forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
kronrod.html
196 lines (165 loc) · 5 KB
/
kronrod.html
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
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
<html>
<head>
<title>
KRONROD - Gauss-Kronrod Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
KRONROD <br> Gauss-Kronrod Quadrature Rules
</h1>
<hr>
<p>
<b>KRONROD</b>
is a C++ library which
computes both a Gauss quadrature rule of order N, and the
Gauss-Kronrod rule of order 2*N+1.
</p>
<p>
A pair of Gauss and Gauss-Kronrod quadrature rules are typically
used to provide an error estimate for an integral. The integral
is estimated using the Gauss rule, and then the Gauss-Kronrod
rule provides a higher precision estimate. The difference between
the two estimates is taken as an approximation to the level
of error.
</p>
<p>
The advantage of using a Gauss and Gauss-Kronrod pair is that
the second rule, which uses 2*N+1 points, actually includes the
N points in the previous Gauss rule. This means that the function
values from that computation can be reused. This efficiency comes
at the cost of a mild reduction in the degree of polynomial precision of the
Gauss-Kronrod rule.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>KRONROD</b> is available in
<a href = "../../c_src/kronrod/kronrod.html">a C version</a> and
<a href = "../../cpp_src/kronrod/kronrod.html">a C++ version</a> and
<a href = "../../f77_src/kronrod/kronrod.html">a FORTRAN77 version</a> and
<a href = "../../f_src/kronrod/kronrod.html">a FORTRAN90 version</a> and
<a href = "../../m_src/kronrod/kronrod.html">a MATLAB version</a> and
<a href = "../../py_src/kronrod/kronrod.html">a PYTHON version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a C++ program which
checks the polynomial exactness
of a 1-dimensional quadrature rule for a finite interval.
</p>
<p>
<a href = "../../cpp_src/patterson_rule/patterson_rule.html">
PATTERSON_RULE</a>,
a C++ program which
computes a Gauss-Patterson quadrature rule.
</p>
<p>
<a href = "../../datasets/quadrature_rules_patterson/quadrature_rules_patterson.html">
QUADRATURE_RULES_PATTERSON</a>,
a dataset directory which
contains Gauss-Patterson quadrature rules for the interval [-1,+1].
</p>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules for 1D domains.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Robert Piessens, Maria Branders,<br>
A Note on the Optimal Addition of Abscissas to Quadrature Formulas
of Gauss and Lobatto,<br>
Mathematics of Computation,<br>
Volume 28, Number 125, January 1974, pages 135-139.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "kronrod.cpp">kronrod.cpp</a>, the source code.
</li>
<li>
<a href = "kronrod.hpp">kronrod.hpp</a>, the include file.
</li>
<li>
<a href = "kronrod.sh">kronrod.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "kronrod_prb.cpp">kronrod_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "kronrod_prb.sh">kronrod_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "kronrod_prb_output.txt">kronrod_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ABWE1</b> calculates a Kronrod abscissa and weight.
</li>
<li>
<b>ABWE2</b> calculates a Gaussian abscissa and two weights.
</li>
<li>
<b>KRONROD</b> adds N+1 points to an N-point Gaussian rule.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 03 August 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>