forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathpiecewise_linear_product_integral.html
296 lines (257 loc) · 8.71 KB
/
piecewise_linear_product_integral.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
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
<html>
<head>
<title>
PIECEWISE_LINEAR_PRODUCT_INTEGRAL - Piecewise Linear Product Integral
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PIECEWISE_LINEAR_PRODUCT_INTEGRAL <br> Piecewise Linear Product Integral
</h1>
<hr>
<p>
<b>PIECEWISE_LINEAR_PRODUCT_INTEGRAL</b>
is a C++ library which
calculates the exact value of the integral of the product of two
piecewise linear functions F(X) and G(X).
</p>
<p>
The piecewise linear function F(X) is defined by
<ul>
<li>
<b>F_NUM</b>, the number of nodes;
</li>
<li>
<b>F_X</b>(1:F_NUM), the X coordinates of the nodes (in ascending order);
</li>
<li>
<b>F_V</b>(1:F_NUM), the value of F at each node;
</li>
</ul>
</p>
<p>
The piecewise linear function G(X) is defined similarly, and there is
no requirement that G(X) uses the same nodes as F(X), or the same number of nodes.
</p>
<p>
The task, then, is to determine the value of
<pre>
Integral ( A <= X <= B ) F(X) * G(X) dx
</pre>
This value can be determined exactly, by breaking the interval ot integration
into subintervals over which F(X) and G(X) are simple linear functions.
</p>
<p>
In finite element programs over 1-dimensional geometries, integrals like
this may occur when assembling the stiffness matrix, but these integrals
are generally treated using quadrature.
</p>
<p>
A more appropriate use for this function occurs when coarsening a finite
element solution, or constructing a piecewise linear least squares
finite element approximant to data that is regarded as a piecewise linear
function.
</p>
<p>
In case 1, we treat the original finite element solution as the function F(X),
and in case 2, we treat the data as a finite element function whose mesh
is implicit in the X coordinates of the given data.
</p>
<p>
Our G_X is, in case 1, the coarse mesh and in case 2 the given mesh.
To determine the finite element coefficient U(I) on this mesh, we
must integrate our data function F(X) against the I-th basis function,
which in this case is simply the I-th "hat function. So to use
our formula for G(X), we set G_V(1:G_NUM) to 0, except that G_V(I) = 1,
and compute
<pre>
U(I) = Integral ( G_X(I-1) <= X <= G_X(I+1) ) F(X) * G(X) dx
</pre>
Doing this for I from 1 to G_NUM allows us to compute the coefficients
of the coarsened solution (case 1) or the least squares approximant to
the piecewise linear function representing the data F.
</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>PIECEWISE_LINEAR_PRODUCT_INTEGRAL</b> is available in
<a href = "../../c_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a C version</a> and
<a href = "../../cpp_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a C++ version</a> and
<a href = "../../f77_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a FORTRAN77 version</a> and
<a href = "../../f_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a FORTRAN90 version</a> and
<a href = "../../m_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../data/fem/fem.html">
FEM</a>,
a data directory which
contains a description and examples of files that describe a finite element model.
</p>
<p>
<a href = "../../cpp_src/fem1d/fem1d.html">
FEM1D</a>,
a C++ program which
applies the finite element method to a 1D linear two point boundary value problem.
</p>
<p>
<a href = "../../cpp_src/fem1d_adaptive/fem1d_adaptive.html">
FEM1D_ADAPTIVE</a>,
a C++ program which
applies the finite element method to a 1D linear two point boundary value problem
using adaptive refinement to improve the solution.
</p>
<p>
<a href = "../../cpp_src/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a C++ program which
applies the finite element method to a 1D nonlinear two point boundary value problem.
</p>
<p>
<a href = "../../cpp_src/fem1d_pmethod/fem1d_pmethod.html">
FEM1D_PMETHOD</a>,
a C++ program which
applies the p-method version of the finite element method to a linear
two point boundary value problem in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_project/fem1d_project.html">
FEM1D_PROJECT</a>,
a C++ program which
projects data into a finite element space, including the least squares
approximation of data, or the projection of a finite element solution
from one mesh to another.
</p>
<p>
<a href = "../../cpp_src/fem1d_sample/fem1d_sample.html">
FEM1D_SAMPLE</a>,
a C++ program which
samples a scalar or vector finite element function of one variable,
defined by FEM files,
returning interpolated values at the sample points.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
TA640.2.Z54.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "piecewise_linear_product_integral.cpp">
piecewise_linear_product_integral.cpp</a>, the source code.
</li>
<li>
<a href = "piecewise_linear_product_integral.hpp">
piecewise_linear_product_integral.hpp</a>, the include file.
</li>
<li>
<a href = "piecewise_linear_product_integral.sh">
piecewise_linear_product_integral.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "piecewise_linear_product_integral_prb.cpp">
piecewise_linear_product_integral_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "piecewise_linear_product_integral_prb.sh">
piecewise_linear_product_integral_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "piecewise_linear_product_integral_prb_output.txt">
piecewise_linear_product_integral_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>PIECEWISE_LINEAR_PRODUCT_INTEGRAL:</b> piecewise linear product integral.
</li>
<li>
<b>PIECEWISE_LINEAR_PRODUCT_QUAD:</b> estimate piecewise linear product integral.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8VEC_BRACKET3</b> finds the interval containing or nearest a given value.
</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 30 April 2009.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>