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
    !>                       convention);
    !>          otherwise:  The upper-right block is made nonpositive (the
    !>                       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
    !>                       convention);
    !>          otherwise:  The upper-right block is made nonpositive (the
    !>                       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
    !>                       convention);
    !>          otherwise:  The upper-right block is made nonpositive (the
    !>                       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
    !>                       convention);
    !>          otherwise:  The upper-right block is made nonpositive (the
    !>                       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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