table of contents
| unbdb1(3) | LAPACK | 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
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 <=
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.
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
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 <=
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.
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
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 <=
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.
subroutine zunbdb1 (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(*) 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
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 <=
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.
Author¶
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