1SLASD4(1)           LAPACK auxiliary routine (version 3.2)           SLASD4(1)
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NAME

6       SLASD4  - subroutine compute the square root of the I-th updated eigen‐
7       value of a positive symmetric rank-one modification to a positive diag‐
8       onal matrix whose entries are given as the squares of the corresponding
9       entries in the array d, and that   0 <= D(i) < D(j) for i < j  and that
10       RHO > 0
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SYNOPSIS

13       SUBROUTINE SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
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15           INTEGER        I, INFO, N
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17           REAL           RHO, SIGMA
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19           REAL           D( * ), DELTA( * ), WORK( * ), Z( * )
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PURPOSE

22       This subroutine computes the square root of the I-th updated eigenvalue
23       of a positive symmetric rank-one modification to  a  positive  diagonal
24       matrix  whose  entries  are  given  as the squares of the corresponding
25       entries in the array d, and that no loss in generality.   The  rank-one
26       modified system is thus
27              diag( D ) * diag( D ) +  RHO *  Z * Z_transpose.
28       where we assume the Euclidean norm of Z is 1.
29       The method consists of approximating the rational functions in the sec‐
30       ular equation by simpler interpolating rational functions.
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ARGUMENTS

33       N      (input) INTEGER
34              The length of all arrays.
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36       I      (input) INTEGER
37              The index of the eigenvalue to be computed.  1 <= I <= N.
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39       D      (input) REAL array, dimension ( N )
40              The original eigenvalues.  It is assumed that they are in order,
41              0 <= D(I) < D(J)  for I < J.
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43       Z      (input) REAL array, dimension (N)
44              The components of the updating vector.
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46       DELTA  (output) REAL array, dimension (N)
47              If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th compo‐
48              nent.  If N = 1, then DELTA(1) = 1.  The vector  DELTA  contains
49              the  information necessary to construct the (singular) eigenvec‐
50              tors.
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52       RHO    (input) REAL
53              The scalar in the symmetric updating formula.
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55       SIGMA  (output) REAL
56              The computed sigma_I, the I-th updated eigenvalue.
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58       WORK   (workspace) REAL array, dimension (N)
59              If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th  compo‐
60              nent.  If N = 1, then WORK( 1 ) = 1.
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62       INFO   (output) INTEGER
63              = 0:  successful exit
64              > 0:  if INFO = 1, the updating process failed.
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PARAMETERS

67       Logical  variable  ORGATI  (origin-at-i?)  is  used  for distinguishing
68       whether D(i) or D(i+1) is treated  as  the  origin.   ORGATI  =  .true.
69       origin  at  i  ORGATI = .false.   origin at i+1 Logical variable SWTCH3
70       (switch-for-3-poles?) is for noting if we are working with THREE poles!
71       MAXIT  is the maximum number of iterations allowed for each eigenvalue.
72       Further Details =============== Based on contributions by Ren-Cang  Li,
73       Computer Science Division, University of California at Berkeley, USA
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77 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       SLASD4(1)