forked from johannesgerer/jburkardt-m
-
Notifications
You must be signed in to change notification settings - Fork 0
/
quadrature_weights_vandermonde_2d.html
274 lines (237 loc) · 8.2 KB
/
quadrature_weights_vandermonde_2d.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
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
<html>
<head>
<title>
QUADRATURE_WEIGHTS_VANDERMONDE_2D - 2D Quadrature Weights by Vandermonde Matrix
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUADRATURE_WEIGHTS_VANDERMONDE_2D <br>
2D Quadrature Weights by Vandermonde Matrix
</h1>
<hr>
<p>
<b>QUADRATURE_WEIGHTS_VANDERMONDE_2D</b>
is a FORTRAN90 library which
illustrates a method for computing the weights W of a
2D interpolatory quadrature rule, assuming that the points (X,Y) have been
specified, by setting up a linear system involving the Vandermonde matrix.
</p>
<p>
We assume that the abscissas (quadrature points) have been chosen,
that the interval [A,B]x[C,D] is known, and that the integrals of polynomials
of total degree 0 through T can be computed.
</p>
<h3 align="center">
The Vandermonde Matrix
</h3>
<p>
We assume that the quadrature formula approximates integrals of the form:
<pre>
I(F) = Integral ( C <= Y <= D ) Integral ( A <= X <= B ) F(X,Y) dX dY
</pre>
by specifying N=(T+1)*(T+2)/2 points (X,Y) and weights W such that
<pre>
Q(F) = Sum ( 1 <= I <= N ) W(I) * F(X(I),Y(I))
</pre>
</p>
<p>
Now let us assume that the points (X,Y) have been specified, but that the
corresponding values W remain to be determined.
</p>
<p>
If we require that the quadrature rule with N points integrates the first
N=(T+1)*(T+2)/2 monomials exactly, then we have N conditions
on the weights W. (This means that we are assuming that N only takes
on appropriate values, namely 1, 3, 6, 10, 15, 21, 28, ...)
</p>
<p>
The K-th condition, for the monomial X^I*Y^J, J = 0 to T, I = 0 to T - J,
has the form:
<pre>
W(1)*X(1)^I*Y(1)^J + W(2)*X(2)^I*Y(2)^j+...+W(N)*X(N)^I*Y(N)^J = (B^(I+1)-A^(I+1))*(D^(J+1)-C(J+1))/(I+1)/(J+1)
</pre>
</p>
<p>
The corresponding matrix is known as a two-dimensional Vandermonde matrix.
It is theoretically guaranteed to be nonsingular as long as the points
(X,Y) are distinct and in "general position". The condition number of
the matrix grows quickly with increasing T.
</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>QUADRATURE_WEIGHTS_VANDERMONDE_2D</b> is available in
<a href = "../../c_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a C version</a> and
<a href = "../../cpp_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a C++ version</a> and
<a href = "../../f77_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/exactness_2d/exactness_2d.html">
EXACTNESS_2D</a>,
a MATLAB library which
investigates the exactness of 2D quadrature rules that estimate the
integral of a function f(x,y) over a 2D domain.
</p>
<p>
<a href = "../../m_src/quadrature_least_squares/quadrature_least_squares.html">
QUADRATURE_LEAST_SQUARES</a>,
a MATLAB library which
computes weights for "sub-interpolatory" quadrature rules,
that is, it estimates integrals by integrating a polynomial that
approximates the function data in a least squares sense.
</p>
<p>
<a href = "../../m_src/quadrature_golub_welsch/quadrature_golub_welsch.html">
QUADRATURE_GOLUB_WELSCH</a>,
a MATLAB library which
computes the points and weights of a Gaussian quadrature rule using the
Golub-Welsch procedure, assuming that the points have been specified.
</p>
<p>
<a href = "../../m_src/quadrature_weights_vandermonde/quadrature_weights_vandermonde.html">
QUADRATURE_WEIGHTS_VANDERMONDE</a>,
a MATLAB library which
computes the weights of a 1D quadrature rule using the Vandermonde
matrix, assuming that the points have been specified.
</p>
<p>
<a href = "../../m_src/quadrule/quadrule.html">
QUADRULE</a>,
a MATLAB library which
defines quadrature rules for 1-dimensional domains.
</p>
<p>
<a href = "../../m_src/toms655/toms655.html">
TOMS655</a>,
a MATLAB library which
computes the weights for interpolatory quadrature rule;<br>
this library is commonly called <b>IQPACK</b>;<br>
this is a MATLAB version of ACM TOMS algorithm 655.
</p>
<p>
<a href = "../../m_src/vandermonde/vandermonde.html">
VANDERMONDE</a>,
a MATLAB library which carries out certain operations associated
with the Vandermonde matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Sylvan Elhay, Jaroslav Kautsky,<br>
Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of
Interpolatory Quadrature,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 4, December 1987, pages 399-415.
</li>
<li>
Jaroslav Kautsky, Sylvan Elhay,<br>
Calculation of the Weights of Interpolatory Quadratures,<br>
Numerische Mathematik,<br>
Volume 40, 1982, pages 407-422.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "qwv_2d.m">qwv_2d.m</a>,
computes 2D quadrature weights using the Vandermonde matrix.
</li>
<li>
<a href = "r8mat_print.m">r8mat_print.m</a>,
prints an R8MAT.
</li>
<li>
<a href = "r8mat_print_some.m">r8mat_print_some.m</a>,
prints some of an R8MAT.
</li>
<li>
<a href = "r8vec_even.m">r8vec_even.m</a>,
returns an R8VEC of evenly spaced values.
</li>
<li>
<a href = "r8vec_print.m">r8vec_print.m</a>,
prints an R8VEC.
</li>
<li>
<a href = "r8vec_print_16.m">r8vec_print_16.m</a>,
prints an R8VEC to 16 decimal places.
</li>
<li>
<a href = "r8vec2_print.m">r8vec2_print.m</a>,
prints a pair of R8VEC's.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "qwv_2d_test.m">qwv_2d_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "qwv_2d_test01.m">qwv_2d_test01.m</a>,
tests QWV_2D for a trio of points.
</li>
<li>
<a href = "qwv_2d_test02.m">qwv_2d_test02.m</a>,
tests QWV_2D for Padua points.
</li>
<li>
<a href = "padua_point_set.m">padua_point_set.m</a>,
sets a 2D Padua point set.
</li>
<li>
<a href = "qwv_2d_test_output.txt">qwv_2d_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 29 May 2014.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>