unbdb(3)

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.

Definition at line 284 of file cunbdb.f.

  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.

Definition at line 284 of file dorbdb.f.

  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.

Definition at line 284 of file sorbdb.f.

  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.

Definition at line 284 of file zunbdb.f.

Author

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