-
Notifications
You must be signed in to change notification settings - Fork 59
/
Copy pathvandermonde_interp_2d.html
247 lines (210 loc) · 7.37 KB
/
vandermonde_interp_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
<html>
<head>
<title>
VANDERMONDE_INTERP_2D - Data Interpolation with Polynomials using the Vandermonde Matrix
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
VANDERMONDE_INTERP_2D <br> Data Interpolation with Polynomials using the Vandermonde Matrix
</h1>
<hr>
<p>
<b>VANDERMONDE_INTERP_2D</b>
is a FORTRAN90 library which
finds P(X,Y), a polynomial interpolant to data Z(X,Y) which depends on two independent
variables X and Y, by setting up and solving a linear system
involving the Vandermonde matrix.
</p>
<p>
This software is primarily intended as an illustration of the problems
that can occur when the interpolation 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>
If the data is available on a product grid, then both the LAGRANGE_INTERP_2D
and VANDERMONDE_INTERP_2D libraries will be trying to compute the same
interpolating function. However, especially for higher degree polynomials,
the Lagrange approach will be superior because it avoids the badly conditioned
Vandermonde matrix associated with the usage of monomials as the basis.
The Lagrange approach uses as a basis a set of Lagrange basis polynomials
l(i,j)(x) which are 1 at node (x(i),y(j)) and zero at the other nodes.
</p>
<p>
<b>VANDERMONDE_INTERP_2D</b> needs access to the QR_SOLVE and R8LIB libraries.
The test code also needs access to the TEST_INTERP_2D library.
</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_INTERP_2D</b> is available in
<a href = "../../c_src/vandermonde_interp_2d/vandermonde_interp_2d.html">a C version</a> and
<a href = "../../cpp_src/vandermonde_interp_2d/vandermonde_interp_2d.html">a C++ version</a> and
<a href = "../../f77_src/vandermonde_interp_2d/vandermonde_interp_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/vandermonde_interp_2d/vandermonde_interp_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/vandermonde_interp_2d/vandermonde_interp_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/lagrange_interp_2d/lagrange_interp_2d.html">
LAGRANGE_INTERP_2D</a>,
a FORTRAN90 library which
defines and evaluates the Lagrange polynomial p(x,y)
which interpolates a set of data depending on a 2D argument
that was evaluated on a product grid,
so that p(x(i),y(j)) = z(i,j).
</p>
<p>
<a href = "../../f_src/pwl_interp_2d/pwl_interp_2d.html">
PWL_INTERP_2D</a>,
a FORTRAN90 library which
evaluates a piecewise linear interpolant to data defined on
a regular 2D grid.
</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/rbf_interp_2d/rbf_interp_2d.html">
RBF_INTERP_2D</a>,
a FORTRAN90 library which
defines and evaluates radial basis function (RBF) interpolants to 2D data.
</p>
<p>
<a href = "../../f_src/shepard_interp_2d/shepard_interp_2d.html">
SHEPARD_INTERP_2D</a>,
a FORTRAN90 library which
defines and evaluates Shepard interpolants to 2D data,
which are based on inverse distance weighting.
</p>
<p>
<a href = "../../f_src/test_interp_2d/test_interp_2d.html">
TEST_INTERP_2D</a>,
a FORTRAN90 library which
defines test problems for interpolation of data z(x,y) of a 2D argument.
</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_interp_2d.f90">vandermonde_interp_2d.f90</a>, the source code.
</li>
<li>
<a href = "vandermonde_interp_2d.sh">vandermonde_interp_2d.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_interp_2d_prb.f90">vandermonde_interp_2d_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "vandermonde_interp_2d_prb.sh">vandermonde_interp_2d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "vandermonde_interp_2d_prb_output.txt">vandermonde_interp_2d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>TRIANGLE_NUM</b> returns the N-th triangular number.
</li>
<li>
<b>VANDERMONDE_INTERP_2D_MATRIX</b> computes a Vandermonde 2D interpolation 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>