slasd4()

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.

Download SLASD4 + dependencies [TGZ] [ZIP] [TXT]

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

[in]	N	N is INTEGER The length of all arrays.
[in]	I	I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.
[in]	D	D is REAL array, dimension ( N ) The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J.
[in]	Z	Z is REAL array, dimension ( N ) The components of the updating vector.
[out]	DELTA	DELTA is REAL 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.
[in]	RHO	RHO is REAL The scalar in the symmetric updating formula.
[out]	SIGMA	SIGMA is REAL The computed sigma_I, the I-th updated eigenvalue.
[out]	WORK	WORK is REAL 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.
[out]	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