unbdb4(3)

NAME
unbdb4 - {un,or}bdb4: step in uncsd2by1

SYNOPSIS

  Functions
subroutine cunbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
CUNBDB4
subroutine dorbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
DORBDB4
subroutine sorbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
SORBDB4
subroutine zunbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
ZUNBDB4

Detailed Description

Function Documentation

  subroutine cunbdb4 (integer m, integer p, integer q, complex, dimension(ldx11,*) x11, integer ldx11, complex, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, complex, dimension(*) taup1, complex, dimension(*) taup2, complex, dimension(*) tauq1, complex, dimension(*) phantom, complex, dimension(*) work, integer lwork, integer info)

CUNBDB4

Purpose:

    !>
    !> CUNBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
    !> matrix X with orthonormal columns:
    !>
    !>                            [ B11 ]
    !>      [ X11 ]   [ P1 |    ] [  0  ]
    !>      [-----] = [---------] [-----] Q1**T .
    !>      [ X21 ]   [    | P2 ] [ B21 ]
    !>                            [  0  ]
    !>
    !> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
    !> M-P, or Q. Routines CUNBDB1, CUNBDB2, and CUNBDB3 handle cases in
    !> which M-Q is not the minimum dimension.
    !>
    !> The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
    !> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
    !> Householder vectors.
    !>
    !> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
    !> implicitly by angles THETA, PHI.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>           The number of rows X11 plus the number of rows in X21.
    !>

    P

    !>          P is INTEGER
    !>           The number of rows in X11. 0 <= P <= M.
    !>

    Q

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

    X11

    !>          X11 is COMPLEX array, dimension (LDX11,Q)
    !>           On entry, the top block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X11) specify reflectors for P1 and
    !>           the rows of triu(X11,1) specify reflectors for Q1.
    !>

    LDX11

    !>          LDX11 is INTEGER
    !>           The leading dimension of X11. LDX11 >= P.
    !>

    X21

    !>          X21 is COMPLEX array, dimension (LDX21,Q)
    !>           On entry, the bottom block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X21) specify reflectors for P2.
    !>

    LDX21

    !>          LDX21 is INTEGER
    !>           The leading dimension of X21. LDX21 >= M-P.
    !>

    THETA

    !>          THETA is REAL array, dimension (Q)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    PHI

    !>          PHI is REAL array, dimension (Q-1)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    TAUP1

    !>          TAUP1 is COMPLEX array, dimension (M-Q)
    !>           The scalar factors of the elementary reflectors that define
    !>           P1.
    !>

    TAUP2

    !>          TAUP2 is COMPLEX array, dimension (M-Q)
    !>           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.
    !>

    PHANTOM

    !>          PHANTOM is COMPLEX array, dimension (M)
    !>           The routine computes an M-by-1 column vector Y that is
    !>           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
    !>           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
    !>           Y(P+1:M), respectively.
    !>

    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 upper-bidiagonal blocks B11, B21 are represented implicitly by
    !>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors.
    !>  See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR
    !>  and CUNGLQ.
    !>

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

Definition at line 210 of file cunbdb4.f.

  subroutine dorbdb4 (integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, double precision, dimension(*) taup1, double precision, dimension(*) taup2, double precision, dimension(*) tauq1, double precision, dimension(*) phantom, double precision, dimension(*) work, integer lwork, integer info)

DORBDB4

Purpose:

    !>
    !> DORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
    !> matrix X with orthonormal columns:
    !>
    !>                            [ B11 ]
    !>      [ X11 ]   [ P1 |    ] [  0  ]
    !>      [-----] = [---------] [-----] Q1**T .
    !>      [ X21 ]   [    | P2 ] [ B21 ]
    !>                            [  0  ]
    !>
    !> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
    !> M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
    !> which M-Q is not the minimum dimension.
    !>
    !> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
    !> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
    !> Householder vectors.
    !>
    !> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
    !> implicitly by angles THETA, PHI.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>           The number of rows X11 plus the number of rows in X21.
    !>

    P

    !>          P is INTEGER
    !>           The number of rows in X11. 0 <= P <= M.
    !>

    Q

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

    X11

    !>          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
    !>           On entry, the top block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X11) specify reflectors for P1 and
    !>           the rows of triu(X11,1) specify reflectors for Q1.
    !>

    LDX11

    !>          LDX11 is INTEGER
    !>           The leading dimension of X11. LDX11 >= P.
    !>

    X21

    !>          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
    !>           On entry, the bottom block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X21) specify reflectors for P2.
    !>

    LDX21

    !>          LDX21 is INTEGER
    !>           The leading dimension of X21. LDX21 >= M-P.
    !>

    THETA

    !>          THETA is DOUBLE PRECISION array, dimension (Q)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    PHI

    !>          PHI is DOUBLE PRECISION array, dimension (Q-1)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    TAUP1

    !>          TAUP1 is DOUBLE PRECISION array, dimension (M-Q)
    !>           The scalar factors of the elementary reflectors that define
    !>           P1.
    !>

    TAUP2

    !>          TAUP2 is DOUBLE PRECISION array, dimension (M-Q)
    !>           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.
    !>

    PHANTOM

    !>          PHANTOM is DOUBLE PRECISION array, dimension (M)
    !>           The routine computes an M-by-1 column vector Y that is
    !>           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
    !>           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
    !>           Y(P+1:M), respectively.
    !>

    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 upper-bidiagonal blocks B11, B21 are represented implicitly by
    !>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors.
    !>  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
    !>  and DORGLQ.
    !>

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

Definition at line 210 of file dorbdb4.f.

  subroutine sorbdb4 (integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real, dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1, real, dimension(*) phantom, real, dimension(*) work, integer lwork, integer info)

SORBDB4

Purpose:

    !>
    !> SORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
    !> matrix X with orthonormal columns:
    !>
    !>                            [ B11 ]
    !>      [ X11 ]   [ P1 |    ] [  0  ]
    !>      [-----] = [---------] [-----] Q1**T .
    !>      [ X21 ]   [    | P2 ] [ B21 ]
    !>                            [  0  ]
    !>
    !> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
    !> M-P, or Q. Routines SORBDB1, SORBDB2, and SORBDB3 handle cases in
    !> which M-Q is not the minimum dimension.
    !>
    !> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
    !> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
    !> Householder vectors.
    !>
    !> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
    !> implicitly by angles THETA, PHI.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>           The number of rows X11 plus the number of rows in X21.
    !>

    P

    !>          P is INTEGER
    !>           The number of rows in X11. 0 <= P <= M.
    !>

    Q

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

    X11

    !>          X11 is REAL array, dimension (LDX11,Q)
    !>           On entry, the top block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X11) specify reflectors for P1 and
    !>           the rows of triu(X11,1) specify reflectors for Q1.
    !>

    LDX11

    !>          LDX11 is INTEGER
    !>           The leading dimension of X11. LDX11 >= P.
    !>

    X21

    !>          X21 is REAL array, dimension (LDX21,Q)
    !>           On entry, the bottom block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X21) specify reflectors for P2.
    !>

    LDX21

    !>          LDX21 is INTEGER
    !>           The leading dimension of X21. LDX21 >= M-P.
    !>

    THETA

    !>          THETA is REAL array, dimension (Q)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    PHI

    !>          PHI is REAL array, dimension (Q-1)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    TAUP1

    !>          TAUP1 is REAL array, dimension (M-Q)
    !>           The scalar factors of the elementary reflectors that define
    !>           P1.
    !>

    TAUP2

    !>          TAUP2 is REAL array, dimension (M-Q)
    !>           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.
    !>

    PHANTOM

    !>          PHANTOM is REAL array, dimension (M)
    !>           The routine computes an M-by-1 column vector Y that is
    !>           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
    !>           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
    !>           Y(P+1:M), respectively.
    !>

    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 upper-bidiagonal blocks B11, B21 are represented implicitly by
    !>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors.
    !>  See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
    !>  and SORGLQ.
    !>

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

Definition at line 211 of file sorbdb4.f.

  subroutine zunbdb4 (integer m, integer p, integer q, complex*16, dimension(ldx11,*) x11, integer ldx11, complex*16, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, complex*16, dimension(*) taup1, complex*16, dimension(*) taup2, complex*16, dimension(*) tauq1, complex*16, dimension(*) phantom, complex*16, dimension(*) work, integer lwork, integer info)

ZUNBDB4

Purpose:

    !>
    !> ZUNBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
    !> matrix X with orthonormal columns:
    !>
    !>                            [ B11 ]
    !>      [ X11 ]   [ P1 |    ] [  0  ]
    !>      [-----] = [---------] [-----] Q1**T .
    !>      [ X21 ]   [    | P2 ] [ B21 ]
    !>                            [  0  ]
    !>
    !> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
    !> M-P, or Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB3 handle cases in
    !> which M-Q is not the minimum dimension.
    !>
    !> The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
    !> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
    !> Householder vectors.
    !>
    !> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
    !> implicitly by angles THETA, PHI.
    !>
    !>

Parameters
    M

    !>          M is INTEGER
    !>           The number of rows X11 plus the number of rows in X21.
    !>

    P

    !>          P is INTEGER
    !>           The number of rows in X11. 0 <= P <= M.
    !>

    Q

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

    X11

    !>          X11 is COMPLEX*16 array, dimension (LDX11,Q)
    !>           On entry, the top block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X11) specify reflectors for P1 and
    !>           the rows of triu(X11,1) specify reflectors for Q1.
    !>

    LDX11

    !>          LDX11 is INTEGER
    !>           The leading dimension of X11. LDX11 >= P.
    !>

    X21

    !>          X21 is COMPLEX*16 array, dimension (LDX21,Q)
    !>           On entry, the bottom block of the matrix X to be reduced. On
    !>           exit, the columns of tril(X21) specify reflectors for P2.
    !>

    LDX21

    !>          LDX21 is INTEGER
    !>           The leading dimension of X21. LDX21 >= M-P.
    !>

    THETA

    !>          THETA is DOUBLE PRECISION array, dimension (Q)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    PHI

    !>          PHI is DOUBLE PRECISION array, dimension (Q-1)
    !>           The entries of the bidiagonal blocks B11, B21 are defined by
    !>           THETA and PHI. See Further Details.
    !>

    TAUP1

    !>          TAUP1 is COMPLEX*16 array, dimension (M-Q)
    !>           The scalar factors of the elementary reflectors that define
    !>           P1.
    !>

    TAUP2

    !>          TAUP2 is COMPLEX*16 array, dimension (M-Q)
    !>           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.
    !>

    PHANTOM

    !>          PHANTOM is COMPLEX*16 array, dimension (M)
    !>           The routine computes an M-by-1 column vector Y that is
    !>           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
    !>           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
    !>           Y(P+1:M), respectively.
    !>

    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 upper-bidiagonal blocks B11, B21 are represented implicitly by
    !>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors.
    !>  See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
    !>  and ZUNGLQ.
    !>

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

Definition at line 210 of file zunbdb4.f.

Author

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