CBDSQR - 計算一個實 (real) NxN 上/下
(upper/lower) 三角 (bidiagonal) 矩陣 B
的單值分解 (singular value decomposition (SVD))
- SUBROUTINE CBDSQR(
- UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC,
RWORK, INFO )
CHARACTER UPLO INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU REAL D( * ), E( *
), RWORK( * ) COMPLEX C( LDC, * ), U( LDU, * ), VT( LDVT, * )
CBDSQR computes the singular value decomposition (SVD) of a real N-by-N (upper
or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P),
where S is a diagonal matrix with non-negative diagonal elements (the singular
values of B), and Q and P are orthogonal matrices.
The routine computes S, and optionally computes U * Q, P' * VT, or Q' * C, for
given complex input matrices U, VT, and C.
See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed
High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note
#3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990)
and
"Accurate singular values and differential qd algorithms," by B.
Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department,
University of California at Berkeley, July 1992 for a detailed description of
the algorithm.
- UPLO (input) CHARACTER*1
- = 'U': B is upper bidiagonal;
= 'L': B is lower bidiagonal.
- N (input) INTEGER
- The order of the matrix B. N >= 0.
- NCVT (input) INTEGER
- The number of columns of the matrix VT. NCVT >= 0.
- NRU (input) INTEGER
- The number of rows of the matrix U. NRU >= 0.
- NCC (input) INTEGER
- The number of columns of the matrix C. NCC >= 0.
- D (input/output) REAL array, dimension (N)
- On entry, the n diagonal elements of the bidiagonal matrix
B. On exit, if INFO=0, the singular values of B in decreasing order.
- E (input/output) REAL array, dimension (N)
- On entry, the elements of E contain the offdiagonal
elements of of the bidiagonal matrix whose SVD is desired. On normal exit
(INFO = 0), E is destroyed. If the algorithm does not converge (INFO >
0), D and E will contain the diagonal and superdiagonal elements of a
bidiagonal matrix orthogonally equivalent to the one given as input. E(N)
is used for workspace.
- VT (input/output) COMPLEX array, dimension (LDVT,
NCVT)
- On entry, an N-by-NCVT matrix VT. On exit, VT is
overwritten by P' * VT. VT is not referenced if NCVT = 0.
- LDVT (input) INTEGER
- The leading dimension of the array VT. LDVT >= max(1,N)
if NCVT > 0; LDVT >= 1 if NCVT = 0.
- U (input/output) COMPLEX array, dimension (LDU, N)
- On entry, an NRU-by-N matrix U. On exit, U is overwritten
by U * Q. U is not referenced if NRU = 0.
- LDU (input) INTEGER
- The leading dimension of the array U. LDU >=
max(1,NRU).
- C (input/output) COMPLEX array, dimension (LDC, NCC)
- On entry, an N-by-NCC matrix C. On exit, C is overwritten
by Q' * C. C is not referenced if NCC = 0.
- LDC (input) INTEGER
- The leading dimension of the array C. LDC >= max(1,N) if
NCC > 0; LDC >=1 if NCC = 0.
- RWORK (workspace) REAL array, dimension (4*N)
- INFO (output) INTEGER
- = 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
> 0: the algorithm did not converge; D and E contain the elements of a
bidiagonal matrix which is orthogonally similar to the input matrix B; if
INFO = i, i elements of E have not converged to zero.
- TOLMUL REAL, default = max(10,min(100,EPS**(-1/8)))
- TOLMUL controls the convergence criterion of the QR loop.
If it is positive, TOLMUL*EPS is the desired relative precision in the
computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is
the desired absolute accuracy in the computed singular values (corresponds
to relative accuracy abs(TOLMUL*EPS) in the largest singular value.
abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for
fast convergence) and .1/EPS (for there to be some accuracy in the
results). Default is to lose at either one eighth or 2 of the available
decimal digits in each computed singular value (whichever is
smaller).
- MAXITR INTEGER, default = 6
- MAXITR controls the maximum number of passes of the
algorithm through its inner loop. The algorithms stops (and so fails to
converge) if the number of passes through the inner loop exceeds
MAXITR*N**2.
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