1DLARRB(1) LAPACK auxiliary routine (version 3.1) DLARRB(1)
2
6 DLARRB - relatively robust representation(RRR) L D L^T, DLARRB does
7 "limited" bisection to refine the eigenvalues of L D L^T,
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10 SUBROUTINE DLARRB( 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 DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPDIAM
17 INTEGER IWORK( * )
19 DOUBLE PRECISION D( * ), LLD( * ), W( * ), WERR( * ), WGAP(
21 * ), WORK( * )
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24 Given the relatively robust representation(RRR) L D L^T, DLARRB 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.
32
35 N (input) INTEGER
36 The order of the matrix.
37 D (input) DOUBLE PRECISION array, dimension (N)
39 The N diagonal elements of the diagonal matrix D.
40 LLD (input) DOUBLE PRECISION 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) DOUBLE PRECISION
51 RTOL2 (input) DOUBLE PRECISION Tolerance for the convergence
52 of the bisection intervals. An interval [LEFT,RIGHT] has con‐
53 verged if RIGHT-LEFT.LT.MAX( RTOL1*GAP,
54 RTOL2*MAX(|LEFT|,|RIGHT|) ) where GAP is the (estimated) dis‐
55 tance to the nearest eigenvalue.
56 OFFSET (input) INTEGER
58 Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
59 through ILAST-OFFSET elements of these arrays are to be used.
60 W (input/output) DOUBLE PRECISION array, dimension (N)
62 On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
63 estimates of the eigenvalues of L D L^T indexed IFIRST throug
64 ILAST. On output, these estimates are refined.
65 WGAP (input/output) DOUBLE PRECISION array, dimension (N-1)
67 On input, the (estimated) gaps between consecutive eigenvalues
68 of L D L^T, i.e., WGAP(I-OFFSET) is the gap between eigenvalues
69 I and I+1. Note that if IFIRST.EQ.ILAST then WGAP(IFIRST-OFF‐
70 SET) must be set to ZERO. On output, these gaps are refined.
71 WERR (input/output) DOUBLE PRECISION array, dimension (N)
73 On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET )
74 are the errors in the estimates of the corresponding elements
75 in W. On output, these errors are refined.
76 WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
78 Workspace.
79 IWORK (workspace) INTEGER array, dimension (2*N)
81 Workspace.
82 PIVMIN (input) DOUBLE PRECISION
84 The minimum pivot in the Sturm sequence.
85 SPDIAM (input) DOUBLE PRECISION
87 The spectral diameter of the matrix.
88 TWIST (input) INTEGER
90 The twist index for the twisted factorization that is used for
91 the negcount. TWIST = N: Compute negcount from L D L^T -
92 LAMBDA I = L+ D+ L+^T
93 TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D-
94 U-^T
95 TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r)
96 N(r)
97 INFO (output) INTEGER
99 Error flag.
100
102 Based on contributions by
103 Beresford Parlett, University of California, Berkeley, USA
104 Jim Demmel, University of California, Berkeley, USA
105 Inderjit Dhillon, University of Texas, Austin, USA
106 Osni Marques, LBNL/NERSC, USA
107 Christof Voemel, University of California, Berkeley, USA
108 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 DLARRB(1)