File: whittaker_shannon.html

package info (click to toggle)
scipy 1.16.0-1exp7
  • links: PTS, VCS
  • area: main
  • in suites: experimental
  • size: 234,820 kB
  • sloc: cpp: 503,145; python: 344,611; ansic: 195,638; javascript: 89,566; fortran: 56,210; cs: 3,081; f90: 1,150; sh: 848; makefile: 785; pascal: 284; csh: 135; lisp: 134; xml: 56; perl: 51
file content (117 lines) | stat: -rw-r--r-- 12,279 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
 
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Whittaker-Shannon 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="cardinal_quintic_b.html" title="Cardinal Quintic B-spline interpolation">
<link rel="next" href="barycentric.html" title="Barycentric Rational 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="cardinal_quintic_b.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="barycentric.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.whittaker_shannon"></a><a class="link" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation">Whittaker-Shannon interpolation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.whittaker_shannon.h0"></a>
      <span class="phrase"><a name="math_toolkit.whittaker_shannon.synopsis"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.synopsis">Synopsis</a>
    </h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">whittaker_shannon</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><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">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  <span class="keyword">class</span> <span class="identifier">whittaker_shannon</span>
  <span class="special">{</span>
  <span class="keyword">public</span><span class="special">:</span>

      <span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>

      <span class="identifier">whittaker_shannon</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">left_endpoint</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">step_size</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">x</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">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  <span class="special">};</span>

<span class="special">}}}</span> <span class="comment">// namespaces</span>
</pre>
<h4>
<a name="math_toolkit.whittaker_shannon.h1"></a>
      <span class="phrase"><a name="math_toolkit.whittaker_shannon.whittaker_shannon_interpolation"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.whittaker_shannon_interpolation">Whittaker-Shannon
      Interpolation</a>
    </h4>
<p>
      The Whittaker-Shannon interpolator takes equispaced data and interpolates between
      them via a sum of sinc functions. This interpolation is stable and infinitely
      smooth, but has linear complexity in the data, making it slow relative to compactly-supported
      b-splines. In addition, we cannot pass an infinite amount of data into the
      class, and must truncate the (perhaps) infinite sinc series to a finite number
      of terms. Since the sinc function has slow <span class="emphasis"><em>1/x</em></span> decay,
      the truncation of the series can incur large error. Hence this interpolator
      works best when operating on samples of compactly-supported functions. Here
      is an example of interpolating a smooth "bump function":
    </p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">bump</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">if</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&gt;=</span> <span class="number">1</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">0.0</span><span class="special">;</span> <span class="special">}</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">(-</span><span class="number">1.0</span><span class="special">/(</span><span class="number">1.0</span><span class="special">-</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">));</span> <span class="special">};</span>

<span class="keyword">double</span> <span class="identifier">t0</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">;</span>
<span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2049</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">2.0</span><span class="special">/(</span><span class="identifier">n</span><span class="special">-</span><span class="number">1.0</span><span class="special">);</span>

<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">(</span><span class="identifier">n</span><span class="special">);</span>
<span class="keyword">for</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="identifier">n</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> <span class="special">{</span>
    <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">t0</span> <span class="special">+</span> <span class="identifier">i</span><span class="special">*</span><span class="identifier">h</span><span class="special">;</span>
    <span class="identifier">v</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">bump</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
<span class="special">}</span>

<span class="keyword">auto</span> <span class="identifier">ws</span> <span class="special">=</span> <span class="identifier">whittaker_shannon</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">v</span><span class="special">),</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">h</span><span class="special">);</span>

<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">ws</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
</pre>
<p>
      The derivative of the interpolant can also be evaluated, but the accuracy is
      not as high:
    </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">yp</span> <span class="special">=</span> <span class="identifier">ws</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
</pre>
<h4>
<a name="math_toolkit.whittaker_shannon.h2"></a>
      <span class="phrase"><a name="math_toolkit.whittaker_shannon.complexity_and_performance"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.complexity_and_performance">Complexity
      and Performance</a>
    </h4>
<p>
      The call to the constructor requires 𝑶(1) operations, simply moving data into
      the class. Each call to the interpolant is 𝑶(<span class="emphasis"><em>n</em></span>), where
      <span class="emphasis"><em>n</em></span> is the number of points to interpolate.
    </p>
</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="cardinal_quintic_b.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="barycentric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>