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

6       DLARRF  -  the  initial representation L D L^T and its cluster of close
7       eigenvalues (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

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

81       Based on contributions by
82          Beresford Parlett, University of California, Berkeley, USA
83          Jim Demmel, University of California, Berkeley, USA
84          Inderjit Dhillon, University of Texas, Austin, USA
85          Osni Marques, LBNL/NERSC, USA
86          Christof Voemel, University of California, Berkeley, USA
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90 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       DLARRF(1)
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