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cbdsqr-f(3)

cbdsqr.f –

Synopsis

Functions/Subroutines

subroutine cbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO)
CBDSQR

Function/Subroutine Documentation

subroutine cbdsqr (characterUPLO, integerN, integerNCVT, integerNRU, integerNCC, real, dimension( * )D, real, dimension( * )E, complex, dimension( ldvt, * )VT, integerLDVT, complex, dimension( ldu, * )U, integerLDU, complex, dimension( ldc, * )C, integerLDC, real, dimension( * )RWORK, integerINFO)

CBDSQR

Purpose:

 CBDSQR 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 UQ instead of Q, and, if right singular
vectors are requested, this subroutine returns P**HVT 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 = UBVT, as computed by CGEBRD, then


A = (UQ) * S * (P**HVT)


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.

 

Parameters:

UPLO

 UPLO is CHARACTER*1
 = 'U': B is upper bidiagonal;
 = 'L': B is lower bidiagonal.

N

 N is INTEGER
 The order of the matrix B. N >= 0.

NCVT

 NCVT is INTEGER
 The number of columns of the matrix VT. NCVT >= 0.

NRU

 NRU is INTEGER
 The number of rows of the matrix U. NRU >= 0.

NCC

 NCC is INTEGER
 The number of columns of the matrix C. NCC >= 0.

D

 D is 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

 E is REAL 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

 VT is COMPLEX 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

 LDVT is INTEGER
 The leading dimension of the array VT.
 LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

U

 U is COMPLEX 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

 LDU is INTEGER
 The leading dimension of the array U. LDU >= max(1,NRU).

C

 C is COMPLEX 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

 LDC is INTEGER
 The leading dimension of the array C.
 LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

RWORK

 RWORK is REAL array, dimension (2*N)
 if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise

INFO

 INFO is 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.

 

Internal Parameters:

 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.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Definition at line 223 of file cbdsqr.f.

Author

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