File: 1D-Interpolation-Example-programs.html

package info (click to toggle)
gsl-ref-html 2.3-1
  • links: PTS
  • area: non-free
  • in suites: bullseye, buster, sid
  • size: 6,876 kB
  • ctags: 4,574
  • sloc: makefile: 35
file content (270 lines) | stat: -rw-r--r-- 9,866 bytes parent folder | download
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
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.

Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".

(a) The Back-Cover Text is: "You have the freedom to copy and modify this
GNU Manual." -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library &ndash; Reference Manual: 1D Interpolation Example programs</title>

<meta name="description" content="GNU Scientific Library &ndash; Reference Manual: 1D Interpolation Example programs">
<meta name="keywords" content="GNU Scientific Library &ndash; Reference Manual: 1D Interpolation Example programs">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Interpolation.html#Interpolation" rel="up" title="Interpolation">
<link href="1D-Interpolation-References-and-Further-Reading.html#g_t1D-Interpolation-References-and-Further-Reading" rel="next" title="1D Interpolation References and Further Reading">
<link href="1D-Higher_002dlevel-Interface.html#g_t1D-Higher_002dlevel-Interface" rel="previous" title="1D Higher-level Interface">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>


</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="g_t1D-Interpolation-Example-programs"></a>
<div class="header">
<p>
Next: <a href="1D-Interpolation-References-and-Further-Reading.html#g_t1D-Interpolation-References-and-Further-Reading" accesskey="n" rel="next">1D Interpolation References and Further Reading</a>, Previous: <a href="1D-Higher_002dlevel-Interface.html#g_t1D-Higher_002dlevel-Interface" accesskey="p" rel="previous">1D Higher-level Interface</a>, Up: <a href="Interpolation.html#Interpolation" accesskey="u" rel="up">Interpolation</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-of-1D-Interpolation"></a>
<h3 class="section">28.7 Examples of 1D Interpolation</h3>

<p>The following program demonstrates the use of the interpolation and
spline functions.  It computes a cubic spline interpolation of the
10-point dataset <em>(x_i, y_i)</em> where <em>x_i = i + \sin(i)/2</em> and
<em>y_i = i + \cos(i^2)</em> for <em>i = 0 \dots 9</em>.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdlib.h&gt;
#include &lt;stdio.h&gt;
#include &lt;math.h&gt;
#include &lt;gsl/gsl_errno.h&gt;
#include &lt;gsl/gsl_spline.h&gt;

int
main (void)
{
  int i;
  double xi, yi, x[10], y[10];

  printf (&quot;#m=0,S=2\n&quot;);

  for (i = 0; i &lt; 10; i++)
    {
      x[i] = i + 0.5 * sin (i);
      y[i] = i + cos (i * i);
      printf (&quot;%g %g\n&quot;, x[i], y[i]);
    }

  printf (&quot;#m=1,S=0\n&quot;);

  {
    gsl_interp_accel *acc 
      = gsl_interp_accel_alloc ();
    gsl_spline *spline 
      = gsl_spline_alloc (gsl_interp_cspline, 10);

    gsl_spline_init (spline, x, y, 10);

    for (xi = x[0]; xi &lt; x[9]; xi += 0.01)
      {
        yi = gsl_spline_eval (spline, xi, acc);
        printf (&quot;%g %g\n&quot;, xi, yi);
      }
    gsl_spline_free (spline);
    gsl_interp_accel_free (acc);
  }
  return 0;
}
</pre></div>

<p>The output is designed to be used with the <small>GNU</small> plotutils
<code>graph</code> program,
</p>
<div class="example">
<pre class="example">$ ./a.out &gt; interp.dat
$ graph -T ps &lt; interp.dat &gt; interp.ps
</pre></div>


<p>The result shows a smooth interpolation of the original points.  The
interpolation method can be changed simply by varying the first argument of
<code>gsl_spline_alloc</code>.
</p>
<p>The next program demonstrates a periodic cubic spline with 4 data
points.  Note that the first and last points must be supplied with 
the same y-value for a periodic spline.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdlib.h&gt;
#include &lt;stdio.h&gt;
#include &lt;math.h&gt;
#include &lt;gsl/gsl_errno.h&gt;
#include &lt;gsl/gsl_spline.h&gt;

int
main (void)
{
  int N = 4;
  double x[4] = {0.00, 0.10,  0.27,  0.30};
  double y[4] = {0.15, 0.70, -0.10,  0.15}; 
             /* Note: y[0] == y[3] for periodic data */

  gsl_interp_accel *acc = gsl_interp_accel_alloc ();
  const gsl_interp_type *t = gsl_interp_cspline_periodic; 
  gsl_spline *spline = gsl_spline_alloc (t, N);

  int i; double xi, yi;

  printf (&quot;#m=0,S=5\n&quot;);
  for (i = 0; i &lt; N; i++)
    {
      printf (&quot;%g %g\n&quot;, x[i], y[i]);
    }

  printf (&quot;#m=1,S=0\n&quot;);
  gsl_spline_init (spline, x, y, N);

  for (i = 0; i &lt;= 100; i++)
    {
      xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
      yi = gsl_spline_eval (spline, xi, acc);
      printf (&quot;%g %g\n&quot;, xi, yi);
    }
  
  gsl_spline_free (spline);
  gsl_interp_accel_free (acc);
  return 0;
}
</pre></div>


<p>The output can be plotted with <small>GNU</small> <code>graph</code>.
</p>
<div class="example">
<pre class="example">$ ./a.out &gt; interp.dat
$ graph -T ps &lt; interp.dat &gt; interp.ps
</pre></div>


<p>The result shows a periodic interpolation of the original points. The
slope of the fitted curve is the same at the beginning and end of the
data, and the second derivative is also.
</p>
<p>The next program illustrates the difference between the cubic spline,
Akima, and Steffen interpolation types on a difficult dataset.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;stdlib.h&gt;
#include &lt;math.h&gt;

#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_spline.h&gt;

int
main(void)
{
  size_t i;
  const size_t N = 9;

  /* this dataset is taken from
   * J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,
   * SIAM J. Sci. Stat. Comput. 4, 4, 1983. */
  const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
                       10.0, 12.0, 15.0, 20.0 };
  const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
                       0.169183, 0.469428, 0.943740,
                       0.998636, 0.999919, 0.999994 };

  gsl_interp_accel *acc = gsl_interp_accel_alloc();
  gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
  gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
  gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);

  gsl_spline_init(spline_cubic, x, y, N);
  gsl_spline_init(spline_akima, x, y, N);
  gsl_spline_init(spline_steffen, x, y, N);

  for (i = 0; i &lt; N; ++i)
    printf(&quot;%g %g\n&quot;, x[i], y[i]);

  printf(&quot;\n\n&quot;);

  for (i = 0; i &lt;= 100; ++i)
    {
      double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
      double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
      double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
      double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);

      printf(&quot;%g %g %g %g\n&quot;, xi, yi_cubic, yi_akima, yi_steffen);
    }

  gsl_spline_free(spline_cubic);
  gsl_spline_free(spline_akima);
  gsl_spline_free(spline_steffen);
  gsl_interp_accel_free(acc);

  return 0;
}
</pre></div>


<p>The cubic method exhibits a local maxima between the 6th and 7th data points
and continues oscillating for the rest of the data. Akima also shows a
local maxima but recovers and follows the data well after the 7th grid point.
Steffen preserves monotonicity in all intervals and does not exhibit oscillations,
at the expense of having a discontinuous second derivative.
</p>
<hr>
<div class="header">
<p>
Next: <a href="1D-Interpolation-References-and-Further-Reading.html#g_t1D-Interpolation-References-and-Further-Reading" accesskey="n" rel="next">1D Interpolation References and Further Reading</a>, Previous: <a href="1D-Higher_002dlevel-Interface.html#g_t1D-Higher_002dlevel-Interface" accesskey="p" rel="previous">1D Higher-level Interface</a>, Up: <a href="Interpolation.html#Interpolation" accesskey="u" rel="up">Interpolation</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>