NAME

unbdb2 - {un,or}bdb2: step in uncsd2by1

SYNOPSIS

Functions

subroutine cunbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
CUNBDB2 subroutine dorbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
DORBDB2 subroutine sorbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
SORBDB2 subroutine zunbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
ZUNBDB2

Detailed Description

Function Documentation

subroutine cunbdb2 (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)

CUNBDB2

Purpose:



 CUNBDB2 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. P must be no larger than M-P,


 Q, or M-Q. Routines CUNBDB1, CUNBDB3, and CUNBDB4 handle cases in


 which P 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 P-by-P 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 <= min(M-P,Q,M-Q).



          Q is INTEGER


           The number of columns in X11 and X21. 0 <= Q <= M.

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-1)


           The scalar factors of the elementary reflectors that define


           P1.

TAUP2



          TAUP2 is COMPLEX array, dimension (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.

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 cunbdb2.f.

subroutine dorbdb2 (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)

DORBDB2

Purpose:



 DORBDB2 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. P must be no larger than M-P,


 Q, or M-Q. Routines DORBDB1, DORBDB3, and DORBDB4 handle cases in


 which P 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 P-by-P 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 <= min(M-P,Q,M-Q).



          Q is INTEGER


           The number of columns in X11 and X21. 0 <= Q <= M.

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-1)


           The scalar factors of the elementary reflectors that define


           P1.

TAUP2



          TAUP2 is DOUBLE PRECISION array, dimension (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.

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 200 of file dorbdb2.f.

subroutine sorbdb2 (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)

SORBDB2

Purpose:



 SORBDB2 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. P must be no larger than M-P,


 Q, or M-Q. Routines SORBDB1, SORBDB3, and SORBDB4 handle cases in


 which P 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 P-by-P 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 <= min(M-P,Q,M-Q).



          Q is INTEGER


           The number of columns in X11 and X21. 0 <= Q <= M.

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-1)


           The scalar factors of the elementary reflectors that define


           P1.

TAUP2



          TAUP2 is REAL array, dimension (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.

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 199 of file sorbdb2.f.

subroutine zunbdb2 (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)

ZUNBDB2

Purpose:



 ZUNBDB2 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. P must be no larger than M-P,


 Q, or M-Q. Routines ZUNBDB1, ZUNBDB3, and ZUNBDB4 handle cases in


 which P 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 P-by-P 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 <= min(M-P,Q,M-Q).



          Q is INTEGER


           The number of columns in X11 and X21. 0 <= Q <= M.

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-1)


           The scalar factors of the elementary reflectors that define


           P1.

TAUP2



          TAUP2 is COMPLEX*16 array, dimension (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.

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 199 of file zunbdb2.f.

Author

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