-
Notifications
You must be signed in to change notification settings - Fork 59
/
vandermonde_approx_1d.html
249 lines (211 loc) · 7.03 KB
/
vandermonde_approx_1d.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
<html>
<head>
<title>
VANDERMONDE_APPROX_1D - Data Approximation with Polynomials using the Vandermonde Matrix
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
VANDERMONDE_APPROX_1D <br> Data Approximation with Polynomials using the Vandermonde Matrix
</h1>
<hr>
<p>
<b>VANDERMONDE_APPROX_1D</b>
is a FORTRAN90 library which
finds a polynomial approximant to 1D data by setting up and
solving an overdetermined linear system involving the Vandermonde matrix.
</p>
<p>
This software is primarily intended as an illustration of the problems
that can occur when the approximatino problem is naively formulated
using the Vandermonde matrix. Unless the data points are well separated,
and the degree of the polynomial is low, the linear system will become
very difficult to store and solve accurately, because the monomials
used as basis vectors by the Vandermonde approach become indistinguishable.
</p>
<p>
<b>VANDERMONDE_INTERP_1D</b> needs access to the QR_SOLVE and R8LIB libraries.
The test code also needs access to the CONDITION and TEST_INTERP libraries.
</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>VANDERMONDE_APPROX_1D</b> is available in
<a href = "../../c_src/vandermonde_approx_1d/vandermonde_approx_1d.html">a C version</a> and
<a href = "../../cpp_src/vandermonde_approx_1d/vandermonde_approx_1d.html">a C++ version</a> and
<a href = "../../f77_src/vandermonde_approx_1d/vandermonde_approx_1d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/vandermonde_approx_1d/vandermonde_approx_1d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/vandermonde_approx_1d/vandermonde_approx_1d.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/chebyshev.html">
CHEBYSHEV</a>,
a FORTRAN90 library which
computes the Chebyshev interpolant/approximant to a given function
over an interval.
</p>
<p>
<a href = "../../f_src/condition/condition.html">
CONDITION</a>,
a FORTRAN90 library which
implements methods of computing or estimating the condition number of a matrix.
</p>
<p>
<a href = "../../f_src/lagrange_approx_1d/lagrange_approx_1d.html">
LAGRANGE_APPROX_1D</a>,
a FORTRAN90 library which
defines and evaluates the Lagrange polynomial p(x) of degree m
which approximates a set of nd data points (x(i),y(i)).
</p>
<p>
<a href = "../../f_src/pwl_approx_1d/pwl_approx_1d.html">
PWL_APPROX_1D</a>,
a FORTRAN90 library which
approximates a set of data using a piecewise linear function.
</p>
<p>
<a href = "../../f_src/qr_solve/qr_solve.html">
QR_SOLVE</a>,
a FORTRAN90 library which
computes the least squares solution of a linear system A*x=b.
</p>
<p>
<a href = "../../f_src/r8lib/r8lib.html">
R8LIB</a>,
a FORTRAN90 library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../f_src/spline/spline.html">
SPLINE</a>,
a FORTRAN90 library which
constructs and evaluates 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/vandermonde_approx_2d/vandermonde_approx_2d.html">
VANDERMONDE_APPROX_2D</a>,
a FORTRAN90 library which
finds a polynomial approximant p(x,y) to data z(x,y) of a 2D argument by setting up and
solving an overdetermined linear system for the polynomial coefficients
involving the Vandermonde matrix.
</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 a function of 1D data
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>
Kendall Atkinson,<br>
An Introduction to Numerical Analysis,<br>
Prentice Hall, 1989,<br>
ISBN: 0471624896,<br>
LC: QA297.A94.1989.
</li>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "vandermonde_approx_1d.f90">vandermonde_approx_1d.f90</a>, the source code.
</li>
<li>
<a href = "vandermonde_approx_1d.sh">vandermonde_approx_1d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "vandermonde_approx_1d_prb.f90">vandermonde_approx_1d_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "vandermonde_approx_1d_prb.sh">vandermonde_approx_1d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "vandermonde_approx_1d_prb_output.txt">vandermonde_approx_1d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>VANDERMONDE_APPROX_1D_COEF</b> computes a 1D polynomial approximant.
</li>
<li>
<b>VANDERMONDE_APPROX_1D_MATRIX</b> computes a Vandermonde 1D approximation matrix.
</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 07 October 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>