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



NAME
       ZUNBDB

SYNOPSIS
       SUBROUTINE ZUNBDB (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
           LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK,
           LWORK, INFO)



PURPOSE
            ZUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
            partitioned unitary matrix X:

                                            [ B11 | B12 0  0 ]
                [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**H
            X = [-----------] = [---------] [----------------] [---------]   .
                [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
                                            [  0  |  0  0  I ]

            X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
            not the case, then X must be transposed and/or permuted. This can be
            done in constant time using the TRANS and SIGNS options. See ZUNCSD
            for details.)

            The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
            (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
            represented implicitly by Householder vectors.

            B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
            implicitly by angles THETA, PHI.




ARGUMENTS
           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 <=
                     MIN(P,M-P,M-Q).

           X11       (input/output)
                     X11 is COMPLEX*16 array, dimension (LDX11,Q)
                     On entry, the top-left block of the unitary matrix to be
                     reduced. On exit, the form depends on TRANS:
                     If TRANS = 'N', then
                        the columns of tril(X11) specify reflectors for P1,
                        the rows of triu(X11,1) specify reflectors for Q1;
                     else TRANS = 'T', and
                        the rows of triu(X11) specify reflectors for P1,
                        the columns of tril(X11,-1) specify reflectors for Q1.

           LDX11     (input)
                     LDX11 is INTEGER
                     The leading dimension of X11. If TRANS = 'N', then LDX11 >=
                     P; else LDX11 >= Q.

           X12       (input/output)
                     X12 is COMPLEX*16 array, dimension (LDX12,M-Q)
                     On entry, the top-right block of the unitary matrix to
                     be reduced. On exit, the form depends on TRANS:
                     If TRANS = 'N', then
                        the rows of triu(X12) specify the first P reflectors for
                        Q2;
                     else TRANS = 'T', and
                        the columns of tril(X12) specify the first P reflectors
                        for Q2.

           LDX12     (input)
                     LDX12 is INTEGER
                     The leading dimension of X12. If TRANS = 'N', then LDX12 >=
                     P; else LDX11 >= M-Q.

           X21       (input/output)
                     X21 is COMPLEX*16 array, dimension (LDX21,Q)
                     On entry, the bottom-left block of the unitary matrix to
                     be reduced. On exit, the form depends on TRANS:
                     If TRANS = 'N', then
                        the columns of tril(X21) specify reflectors for P2;
                     else TRANS = 'T', and
                        the rows of triu(X21) specify reflectors for P2.

           LDX21     (input)
                     LDX21 is INTEGER
                     The leading dimension of X21. If TRANS = 'N', then LDX21 >=
                     M-P; else LDX21 >= Q.

           X22       (input/output)
                     X22 is COMPLEX*16 array, dimension (LDX22,M-Q)
                     On entry, the bottom-right block of the unitary matrix to
                     be reduced. On exit, the form depends on TRANS:
                     If TRANS = 'N', then
                        the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
                        M-P-Q reflectors for Q2,
                     else TRANS = 'T', and
                        the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
                        M-P-Q reflectors for P2.

           LDX22     (input)
                     LDX22 is INTEGER
                     The leading dimension of X22. If TRANS = 'N', then LDX22 >=
                     M-P; else LDX22 >= M-Q.

           THETA     (output)
                     THETA is DOUBLE PRECISION array, dimension (Q)
                     The entries of the bidiagonal blocks B11, B12, B21, B22 can
                     be computed from the angles THETA and PHI. See Further
                     Details.

           PHI       (output)
                     PHI is DOUBLE PRECISION array, dimension (Q-1)
                     The entries of the bidiagonal blocks B11, B12, B21, B22 can
                     be computed from the angles THETA and PHI. See Further
                     Details.

           TAUP1     (output)
                     TAUP1 is COMPLEX*16 array, dimension (P)
                     The scalar factors of the elementary reflectors that define
                     P1.

           TAUP2     (output)
                     TAUP2 is COMPLEX*16 array, dimension (M-P)
                     The scalar factors of the elementary reflectors that define
                     P2.

           TAUQ1     (output)
                     TAUQ1 is COMPLEX*16 array, dimension (Q)
                     The scalar factors of the elementary reflectors that define
                     Q1.

           TAUQ2     (output)
                     TAUQ2 is COMPLEX*16 array, dimension (M-Q)
                     The scalar factors of the elementary reflectors that define
                     Q2.

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (LWORK)

           LWORK     (input)
                     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.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.






FURTHER DETAILS
             The bidiagonal blocks B11, B12, B21, and B22 are represented
             implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
             PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
             lower bidiagonal. 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
             ZUNCSD for details.

             P1, P2, Q1, and Q2 are represented as products of elementary
             reflectors. See ZUNCSD for details on generating P1, P2, Q1, and Q2
             using ZUNGQR and ZUNGLQ.



LAPACK routine                  31 October 2017                      ZUNBDB(3)