dbdsqr(3)

NAME

dbdsqr.f –

SYNOPSIS

Functions/Subroutines

subroutine dbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)
DBDSQR

Function/Subroutine Documentation

subroutine dbdsqr (characterUPLO, integerN, integerNCVT, integerNRU, integerNCC, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( ldvt, * )VT, integerLDVT, double precision, dimension( ldu, * )U, integerLDU, double precision, dimension( ldc, * )C, integerLDC, double precision, dimension( * )WORK, integerINFO)

DBDSQR

Purpose:

DBDSQR 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**T

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**TVT instead of PT, 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 = UBVT, as computed by DGEBRD, then A = (U*Q) * S * (PTVT) is the SVD of A. Optionally, the subroutine may also compute Q**TC 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. 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 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 is 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 is 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 is DOUBLE PRECISION array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**T * 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 is DOUBLE PRECISION 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 is DOUBLE PRECISION array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**T * 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.

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

INFO


INFO is INTEGER = 0 : successful exit < 0 : If INFO = -i, the i - th 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.

Internal 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

: November 2011

Definition at line 230 of file dbdsqr.f.

Author