unbdb1(3)

Library Functions Manual

unbdb1(3)

NAME

unbdb1 - {un,or}bdb1: step in uncsd2by1

SYNOPSIS

Functions

subroutine cunbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
CUNBDB1 subroutine dorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
DORBDB1 subroutine sorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
SORBDB1 subroutine zunbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
ZUNBDB1

Detailed Description

Function Documentation

subroutine cunbdb1 (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() work, integer lwork, integer info)

CUNBDB1

Purpose:

 CUNBDB1 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. Q must be no larger than P,
 M-P, or M-Q. Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in
 which 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 Q-by-Q bidiagonal matrices represented implicitly by
 angles THETA, PHI.

Parameters

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

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

          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 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 (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.

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 200 of file cunbdb1.f.

subroutine dorbdb1 (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() work, integer lwork, integer info)

DORBDB1

Purpose:

 DORBDB1 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. Q must be no larger than P,
 M-P, or M-Q. Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in
 which 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 Q-by-Q bidiagonal matrices represented implicitly by
 angles THETA, PHI.

Parameters

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

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

          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 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 (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.

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 201 of file dorbdb1.f.

subroutine sorbdb1 (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() work, integer lwork, integer info)

SORBDB1

Purpose:

 SORBDB1 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. Q must be no larger than P,
 M-P, or M-Q. Routines SORBDB2, SORBDB3, and SORBDB4 handle cases in
 which 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 Q-by-Q bidiagonal matrices represented implicitly by
 angles THETA, PHI.

Parameters

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

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

          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 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 (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.

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 201 of file sorbdb1.f.

subroutine zunbdb1 (integer m, integer p, integer q, complex16, dimension(ldx11,) x11, integer ldx11, complex16, dimension(ldx21,) x21, integer ldx21, double precision, dimension() theta, double precision, dimension() phi, complex16, dimension() taup1, complex16, dimension() taup2, complex16, dimension() tauq1, complex16, dimension() work, integer lwork, integer info)

ZUNBDB1

Purpose:

 ZUNBDB1 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. Q must be no larger than P,
 M-P, or M-Q. Routines ZUNBDB2, ZUNBDB3, and ZUNBDB4 handle cases in
 which 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 Q-by-Q bidiagonal matrices represented implicitly by
 angles THETA, PHI.

Parameters

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

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

          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 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 (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.

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 201 of file zunbdb1.f.

Author

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