forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
chebyshev.html
276 lines (236 loc) · 7.59 KB
/
chebyshev.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
<html>
<head>
<title>
CHEBYSHEV - Interpolation Using Chebyshev Polynomials
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CHEBYSHEV <br> Interpolation Using Chebyshev Polynomials
</h1>
<hr>
<p>
<b>CHEBYSHEV</b>
is a FORTRAN90 library which
constructs the Chebyshev interpolant to a function.
</p>
<p>
Note that the user is not free to choose the interpolation points.
Instead, the function f(x) will be evaluated at points chosen by the algorithm.
In the standard case, in which the interpolation interval is [-1,+1],
these points will be the zeros of the Chebyshev polynomial of order N.
However, the algorithm can also be applied to an interval of the form [a,b],
in which case the evaluation points are linearly mapped from [-1,+1].
</p>
<p>
The resulting interpolant is defined by a set of N coefficients c(),
and has the form:
<pre>
C(f)(x) = sum ( 1 <= i <= n ) c(i) T(i-1,x) - 0.5 * c(1)
</pre>
where T(i-1,x) is the (i-1)-th Chebyshev polynomial.
</p>
<p>
Within the interval [-1,+1], or the generalized interval [a,b], the
interpolant actually remains bounded by the sum of the absolute values
of the coefficients c(). It is therefore common to use Chebyshev
interpolants as approximating functions over a given interval.
</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>CHEBYSHEV</b> is available in
<a href = "../../c_src/chebyshev/chebyshev.html">a C version</a> and
<a href = "../../cpp_src/chebyshev/chebyshev.html">a C++ version</a> and
<a href = "../../f77_src/chebyshev/chebyshev.html">a FORTRAN77 version</a> and
<a href = "../../f_src/chebyshev/chebyshev.html">a FORTRAN90 version</a> and
<a href = "../../m_src/chebyshev/chebyshev.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/bernstein/bernstein.html">
BERNSTEIN</a>,
a FORTRAN90 library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../f_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/divdif/divdif.html">
DIVDIF</a>,
a FORTRAN90 library which
computes interpolants by divided differences.
</p>
<p>
<a href = "../../f_src/hermite/hermite.html">
HERMITE</a>,
a FORTRAN90 library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
</p>
<p>
<a href = "../../f_src/hermite_cubic/hermite_cubic.html">
HERMITE_CUBIC</a>,
a FORTRAN90 library which
can compute the value, derivatives or integral of a Hermite cubic polynomial,
or manipulate an interpolating function made up of piecewise Hermite cubic
polynomials.
</p>
<p>
<a href = "../../f_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a FORTRAN90 library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
</p>
<p>
<a href = "../../f_src/rbf_interp/rbf_interp.html">
RBF_INTERP</a>,
a FORTRAN90 library which
defines and evaluates radial basis interpolants to multidimensional data.
</p>
<p>
<a href = "../../f_src/spline/spline.html">
SPLINE</a>,
a FORTRAN90 library which
includes many routines to construct
and evaluate spline interpolants and approximants.
</p>
<p>
<a href = "../../f_src/test_approx/test_approx.html">
TEST_APPROX</a>,
a FORTRAN90 library which
defines test problems for approximation,
provided as a set of (x,y) data.
</p>
<p>
<a href = "../../f_src/test_interp_1d/test_interp_1d.html">
TEST_INTERP_1D</a>,
a FORTRAN90 library which
defines test problems for interpolation of data y(x),
depending on a 1D argument.
</p>
<p>
<a href = "../../f_src/toms446/toms446.html">
TOMS446</a>,
a FORTRAN90 library which
manipulates Chebyshev series for interpolation and approximation;<br>
this is a version of ACM TOMS algorithm 446,
by Roger Broucke.
</p>
<p>
<a href = "../../f_src/vandermonde_interp_1d/vandermonde_interp_1d.html">
VANDERMONDE_INTERP_1D</a>,
a FORTRAN90 library which
finds a polynomial interpolant to data y(x) of a 1D argument,
by setting up and solving a linear system for the polynomial coefficients,
involving the Vandermonde matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Roger Broucke,<br>
Algorithm 446:
Ten Subroutines for the Manipulation of Chebyshev Series,<br>
Communications of the ACM,<br>
Volume 16, Number 4, April 1973, pages 254-256.
</li>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
William Press, Brian Flannery, Saul Teukolsky, William Vetterling,<br>
Numerical Recipes in C: The Art of Scientific Computing,<br>
Cambridge University Press, 1988,<br>
ISBN: 0-521-35465-X,<br>
LC: QA76.73.C15N865.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "chebyshev.f90">chebyshev.f90</a>, the source code.
</li>
<li>
<a href = "chebyshev.sh">chebyshev.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "chebyshev_prb.f90">chebyshev_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "chebyshev_prb.sh">chebyshev_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "chebyshev_prb_output.txt">chebyshev_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CHEBYSHEV_COEFFICIENTS</b> determines Chebyshev interpolation coefficients.
</li>
<li>
<b>CHEBYSHEV_INTERPOLANT</b> evaluates a Chebyshev interpolant.
</li>
<li>
<b>CHEBYSHEV_ZEROS</b> returns zeroes of the Chebyshev polynomial T(N,X).
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 14 September 2011.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>