1SLASD4(1)           LAPACK auxiliary routine (version 3.1)           SLASD4(1)
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NAME

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

12       SUBROUTINE SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
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14           INTEGER        I, INFO, N
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16           REAL           RHO, SIGMA
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18           REAL           D( * ), DELTA( * ), WORK( * ), Z( * )
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PURPOSE

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

36       N      (input) INTEGER
37              The length of all arrays.
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39       I      (input) INTEGER
40              The index of the eigenvalue to be computed.  1 <= I <= N.
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42       D      (input) REAL array, dimension ( N )
43              The original eigenvalues.  It is assumed that they are in order,
44              0 <= D(I) < D(J)  for I < J.
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46       Z      (input) REAL array, dimension (N)
47              The components of the updating vector.
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49       DELTA  (output) REAL array, dimension (N)
50              If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th compo‐
51              nent.   If  N = 1, then DELTA(1) = 1.  The vector DELTA contains
52              the information necessary to construct the (singular)  eigenvec‐
53              tors.
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55       RHO    (input) REAL
56              The scalar in the symmetric updating formula.
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58       SIGMA  (output) REAL
59              The computed sigma_I, the I-th updated eigenvalue.
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61       WORK   (workspace) REAL array, dimension (N)
62              If  N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th compo‐
63              nent.  If N = 1, then WORK( 1 ) = 1.
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65       INFO   (output) INTEGER
66              = 0:  successful exit
67              > 0:  if INFO = 1, the updating process failed.
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PARAMETERS

70       Logical variable  ORGATI  (origin-at-i?)  is  used  for  distinguishing
71       whether D(i) or D(i+1) is treated as the origin.
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73       ORGATI = .true.    origin at i ORGATI = .false.   origin at i+1
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75       Logical  variable  SWTCH3 (switch-for-3-poles?) is for noting if we are
76       working with THREE poles!
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78       MAXIT is the maximum number of iterations allowed for each eigenvalue.
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80       Further Details ===============
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82       Based on contributions by Ren-Cang Li, Computer Science Division,  Uni‐
83       versity of California at Berkeley, USA
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87 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       SLASD4(1)