CUNBDB (3) Linux Manual Page
cunbdb.f –
Synopsis
Functions/Subroutines
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
Function/Subroutine Documentation
subroutine cunbdb (characterTRANS, characterSIGNS, integerM, integerP, integerQ, complex, dimension( ldx11, * )X11, integerLDX11, complex, dimension( ldx12, * )X12, integerLDX12, complex, dimension( ldx21, * )X21, integerLDX21, complex, dimension( ldx22, * )X22, integerLDX22, real, dimension( * )THETA, real, dimension( * )PHI, complex, dimension( * )TAUP1, complex, dimension( * )TAUP2, complex, dimension( * )TAUQ1, complex, dimension( * )TAUQ2, complex, dimension( * )WORK, integerLWORK, integerINFO)
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
SIGNS
= ‘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 is CHARACTER
M
= ‘O’: The lower-left block is made nonpositive (the
"other" convention);
otherwise: The upper-right block is made nonpositive (the
"default" convention).M is INTEGER
P
The number of rows and columns in X.P is INTEGER
Q
The number of rows in X11 and X12. 0 <= P <= M.Q is INTEGER
X11
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q).X11 is COMPLEX array, dimension (LDX11,Q)
LDX11
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 is INTEGER
X12
The leading dimension of X11. If TRANS = ‘N’, then LDX11 >=
P; else LDX11 >= Q.X12 is COMPLEX array, dimension (LDX12,M-Q)
LDX12
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 is INTEGER
X21
The leading dimension of X12. If TRANS = ‘N’, then LDX12 >=
P; else LDX11 >= M-Q.X21 is COMPLEX array, dimension (LDX21,Q)
LDX21
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 is INTEGER
X22
The leading dimension of X21. If TRANS = ‘N’, then LDX21 >=
M-P; else LDX21 >= Q.X22 is COMPLEX array, dimension (LDX22,M-Q)
LDX22
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 is INTEGER
THETA
The leading dimension of X22. If TRANS = ‘N’, then LDX22 >=
M-P; else LDX22 >= M-Q.THETA is REAL array, dimension (Q)
PHI
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.PHI is REAL array, dimension (Q-1)
TAUP1
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.TAUP1 is COMPLEX array, dimension (P)
TAUP2
The scalar factors of the elementary reflectors that define
P1.TAUP2 is COMPLEX array, dimension (M-P)
TAUQ1
The scalar factors of the elementary reflectors that define
P2.TAUQ1 is COMPLEX array, dimension (Q)
TAUQ2
The scalar factors of the elementary reflectors that define
Q1.TAUQ2 is COMPLEX array, dimension (M-Q)
WORK
The scalar factors of the elementary reflectors that define
Q2.WORK is COMPLEX array, dimension (LWORK)
LWORKLWORK is INTEGER
INFO
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 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.
Date:
- November 2013
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.
Definition at line 286 of file cunbdb.f.