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
|
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Cardinal Trigonometric interpolation</title>
<link rel="stylesheet" href="../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../index.html" title="Math Toolkit 4.2.1">
<link rel="up" href="../interpolation.html" title="Chapter 13. Interpolation">
<link rel="prev" href="bezier_polynomial.html" title="Bezier Polynomials">
<link rel="next" href="cubic_hermite.html" title="Cubic Hermite interpolation">
<meta name="viewport" content="width=device-width, initial-scale=1">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
<td align="center"><a href="../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="bezier_polynomial.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cubic_hermite.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.cardinal_trigonometric"></a><a class="link" href="cardinal_trigonometric.html" title="Cardinal Trigonometric interpolation">Cardinal Trigonometric
interpolation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.cardinal_trigonometric.h0"></a>
<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.synopsis"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">cardinal_trigonometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">interpolators</span> <span class="special">{</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">cardinal_trigonometric</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="identifier">cardinal_trigonometric</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">h</span><span class="special">);</span>
<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">double_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">period</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">integrate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">squared_l2</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}}}</span>
</pre>
<h4>
<a name="math_toolkit.cardinal_trigonometric.h1"></a>
<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.cardinal_trigonometric_interpola"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.cardinal_trigonometric_interpola">Cardinal
Trigonometric Interpolation</a>
</h4>
<p>
The cardinal trigonometric interpolation problem takes uniformly spaced samples
<span class="emphasis"><em>y</em></span><sub>j</sub> of a periodic function <span class="emphasis"><em>f</em></span> defined
via <span class="emphasis"><em>y</em></span><sub><span class="emphasis"><em>j</em></span></sub> = <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span><sub>0</sub> +
<span class="emphasis"><em>j</em></span> <span class="emphasis"><em>h</em></span>) and represents them as a linear
combination of sines and cosines. If the period of <span class="emphasis"><em>f</em></span> is
<span class="emphasis"><em>T</em></span>, and the number of data points is <span class="emphasis"><em>n = 2m</em></span>
or <span class="emphasis"><em>n = 2m+1</em></span>, we hope to have
</p>
<p>
<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>t</em></span>) ≈ <span class="emphasis"><em>a</em></span><sub>0</sub>/2
+ ∑<sub><span class="emphasis"><em>k</em></span> = 1</sub><sup><span class="emphasis"><em>m</em></span></sup> <span class="emphasis"><em>a</em></span><sub><span class="emphasis"><em>k</em></span></sub> cos(2π
<span class="emphasis"><em>k</em></span> (<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>) /T)
+ <span class="emphasis"><em>b</em></span><sub><span class="emphasis"><em>k</em></span></sub> sin(2π <span class="emphasis"><em>k</em></span>
(<span class="emphasis"><em>t</em></span>-<span class="emphasis"><em>t</em></span><sub>0</sub>)/T)
</p>
<p>
Convergence rates depend on the number of continuous derivatives of <span class="emphasis"><em>f</em></span>;
see either Atkinson or Kress for details.
</p>
<p>
A simple use of this interpolator is shown below:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">cardinal_trigonometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cardinal_trigonometric</span><span class="special">;</span>
<span class="special">...</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="number">17</span><span class="special">,</span> <span class="number">3.2</span><span class="special">);</span>
<span class="keyword">auto</span> <span class="identifier">ct</span> <span class="special">=</span> <span class="identifier">cardinal_trigonometric</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="comment">/*t0 = */</span> <span class="number">0.0</span><span class="special">,</span> <span class="comment">/* h = */</span> <span class="number">0.125</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ct(1.3) = "</span> <span class="special"><<</span> <span class="identifier">ct</span><span class="special">(</span><span class="number">1.3</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// Derivative:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">ct</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">1.2</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
<span class="comment">// Second derivative:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">ct</span><span class="special">.</span><span class="identifier">double_prime</span><span class="special">(</span><span class="number">1.2</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
The period is always given by <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()*</span><span class="identifier">h</span></code>. Off-by-one errors are common in programming,
and hence if you wonder what this interpolator believes the period to be, you
can query it with the <code class="computeroutput"><span class="special">.</span><span class="identifier">period</span><span class="special">()</span></code> member function.
</p>
<p>
In addition, the transform into the trigonometric basis gives a trivial way
to compute the integral of the function over a period; this is done via the
<code class="computeroutput"><span class="special">.</span><span class="identifier">integrate</span><span class="special">()</span></code> member function. Evaluation of the square
of the L<sup>2</sup> norm is trivial in this basis; it is computed by the <code class="computeroutput"><span class="special">.</span><span class="identifier">squared_l2</span><span class="special">()</span></code> member function.
</p>
<p>
Below is a graph of a <span class="emphasis"><em>C</em></span><sup>∞</sup> bump function approximated
by trigonometric series. The graphs are visually indistinguishable at 20 samples.
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/fourier_bump.svg"></object></span>
</p>
<h4>
<a name="math_toolkit.cardinal_trigonometric.h2"></a>
<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.caveats"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.caveats">Caveats</a>
</h4>
<p>
This routine depends on FFTW3, and hence will only compile in float, double,
long double, and quad precision, unlike the large bulk of the library which
is compatible with arbitrary precision arithmetic. The FFTW linker flags must
be added to the compile step, i.e., <code class="computeroutput"><span class="special">-</span><span class="identifier">lm</span> <span class="special">-</span><span class="identifier">lfftw3</span></code>
for double precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lm</span>
<span class="special">-</span><span class="identifier">lfftw3f</span></code>
for float, so on.
</p>
<p>
Evaluation of derivatives is done by differentiation of Horner's method. As
always, differentiation amplifies noise; and because some rounding error is
produced by computation of the Fourier coefficients, this error is amplified
by differentiation.
</p>
<h4>
<a name="math_toolkit.cardinal_trigonometric.h3"></a>
<span class="phrase"><a name="math_toolkit.cardinal_trigonometric.references"></a></span><a class="link" href="cardinal_trigonometric.html#math_toolkit.cardinal_trigonometric.references">References</a>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Atkinson, Kendall, and Weimin Han. <span class="emphasis"><em>Theoretical numerical analysis.</em></span>
Vol. 39. Berlin: Springer, 2005.
</li>
<li class="listitem">
Kress, Rainer. <span class="emphasis"><em>Numerical Analysis.</em></span> 1998. Academic
Edition 1.
</li>
</ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="bezier_polynomial.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cubic_hermite.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
|
