- SUBROUTINE ZBDSQR(
- 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
- DOUBLE PRECISION D( * ), E( * ), RWORK( * )
- COMPLEX*16 C( LDC, * ), U( LDU, * ), VT( LDVT, * )
PURPOSEZBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form
B = Q * S * P**H
where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**H*VT instead of P**H, for given complex input matrices U and VT. When U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by ZGEBRD, then
A = (U*Q) * S * (P**H*VT)
is the SVD of A. Optionally, the subroutine may also compute Q**H*C for a given complex input matrix 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N-1)
- On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.
- VT (input/output) COMPLEX*16 array, dimension (LDVT, NCVT)
- On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**H * VT. 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*16 array, dimension (LDU, N)
- On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.
- LDU (input) INTEGER
- The leading dimension of the array U. LDU >= max(1,NRU).
- C (input/output) COMPLEX*16 array, dimension (LDC, NCC)
- On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**H * C. 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) DOUBLE PRECISION array, dimension (2*N)
- if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise
- 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 DOUBLE PRECISION, 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.