LAPACK

NAME¶

lasd4 - lasd4: D&C step: secular equation nonlinear solver

SYNOPSIS¶

Functions¶

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

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.