1SLAED4(1)                LAPACK routine (version 3.1)                SLAED4(1)
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NAME

6       SLAED4  -  compute  the I-th updated eigenvalue of a symmetric rank-one
7       modification to a diagonal matrix whose elements are given in the array
8       d, and that   D(i) < D(j) for i < j  and that RHO > 0
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SYNOPSIS

11       SUBROUTINE SLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
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13           INTEGER        I, INFO, N
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15           REAL           DLAM, RHO
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17           REAL           D( * ), DELTA( * ), Z( * )
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PURPOSE

20       This  subroutine  computes  the  I-th updated eigenvalue of a symmetric
21       rank-one modification to a diagonal matrix whose elements are given  in
22       the  array  d,  and  that no loss in generality.  The rank-one modified
23       system is thus
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25                  diag( D )  +  RHO *  Z * Z_transpose.
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27       where we assume the Euclidean norm of Z is 1.
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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

34       N      (input) INTEGER
35              The length of all arrays.
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37       I      (input) INTEGER
38              The index of the eigenvalue to be computed.  1 <= I <= N.
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40       D      (input) REAL array, dimension (N)
41              The original eigenvalues.  It is assumed that they are in order,
42              D(I) < D(J)  for I < J.
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44       Z      (input) REAL array, dimension (N)
45              The components of the updating vector.
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47       DELTA  (output) REAL array, dimension (N)
48              If N .GT. 2, DELTA contains (D(j) - lambda_I) in its  j-th  com‐
49              ponent.   If  N = 1, then DELTA(1) = 1. If N = 2, see SLAED5 for
50              detail. The vector DELTA contains the information  necessary  to
51              construct the eigenvectors by SLAED3 and SLAED9.
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53       RHO    (input) REAL
54              The scalar in the symmetric updating formula.
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56       DLAM   (output) REAL
57              The computed lambda_I, the I-th updated eigenvalue.
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59       INFO   (output) INTEGER
60              = 0:  successful exit
61              > 0:  if INFO = 1, the updating process failed.
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PARAMETERS

64       Logical  variable  ORGATI  (origin-at-i?)  is  used  for distinguishing
65       whether D(i) or D(i+1) is treated as the origin.
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67       ORGATI = .true.    origin at i ORGATI = .false.   origin at i+1
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69       Logical variable SWTCH3 (switch-for-3-poles?) is for noting if  we  are
70       working with THREE poles!
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72       MAXIT is the maximum number of iterations allowed for each eigenvalue.
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74       Further Details ===============
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76       Based  on contributions by Ren-Cang Li, Computer Science Division, Uni‐
77       versity of California at Berkeley, USA
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81 LAPACK routine (version 3.1)    November 2006                       SLAED4(1)