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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