lasd4 - Man Page

lasd4: D&C step: secular equation nonlinear solver

subroutine dlasd4 (n, i, d, z, delta, rho, sigma, work, info)
DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
subroutine slasd4 (n, i, d, z, delta, rho, sigma, work, info)
SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.

Detailed Description

Function Documentation

subroutine dlasd4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision sigma, double precision, dimension( * ) work, integer info)

DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.

Purpose:

 This subroutine computes the square root of the I-th updated
 eigenvalue of a positive symmetric rank-one modification to
 a positive diagonal matrix whose entries are given as the squares
 of the corresponding entries in the array d, and that

        0 <= D(i) < D(j)  for  i < j

 and that RHO > 0. This is arranged by the calling routine, and is
 no loss in generality.  The rank-one modified system is thus

        diag( D ) * diag( D ) +  RHO * Z * Z_transpose.

 where we assume the Euclidean norm of Z is 1.

 The method consists of approximating the rational functions in the
 secular equation by simpler interpolating rational functions.

Parameters

          N is INTEGER
         The length of all arrays.

          I is INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

          D is DOUBLE PRECISION array, dimension ( N )
         The original eigenvalues.  It is assumed that they are in
         order, 0 <= D(I) < D(J)  for I < J.

          Z is DOUBLE PRECISION array, dimension ( N )
         The components of the updating vector.

DELTA

          DELTA is DOUBLE PRECISION array, dimension ( N )
         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
         contains the information necessary to construct the
         (singular) eigenvectors.

RHO

          RHO is DOUBLE PRECISION
         The scalar in the symmetric updating formula.

SIGMA

          SIGMA is DOUBLE PRECISION
         The computed sigma_I, the I-th updated eigenvalue.

WORK

          WORK is DOUBLE PRECISION array, dimension ( N )
         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
         component.  If N = 1, then WORK( 1 ) = 1.

INFO

          INFO is INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

Internal Parameters:

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.

            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1

  Logical variable SWTCH3 (switch-for-3-poles?) is for noting
  if we are working with THREE poles!

  MAXIT is the maximum number of iterations allowed for each
  eigenvalue.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 152 of file dlasd4.f.

subroutine slasd4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real sigma, real, dimension( * ) work, integer info)

SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.