dbdsqr (l)  Linux Man Pages
dbdsqr: computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real NbyN (upper or lower) bidiagonal matrix B using the implicit zeroshift QR algorithm
NAME
DBDSQR  computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real NbyN (upper or lower) bidiagonal matrix B using the implicit zeroshift QR algorithmSYNOPSIS
 SUBROUTINE DBDSQR(
 UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO )
 CHARACTER UPLO
 INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
 DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
PURPOSE
DBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real NbyN (upper or lower) bidiagonal matrix B using the implicit zeroshift QR algorithm. The SVD of B has the formB
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**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by DGEBRD, then
A
is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real 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. 873912, Sept 1990) and
"Accurate singular values and differential qd algorithms," by B. Parlett and V. Fernando, Technical Report CPAM554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm.
ARGUMENTS
 UPLO (input) CHARACTER*1

= aqUaq: B is upper bidiagonal;
= aqLaq: 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 (N1)
 On entry, the N1 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) DOUBLE PRECISION array, dimension (LDVT, NCVT)
 On entry, an NbyNCVT matrix VT. On exit, VT is overwritten by P**T * 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) DOUBLE PRECISION array, dimension (LDU, N)
 On entry, an NRUbyN 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) DOUBLE PRECISION array, dimension (LDC, NCC)
 On entry, an NbyNCC matrix C. On exit, C is overwritten by Q**T * 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.
 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
 INFO (output) INTEGER

= 0: successful exit
< 0: If INFO = i, the ith argument had an illegal value
> 0: if NCVT = NRU = NCC = 0, = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop) = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks) else NCVT = NRU = NCC = 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.
PARAMETERS
 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.