Provided by: liblapack-doc_3.12.0-3build1.1_all bug

NAME

       unbdb - {un,or}bdb: bidiagonalize partitioned unitary matrix, step in uncsd

SYNOPSIS

   Functions
       subroutine cunbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22,
           theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
           CUNBDB
       subroutine dorbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22,
           theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
           DORBDB
       subroutine sorbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22,
           theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
           SORBDB
       subroutine zunbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22,
           theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
           ZUNBDB

Detailed Description

Function Documentation

   subroutine cunbdb (character trans, character signs, integer m, integer p, integer q, complex,
       dimension( ldx11, * ) x11, integer ldx11, complex, dimension( ldx12, * ) x12, integer
       ldx12, complex, dimension( ldx21, * ) x21, integer ldx21, complex, dimension( ldx22, * )
       x22, integer ldx22, real, dimension( * ) theta, real, dimension( * ) phi, complex,
       dimension( * ) taup1, complex, dimension( * ) taup2, complex, dimension( * ) tauq1,
       complex, dimension( * ) tauq2, complex, dimension( * ) work, integer lwork, integer info)
       CUNBDB

       Purpose:

            CUNBDB 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 CUNCSD
            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.

       Parameters
           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.

           SIGNS

                     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

                     M is INTEGER
                     The number of rows and columns in X.

           P

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

           Q

                     Q is INTEGER
                     The number of columns in X11 and X21. 0 <= Q <=
                     MIN(P,M-P,M-Q).

           X11

                     X11 is COMPLEX 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

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

           X12

                     X12 is COMPLEX 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

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

           X21

                     X21 is COMPLEX 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

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

           X22

                     X22 is COMPLEX 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

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

           THETA

                     THETA is REAL 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

                     PHI is REAL 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

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

           TAUP2

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

           TAUQ1

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

           TAUQ2

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

           WORK

                     WORK is COMPLEX array, dimension (LWORK)

           LWORK

                     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

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       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
             [1] or CUNCSD for details.

             P1, P2, Q1, and Q2 are represented as products of elementary
             reflectors. See CUNCSD for details on generating P1, P2, Q1, and Q2
             using CUNGQR and CUNGLQ.

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

   subroutine dorbdb (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( * ) phi, double precision, dimension( * ) taup1,
       double precision, dimension( * ) taup2, double precision, dimension( * ) tauq1, double
       precision, dimension( * ) tauq2, double precision, dimension( * ) work, integer lwork,
       integer info)
       DORBDB

       Purpose:

            DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
            partitioned orthogonal matrix X:

                                            [ B11 | B12 0  0 ]
                [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
            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 DORCSD
            for details.)

            The orthogonal 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.

       Parameters
           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.

           SIGNS

                     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

                     M is INTEGER
                     The number of rows and columns in X.

           P

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

           Q

                     Q is INTEGER
                     The number of columns in X11 and X21. 0 <= Q <=
                     MIN(P,M-P,M-Q).

           X11

                     X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
                     On entry, the top-left block of the orthogonal 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

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

           X12

                     X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
                     On entry, the top-right block of the orthogonal 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

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

           X21

                     X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
                     On entry, the bottom-left block of the orthogonal 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

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

           X22

                     X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
                     On entry, the bottom-right block of the orthogonal 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

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

           THETA

                     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

                     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

                     TAUP1 is DOUBLE PRECISION array, dimension (P)
                     The scalar factors of the elementary reflectors that define
                     P1.

           TAUP2

                     TAUP2 is DOUBLE PRECISION array, dimension (M-P)
                     The scalar factors of the elementary reflectors that define
                     P2.

           TAUQ1

                     TAUQ1 is DOUBLE PRECISION array, dimension (Q)
                     The scalar factors of the elementary reflectors that define
                     Q1.

           TAUQ2

                     TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
                     The scalar factors of the elementary reflectors that define
                     Q2.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (LWORK)

           LWORK

                     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

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       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
             [1] or DORCSD for details.

             P1, P2, Q1, and Q2 are represented as products of elementary
             reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
             using DORGQR and DORGLQ.

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

   subroutine sorbdb (character trans, character signs, integer m, integer p, integer q, real,
       dimension( ldx11, * ) x11, integer ldx11, real, dimension( ldx12, * ) x12, integer ldx12,
       real, dimension( ldx21, * ) x21, integer ldx21, real, dimension( ldx22, * ) x22, integer
       ldx22, real, dimension( * ) theta, real, dimension( * ) phi, real, dimension( * ) taup1,
       real, dimension( * ) taup2, real, dimension( * ) tauq1, real, dimension( * ) tauq2, real,
       dimension( * ) work, integer lwork, integer info)
       SORBDB

       Purpose:

            SORBDB simultaneously bidiagonalizes the blocks of an M-by-M
            partitioned orthogonal matrix X:

                                            [ B11 | B12 0  0 ]
                [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
            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 SORCSD
            for details.)

            The orthogonal 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.

       Parameters
           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.

           SIGNS

                     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

                     M is INTEGER
                     The number of rows and columns in X.

           P

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

           Q

                     Q is INTEGER
                     The number of columns in X11 and X21. 0 <= Q <=
                     MIN(P,M-P,M-Q).

           X11

                     X11 is REAL array, dimension (LDX11,Q)
                     On entry, the top-left block of the orthogonal 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

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

           X12

                     X12 is REAL array, dimension (LDX12,M-Q)
                     On entry, the top-right block of the orthogonal 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

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

           X21

                     X21 is REAL array, dimension (LDX21,Q)
                     On entry, the bottom-left block of the orthogonal 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

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

           X22

                     X22 is REAL array, dimension (LDX22,M-Q)
                     On entry, the bottom-right block of the orthogonal 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

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

           THETA

                     THETA is REAL 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

                     PHI is REAL 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

                     TAUP1 is REAL array, dimension (P)
                     The scalar factors of the elementary reflectors that define
                     P1.

           TAUP2

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

           TAUQ1

                     TAUQ1 is REAL array, dimension (Q)
                     The scalar factors of the elementary reflectors that define
                     Q1.

           TAUQ2

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

           WORK

                     WORK is REAL array, dimension (LWORK)

           LWORK

                     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

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       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
             [1] or SORCSD for details.

             P1, P2, Q1, and Q2 are represented as products of elementary
             reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2
             using SORGQR and SORGLQ.

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

   subroutine zunbdb (character trans, character signs, integer m, integer p, integer q,
       complex*16, dimension( ldx11, * ) x11, integer ldx11, complex*16, dimension( ldx12, * )
       x12, integer ldx12, complex*16, dimension( ldx21, * ) x21, integer ldx21, complex*16,
       dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double
       precision, dimension( * ) phi, complex*16, dimension( * ) taup1, complex*16, dimension( *
       ) taup2, complex*16, dimension( * ) tauq1, complex*16, dimension( * ) tauq2, complex*16,
       dimension( * ) work, integer lwork, integer info)
       ZUNBDB

       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.

       Parameters
           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.

           SIGNS

                     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

                     M is INTEGER
                     The number of rows and columns in X.

           P

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

           Q

                     Q is INTEGER
                     The number of columns in X11 and X21. 0 <= Q <=
                     MIN(P,M-P,M-Q).

           X11

                     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

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

           X12

                     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

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

           X21

                     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

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

           X22

                     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

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

           THETA

                     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

                     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

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

           TAUP2

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

           TAUQ1

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

           TAUQ2

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

           WORK

                     WORK is COMPLEX*16 array, dimension (LWORK)

           LWORK

                     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

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       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
             [1] or 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.

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

Author

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