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

6       DLARRF - 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 DLARRF( 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           DOUBLE         PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
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18           DOUBLE         PRECISION D( * ), DPLUS( * ),  L(  *  ),  LD(  *  ),
19                          LPLUS( * ), W( * ), 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  ), DLARRF 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) DOUBLE PRECISION array, dimension (N)
34               The N diagonal elements of the diagonal matrix D.
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36       L       (input) DOUBLE PRECISION array, dimension (N-1)
37               The (N-1) subdiagonal elements of the unit bidiagonal matrix L.
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39       LD      (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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)  DOUBLE  PRECISION array, dimension >=  (CLEND-
54       CLSTRT+1)
55               The separation from the right neighbor eigenvalue in W.
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57       WERR    (input) DOUBLE PRECISION 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) DOUBLE PRECISION
72               The shift used to form L(+) D(+) L(+)^T.
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74       DPLUS   (output) DOUBLE PRECISION array, dimension (N)
75               The N diagonal elements of the diagonal matrix D(+).
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77       LPLUS   (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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                       DLARRF(1)
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