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

6       SLARRF - initial representation L D L^T and its cluster of close eigen‐
7       values (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ..
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SYNOPSIS

10       SUBROUTINE SLARRF( N, D, L, LD, CLSTRT, CLEND, W, WGAP,  WERR,  SPDIAM,
11                          CLGAPL,  CLGAPR,  PIVMIN, SIGMA, DPLUS, LPLUS, WORK,
12                          INFO )
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14           INTEGER        CLSTRT, CLEND, INFO, N
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16           REAL           CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
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18           REAL           D( * ), DPLUS( * ), L( * ), LD( * ), LPLUS( * ),  W(
19                          * ), WGAP( * ), WERR( * ), WORK( * )
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PURPOSE

22       Given  the  initial representation L D L^T and its cluster of close ei‐
23       genvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...   W(
24       CLEND  ), SLARRF finds a new relatively robust representation L D L^T -
25       SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of
26       L(+) D(+) L(+)^T is relatively isolated.
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ARGUMENTS

30       N       (input) INTEGER
31               The order of the matrix (subblock, if the matrix splitted).
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33       D       (input) REAL             array, dimension (N)
34               The N diagonal elements of the diagonal matrix D.
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36       L       (input) REAL             array, dimension (N-1)
37               The (N-1) subdiagonal elements of the unit bidiagonal matrix L.
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39       LD      (input) REAL             array, dimension (N-1)
40               The (N-1) elements L(i)*D(i).
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42       CLSTRT  (input) INTEGER
43               The index of the first eigenvalue in the cluster.
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45       CLEND   (input) INTEGER
46               The index of the last eigenvalue in the cluster.
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48       W       (input) REAL             array, dimension >=  (CLEND-CLSTRT+1)
49               The  eigenvalue  APPROXIMATIONS  of L D L^T in ascending order.
50               W( CLSTRT ) through W( CLEND ) form the cluster  of  relatively
51               close eigenalues.
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53       WGAP     (input/output)  REAL              array, dimension >=  (CLEND-
54       CLSTRT+1)
55               The separation from the right neighbor eigenvalue in W.
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57       WERR    (input) REAL             array, dimension >=  (CLEND-CLSTRT+1)
58               WERR contain the semiwidth of the uncertainty interval  of  the
59               corresponding eigenvalue APPROXIMATION in W
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61               SPDIAM  (input) estimate of the spectral diameter obtained from
62               the Gerschgorin intervals
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64               CLGAPL, CLGAPR (input) absolute gap on each end of the cluster.
65               Set  by the calling routine to protect against shifts too close
66               to eigenvalues outside the cluster.
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68       PIVMIN  (input) DOUBLE PRECISION
69               The minimum pivot allowed in the Sturm sequence.
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71       SIGMA   (output) REAL
72               The shift used to form L(+) D(+) L(+)^T.
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74       DPLUS   (output) REAL             array, dimension (N)
75               The N diagonal elements of the diagonal matrix D(+).
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77       LPLUS   (output) REAL             array, dimension (N-1)
78               The first (N-1) elements of LPLUS contain the subdiagonal  ele‐
79               ments of the unit bidiagonal matrix L(+).
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81       WORK    (workspace) REAL             array, dimension (2*N)
82               Workspace.
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FURTHER DETAILS

85       Based on contributions by
86          Beresford Parlett, University of California, Berkeley, USA
87          Jim Demmel, University of California, Berkeley, USA
88          Inderjit Dhillon, University of Texas, Austin, USA
89          Osni Marques, LBNL/NERSC, USA
90          Christof Voemel, University of California, Berkeley, USA
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95 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       SLARRF(1)
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