table of contents
| unbdb(3) | LAPACK | unbdb(3) |
NAME¶
unbdb - {un,or}bdb: bidiagonalize partitioned unitary matrix, step in uncsd
SYNOPSIS¶
Functions¶
subroutine cunbdb (trans, signs, m, p, q, x11, ldx11, x12,
ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work,
lwork, info)
CUNBDB subroutine dorbdb (trans, signs, m, p, q, x11, ldx11,
x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2,
work, lwork, info)
DORBDB subroutine sorbdb (trans, signs, m, p, q, x11, ldx11,
x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2,
work, lwork, info)
SORBDB subroutine zunbdb (trans, signs, m, p, q, x11, ldx11,
x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2,
work, lwork, info)
ZUNBDB
Detailed Description¶
Function Documentation¶
subroutine cunbdb (character trans, character signs, integer m, integer p, integer q, complex, dimension( ldx11, * ) x11, integer ldx11, complex, dimension( ldx12, * ) x12, integer ldx12, complex, dimension( ldx21, * ) x21, integer ldx21, complex, dimension( ldx22, * ) x22, integer ldx22, real, dimension( * ) theta, real, dimension( * ) phi, complex, dimension( * ) taup1, complex, dimension( * ) taup2, complex, dimension( * ) tauq1, complex, dimension( * ) tauq2, complex, dimension( * ) work, integer lwork, integer info)¶
CUNBDB
Purpose:
CUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned unitary matrix X:
[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**H
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]
X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See CUNCSD
for details.)
The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.
B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS
SIGNS is CHARACTER
= 'O': The lower-left block is made nonpositive (the
'other' convention);
otherwise: The upper-right block is made nonpositive (the
'default' convention).
M
M is INTEGER
The number of rows and columns in X.
P
P is INTEGER
The number of rows in X11 and X12. 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-left block of the unitary matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = 'T', and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.
LDX11
LDX11 is INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q.
X12
X12 is COMPLEX array, dimension (LDX12,M-Q)
On entry, the top-right block of the unitary matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X12) specify the first P reflectors for
Q2;
else TRANS = 'T', and
the columns of tril(X12) specify the first P reflectors
for Q2.
LDX12
LDX12 is INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q.
X21
X21 is COMPLEX array, dimension (LDX21,Q)
On entry, the bottom-left block of the unitary matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X21) specify reflectors for P2;
else TRANS = 'T', and
the rows of triu(X21) specify reflectors for P2.
LDX21
LDX21 is INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q.
X22
X22 is COMPLEX array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the unitary matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = 'T', and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.
LDX22
LDX22 is INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.
PHI
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles 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.
TAUQ2
TAUQ2 is COMPLEX array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.
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 bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. 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, Q1, and Q2 are represented as products of elementary
reflectors. See CUNCSD for details on generating P1, P2, Q1, and Q2
using CUNGQR and CUNGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine dorbdb (character trans, character signs, integer m, integer p, integer q, double precision, dimension( ldx11, * ) x11, integer ldx11, double precision, dimension( ldx12, * ) x12, integer ldx12, double precision, dimension( ldx21, * ) x21, integer ldx21, double precision, dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double precision, dimension( * ) phi, double precision, dimension( * ) taup1, double precision, dimension( * ) taup2, double precision, dimension( * ) tauq1, double precision, dimension( * ) tauq2, double precision, dimension( * ) work, integer lwork, integer info)¶
DORBDB
Purpose:
DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned orthogonal matrix X:
[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]
X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See DORCSD
for details.)
The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.
B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS
SIGNS is CHARACTER
= 'O': The lower-left block is made nonpositive (the
'other' convention);
otherwise: The upper-right block is made nonpositive (the
'default' convention).
M
M is INTEGER
The number of rows and columns in X.
P
P is INTEGER
The number of rows in X11 and X12. 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-left block of the orthogonal matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = 'T', and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.
LDX11
LDX11 is INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q.
X12
X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
On entry, the top-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X12) specify the first P reflectors for
Q2;
else TRANS = 'T', and
the columns of tril(X12) specify the first P reflectors
for Q2.
LDX12
LDX12 is INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q.
X21
X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom-left block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X21) specify reflectors for P2;
else TRANS = 'T', and
the rows of triu(X21) specify reflectors for P2.
LDX21
LDX21 is INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q.
X22
X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = 'T', and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.
LDX22
LDX22 is INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA
THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.
PHI
PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles 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.
TAUQ2
TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.
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 bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. 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, Q1, and Q2 are represented as products of elementary
reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
using DORGQR and DORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine sorbdb (character trans, character signs, integer m, integer p, integer q, real, dimension( ldx11, * ) x11, integer ldx11, real, dimension( ldx12, * ) x12, integer ldx12, real, dimension( ldx21, * ) x21, integer ldx21, real, dimension( ldx22, * ) x22, integer ldx22, real, dimension( * ) theta, real, dimension( * ) phi, real, dimension( * ) taup1, real, dimension( * ) taup2, real, dimension( * ) tauq1, real, dimension( * ) tauq2, real, dimension( * ) work, integer lwork, integer info)¶
SORBDB
Purpose:
SORBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned orthogonal matrix X:
[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]
X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See SORCSD
for details.)
The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.
B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS
SIGNS is CHARACTER
= 'O': The lower-left block is made nonpositive (the
'other' convention);
otherwise: The upper-right block is made nonpositive (the
'default' convention).
M
M is INTEGER
The number of rows and columns in X.
P
P is INTEGER
The number of rows in X11 and X12. 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-left block of the orthogonal matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = 'T', and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.
LDX11
LDX11 is INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q.
X12
X12 is REAL array, dimension (LDX12,M-Q)
On entry, the top-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X12) specify the first P reflectors for
Q2;
else TRANS = 'T', and
the columns of tril(X12) specify the first P reflectors
for Q2.
LDX12
LDX12 is INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q.
X21
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom-left block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X21) specify reflectors for P2;
else TRANS = 'T', and
the rows of triu(X21) specify reflectors for P2.
LDX21
LDX21 is INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q.
X22
X22 is REAL array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = 'T', and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.
LDX22
LDX22 is INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA
THETA is REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.
PHI
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles 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.
TAUQ2
TAUQ2 is REAL array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.
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 bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. 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, Q1, and Q2 are represented as products of elementary
reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2
using SORGQR and SORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):33-65, 2009.
subroutine zunbdb (character trans, character signs, integer m, integer p, integer q, complex*16, dimension( ldx11, * ) x11, integer ldx11, complex*16, dimension( ldx12, * ) x12, integer ldx12, complex*16, dimension( ldx21, * ) x21, integer ldx21, complex*16, dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double precision, dimension( * ) phi, complex*16, dimension( * ) taup1, complex*16, dimension( * ) taup2, complex*16, dimension( * ) tauq1, complex*16, dimension( * ) tauq2, complex*16, dimension( * ) work, integer lwork, integer info)¶
ZUNBDB
Purpose:
ZUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned unitary matrix X:
[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**H
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]
X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See ZUNCSD
for details.)
The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.
B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS
SIGNS is CHARACTER
= 'O': The lower-left block is made nonpositive (the
'other' convention);
otherwise: The upper-right block is made nonpositive (the
'default' convention).
M
M is INTEGER
The number of rows and columns in X.
P
P is INTEGER
The number of rows in X11 and X12. 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-left block of the unitary matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = 'T', and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.
LDX11
LDX11 is INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q.
X12
X12 is COMPLEX*16 array, dimension (LDX12,M-Q)
On entry, the top-right block of the unitary matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X12) specify the first P reflectors for
Q2;
else TRANS = 'T', and
the columns of tril(X12) specify the first P reflectors
for Q2.
LDX12
LDX12 is INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q.
X21
X21 is COMPLEX*16 array, dimension (LDX21,Q)
On entry, the bottom-left block of the unitary matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X21) specify reflectors for P2;
else TRANS = 'T', and
the rows of triu(X21) specify reflectors for P2.
LDX21
LDX21 is INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q.
X22
X22 is COMPLEX*16 array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the unitary matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = 'T', and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.
LDX22
LDX22 is INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA
THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.
PHI
PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles 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.
TAUQ2
TAUQ2 is COMPLEX*16 array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.
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 bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. 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, Q1, and Q2 are represented as products of elementary
reflectors. See ZUNCSD for details on generating P1, P2, Q1, and Q2
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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