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.
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.
Arguments
| TRANS |
(input) 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 |
(input) CHARACTER
= 'O': The lower-left block is made nonpositive (the
"other" convention); otherwise: The upper-right block is made nonpositive (the "default" convention). |
| M |
(input) INTEGER
The number of rows and columns in X.
|
| P |
(input) INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
|
| Q |
(input) INTEGER
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q). |
| X11 |
(input/output) 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 |
(input) INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q. |
| X12 |
(input/output) CMPLX 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 |
(input) INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q. |
| X21 |
(input/output) 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 |
(input) INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q. |
| X22 |
(input/output) 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 |
(input) INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q. |
| THETA |
(output) 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 |
(output) 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 |
(output) COMPLEX array, dimension (P)
The scalar factors of the elementary reflectors that define
P1. |
| TAUP2 |
(output) COMPLEX array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2. |
| TAUQ1 |
(output) COMPLEX array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1. |
| TAUQ2 |
(output) COMPLEX array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2. |
| WORK |
(workspace) COMPLEX array, dimension (LWORK)
|
| LWORK |
(input) 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 |
(output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
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.
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.
Reference
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
Algorithms, 50(1):33-65, 2009.
Algorithms, 50(1):33-65, 2009.