017                      SORCSD(3)