1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
|
---
:name: slarrb
:md5sum: ac43c9637a4b29c89256e616a3b9c674
:category: :subroutine
:arguments:
- n:
:type: integer
:intent: input
- d:
:type: real
:intent: input
:dims:
- n
- lld:
:type: real
:intent: input
:dims:
- n-1
- ifirst:
:type: integer
:intent: input
- ilast:
:type: integer
:intent: input
- rtol1:
:type: real
:intent: input
- rtol2:
:type: real
:intent: input
- offset:
:type: integer
:intent: input
- w:
:type: real
:intent: input/output
:dims:
- n
- wgap:
:type: real
:intent: input/output
:dims:
- n-1
- werr:
:type: real
:intent: input/output
:dims:
- n
- work:
:type: real
:intent: workspace
:dims:
- 2*n
- iwork:
:type: integer
:intent: workspace
:dims:
- 2*n
- pivmin:
:type: real
:intent: input
- spdiam:
:type: real
:intent: input
- twist:
:type: integer
:intent: input
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE SLARRB( N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* Given the relatively robust representation(RRR) L D L^T, SLARRB\n\
* does \"limited\" bisection to refine the eigenvalues of L D L^T,\n\
* W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial\n\
* guesses for these eigenvalues are input in W, the corresponding estimate\n\
* of the error in these guesses and their gaps are input in WERR\n\
* and WGAP, respectively. During bisection, intervals\n\
* [left, right] are maintained by storing their mid-points and\n\
* semi-widths in the arrays W and WERR respectively.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix.\n\
*\n\
* D (input) REAL array, dimension (N)\n\
* The N diagonal elements of the diagonal matrix D.\n\
*\n\
* LLD (input) REAL array, dimension (N-1)\n\
* The (N-1) elements L(i)*L(i)*D(i).\n\
*\n\
* IFIRST (input) INTEGER\n\
* The index of the first eigenvalue to be computed.\n\
*\n\
* ILAST (input) INTEGER\n\
* The index of the last eigenvalue to be computed.\n\
*\n\
* RTOL1 (input) REAL \n\
* RTOL2 (input) REAL \n\
* Tolerance for the convergence of the bisection intervals.\n\
* An interval [LEFT,RIGHT] has converged if\n\
* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )\n\
* where GAP is the (estimated) distance to the nearest\n\
* eigenvalue.\n\
*\n\
* OFFSET (input) INTEGER\n\
* Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET\n\
* through ILAST-OFFSET elements of these arrays are to be used.\n\
*\n\
* W (input/output) REAL array, dimension (N)\n\
* On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are\n\
* estimates of the eigenvalues of L D L^T indexed IFIRST throug\n\
* ILAST.\n\
* On output, these estimates are refined.\n\
*\n\
* WGAP (input/output) REAL array, dimension (N-1)\n\
* On input, the (estimated) gaps between consecutive\n\
* eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between\n\
* eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST\n\
* then WGAP(IFIRST-OFFSET) must be set to ZERO.\n\
* On output, these gaps are refined.\n\
*\n\
* WERR (input/output) REAL array, dimension (N)\n\
* On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are\n\
* the errors in the estimates of the corresponding elements in W.\n\
* On output, these errors are refined.\n\
*\n\
* WORK (workspace) REAL array, dimension (2*N)\n\
* Workspace.\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (2*N)\n\
* Workspace.\n\
*\n\
* PIVMIN (input) REAL\n\
* The minimum pivot in the Sturm sequence.\n\
*\n\
* SPDIAM (input) REAL\n\
* The spectral diameter of the matrix.\n\
*\n\
* TWIST (input) INTEGER\n\
* The twist index for the twisted factorization that is used\n\
* for the negcount.\n\
* TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T\n\
* TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T\n\
* TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)\n\
*\n\
* INFO (output) INTEGER\n\
* Error flag.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* Based on contributions by\n\
* Beresford Parlett, University of California, Berkeley, USA\n\
* Jim Demmel, University of California, Berkeley, USA\n\
* Inderjit Dhillon, University of Texas, Austin, USA\n\
* Osni Marques, LBNL/NERSC, USA\n\
* Christof Voemel, University of California, Berkeley, USA\n\
*\n\
* =====================================================================\n\
*\n"
|
