1SLARRB(1) LAPACK auxiliary routine (version 3.1) SLARRB(1)
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6 SLARRB - 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( * )
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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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35 N (input) INTEGER
36 The order of the matrix.
37 D (input) REAL array, dimension (N)
39 The N diagonal elements of the diagonal matrix D.
40 LLD (input) REAL array, dimension (N-1)
42 The (N-1) elements L(i)*L(i)*D(i).
43 IFIRST (input) INTEGER
45 The index of the first eigenvalue to be computed.
46 ILAST (input) INTEGER
48 The index of the last eigenvalue to be computed.
49 RTOL1 (input) REAL
51 RTOL2 (input) REAL Tolerance for the convergence of the
52 bisection intervals. An interval [LEFT,RIGHT] has converged if
53 RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) where
54 GAP is the (estimated) distance to the nearest eigenvalue.
55 OFFSET (input) INTEGER
57 Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
58 through ILAST-OFFSET elements of these arrays are to be used.
59 W (input/output) REAL array, dimension (N)
61 On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
62 estimates of the eigenvalues of L D L^T indexed IFIRST throug
63 ILAST. On output, these estimates are refined.
64 WGAP (input/output) REAL array, dimension (N-1)
66 On input, the (estimated) gaps between consecutive eigenvalues
67 of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues
68 I and I+1. Note that if IFIRST.EQ.ILAST then WGAP(IFIRST-OFF‐
69 SET) must be set to ZERO. On output, these gaps are refined.
70 WERR (input/output) REAL array, dimension (N)
72 On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET )
73 are the errors in the estimates of the corresponding elements
74 in W. On output, these errors are refined.
75 WORK (workspace) REAL array, dimension (2*N)
77 Workspace.
78 IWORK (workspace) INTEGER array, dimension (2*N)
80 Workspace.
81 PIVMIN (input) DOUBLE PRECISION
83 The minimum pivot in the Sturm sequence.
84 SPDIAM (input) DOUBLE PRECISION
86 The spectral diameter of the matrix.
87 TWIST (input) INTEGER
89 The twist index for the twisted factorization that is used for
90 the negcount. TWIST = N: Compute negcount from L D L^T -
91 LAMBDA I = L+ D+ L+^T
92 TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D-
93 U-^T
94 TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r)
95 N(r)
96 INFO (output) INTEGER
98 Error flag.
99
101 Based on contributions by
102 Beresford Parlett, University of California, Berkeley, USA
103 Jim Demmel, University of California, Berkeley, USA
104 Inderjit Dhillon, University of Texas, Austin, USA
105 Osni Marques, LBNL/NERSC, USA
106 Christof Voemel, University of California, Berkeley, USA
107 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 SLARRB(1)