SORCSD(3) LAPACK routine of NEC Numeric Library Collection SORCSD(3)
NAME
SORCSD
SYNOPSIS
RECURSIVE SUBROUTINE SORCSD (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS,
SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22,
THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK,
IWORK, INFO)
PURPOSE
SORCSD computes the CS decomposition of an M-by-M partitioned
orthogonal matrix X:
[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
X = [-----------] = [---------] [---------------------] [---------] .
[ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
[ 0 S 0 | 0 C 0 ]
[ 0 0 I | 0 0 0 ]
X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
(M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
which R = MIN(P,M-P,Q,M-Q).
ARGUMENTS
JOBU1 (input)
JOBU1 is CHARACTER
= 'Y': U1 is computed;
otherwise: U1 is not computed.
JOBU2 (input)
JOBU2 is CHARACTER
= 'Y': U2 is computed;
otherwise: U2 is not computed.
JOBV1T (input)
JOBV1T is CHARACTER
= 'Y': V1T is computed;
otherwise: V1T is not computed.
JOBV2T (input)
JOBV2T is CHARACTER
= 'Y': V2T is computed;
otherwise: V2T is not computed.
TRANS (input)
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 (input)
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 (input)
M is INTEGER
The number of rows and columns in X.
P (input)
P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
Q (input)
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11 (input/output)
X11 is REAL array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX11 (input)
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).
X12 (input/output)
X12 is REAL array, dimension (LDX12,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX12 (input)
LDX12 is INTEGER
The leading dimension of X12. LDX12 >= MAX(1,P).
X21 (input/output)
X21 is REAL array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX21 (input)
LDX21 is INTEGER
The leading dimension of X11. LDX21 >= MAX(1,M-P).
X22 (input/output)
X22 is REAL array, dimension (LDX22,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX22 (input)
LDX22 is INTEGER
The leading dimension of X11. LDX22 >= MAX(1,M-P).
THETA (output)
THETA is REAL array, dimension (R), in which R =
MIN(P,M-P,Q,M-Q).
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
U1 (output)
U1 is REAL array, dimension (P)
If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
LDU1 (input)
LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
MAX(1,P).
U2 (output)
U2 is REAL array, dimension (M-P)
If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.
LDU2 (input)
LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
MAX(1,M-P).
V1T (output)
V1T is REAL array, dimension (Q)
If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.
LDV1T (input)
LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
MAX(1,Q).
V2T (output)
V2T is REAL array, dimension (M-Q)
If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
matrix V2**T.
LDV2T (input)
LDV2T is INTEGER
The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
MAX(1,M-Q).
WORK (output)
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI's.
LWORK (input)
LWORK is INTEGER
The dimension of the array WORK.
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.
IWORK (output)
IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
INFO (output)
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: SBBCSD did not converge. See the description of WORK
above for details.
LAPACK routine 31 October 2017 SORCSD(3)