1SLARRB(1) LAPACK auxiliary routine (version 3.2) SLARRB(1)
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6 SLARRB - the relatively robust representation(RRR) L D L^T, SLARRB does
7 "limited" bisection to refine the eigenvalues of L D L^T,
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10 SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W,
11 WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO
12 )
13 INTEGER IFIRST, ILAST, INFO, N, OFFSET, TWIST
15 REAL PIVMIN, RTOL1, RTOL2, SPDIAM
17 INTEGER IWORK( * )
19 REAL D( * ), LLD( * ), W( * ), WERR( * ), WGAP( * ),
21 WORK( * )
22
24 Given the relatively robust representation(RRR) L D L^T, SLARRB does
25 "limited" bisection to refine the eigenvalues of L D L^T, W( IFIRST-
26 OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial guesses
27 for these eigenvalues are input in W, the corresponding estimate of the
28 error in these guesses and their gaps are input in WERR and WGAP,
29 respectively. During bisection, intervals
30 [left, right] are maintained by storing their mid-points and semi-
31 widths in the arrays W and WERR respectively.
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34 N (input) INTEGER
35 The order of the matrix.
36 D (input) REAL array, dimension (N)
38 The N diagonal elements of the diagonal matrix D.
39 LLD (input) REAL array, dimension (N-1)
41 The (N-1) elements L(i)*L(i)*D(i).
42 IFIRST (input) INTEGER
44 The index of the first eigenvalue to be computed.
45 ILAST (input) INTEGER
47 The index of the last eigenvalue to be computed.
48 RTOL1 (input) REAL
50 RTOL2 (input) REAL Tolerance for the convergence of the
51 bisection intervals. An interval [LEFT,RIGHT] has converged if
52 RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) where
53 GAP is the (estimated) distance to the nearest eigenvalue.
54 OFFSET (input) INTEGER
56 Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
57 through ILAST-OFFSET elements of these arrays are to be used.
58 W (input/output) REAL array, dimension (N)
60 On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
61 estimates of the eigenvalues of L D L^T indexed IFIRST throug
62 ILAST. On output, these estimates are refined.
63 WGAP (input/output) REAL array, dimension (N-1)
65 On input, the (estimated) gaps between consecutive eigenvalues
66 of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues
67 I and I+1. Note that if IFIRST.EQ.ILAST then WGAP(IFIRST-OFF‐
68 SET) must be set to ZERO. On output, these gaps are refined.
69 WERR (input/output) REAL array, dimension (N)
71 On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET )
72 are the errors in the estimates of the corresponding elements
73 in W. On output, these errors are refined.
74 WORK (workspace) REAL array, dimension (2*N)
76 Workspace.
77 IWORK (workspace) INTEGER array, dimension (2*N)
79 Workspace.
80 PIVMIN (input) DOUBLE PRECISION
82 The minimum pivot in the Sturm sequence.
83 SPDIAM (input) DOUBLE PRECISION
85 The spectral diameter of the matrix.
86 TWIST (input) INTEGER
88 The twist index for the twisted factorization that is used for
89 the negcount. TWIST = N: Compute negcount from L D L^T -
90 LAMBDA I = L+ D+ L+^T
91 TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D-
92 U-^T
93 TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r)
94 N(r)
95 INFO (output) INTEGER
97 Error flag.
98
100 Based on contributions by
101 Beresford Parlett, University of California, Berkeley, USA
102 Jim Demmel, University of California, Berkeley, USA
103 Inderjit Dhillon, University of Texas, Austin, USA
104 Osni Marques, LBNL/NERSC, USA
105 Christof Voemel, University of California, Berkeley, USA
106 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 SLARRB(1)