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
|
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Quintic Hermite 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="pchip.html" title="PCHIP interpolation">
<link rel="next" href="bilinear_uniform.html" title="Bilinear Uniform 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="pchip.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="bilinear_uniform.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.quintic_hermite"></a><a class="link" href="quintic_hermite.html" title="Quintic Hermite interpolation">Quintic Hermite interpolation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.quintic_hermite.h0"></a>
<span class="phrase"><a name="math_toolkit.quintic_hermite.synopsis"></a></span><a class="link" href="quintic_hermite.html#math_toolkit.quintic_hermite.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">quintic_hermite</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="identifier">math</span><span class="special">::</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">quintic_hermite</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="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="identifier">quintic_hermite</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">dydx</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">d2ydx2</span><span class="special">)</span>
<span class="keyword">inline</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="keyword">inline</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="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="keyword">friend</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="keyword">operator</span><span class="special"><<(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">quintic_hermite</span> <span class="special">&</span> <span class="identifier">m</span><span class="special">);</span>
<span class="keyword">void</span> <span class="identifier">push_back</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dydx</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">d2ydx2</span><span class="special">);</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_quintic_hermite</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="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="identifier">cardinal_quintic_hermite</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">dydx</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">d2ydx2</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dx</span><span class="special">);</span>
<span class="keyword">inline</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="keyword">inline</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="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</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_quintic_hermite_aos</span> <span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">using</span> <span class="identifier">Point</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
<span class="identifier">cardinal_quintic_hermite_aos</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">data</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dx</span><span class="special">)</span>
<span class="keyword">inline</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="keyword">inline</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="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<h4>
<a name="math_toolkit.quintic_hermite.h1"></a>
<span class="phrase"><a name="math_toolkit.quintic_hermite.quintic_hermite_interpolation"></a></span><a class="link" href="quintic_hermite.html#math_toolkit.quintic_hermite.quintic_hermite_interpolation">Quintic
Hermite Interpolation</a>
</h4>
<p>
The quintic Hermite interpolator takes a list of possibly non-uniformly spaced
abscissas, ordinates, and their velocities and accelerations which are used
to construct a quintic interpolating polynomial between segments. This is useful
for taking solution skeletons from ODE steppers and turning them into a continuous
function, provided that the right-hand side <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>,
<span class="emphasis"><em>y</em></span>) is differentiable along the solution path. The interpolant
is <span class="emphasis"><em>C</em></span><sup>2</sup> and its evaluation has 𝑶(log(<span class="emphasis"><em>N</em></span>))
complexity. An example usage is as follows:
</p>
<pre class="programlisting"><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">x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span> <span class="number">4</span><span class="special">,</span> <span class="number">5</span><span class="special">,</span> <span class="number">6</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">y</span><span class="special">{</span><span class="number">7</span><span class="special">,</span><span class="number">8</span><span class="special">,</span><span class="number">9</span><span class="special">,</span><span class="number">10</span><span class="special">,</span><span class="number">11</span><span class="special">,</span><span class="number">12</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">dydx</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</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">d2ydx2</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</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">quintic_hermite</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">spline</span> <span class="special">=</span> <span class="identifier">quintic_hermite</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">x</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">y</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">dydx</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">d2ydx2</span><span class="special">));</span>
<span class="comment">// evaluate at a point:</span>
<span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span>
<span class="comment">// evaluate derivative at a point:</span>
<span class="keyword">double</span> <span class="identifier">zprime</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span>
</pre>
<p>
Periodically, it is helpful to see what data the interpolator has. This can
be achieved via
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">spline</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
Note that the interpolator is pimpl'd, so that copying the class is cheap,
and hence it can be shared between threads. (The call operator and <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span><span class="special">()</span></code>
are threadsafe.)
</p>
<p>
The interpolator can be updated in constant time. Hence we can use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span></code>
to do real-time interpolation.
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">circular_buffer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="special">...</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">};</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_y</span><span class="special">{</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">};</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_dydx</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_d2ydx2</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="keyword">auto</span> <span class="identifier">circular_akima</span> <span class="special">=</span> <span class="identifier">quintic_hermite</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">initial_x</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">initial_y</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">initial_dydx</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">initial_d2ydx2</span><span class="special">));</span>
<span class="comment">// interpolate via call operation:</span>
<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_akima</span><span class="special">(</span><span class="number">3.5</span><span class="special">);</span>
<span class="comment">// add new data:</span>
<span class="identifier">circular_akima</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">8</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">);</span>
<span class="comment">// interpolate at 4.5:</span>
<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_akima</span><span class="special">(</span><span class="number">4.5</span><span class="special">);</span>
</pre>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/quintic_sine_approximation.svg"></object></span>
</p>
<p>
For equispaced data, we can use <code class="computeroutput"><span class="identifier">cardinal_quintic_hermite</span></code>
or <code class="computeroutput"><span class="identifier">cardinal_quintic_hermite_aos</span></code>
to get constant-time evaluation. This is useful in memory-constrained or performance
critical applications where data is equispaced.
</p>
<h4>
<a name="math_toolkit.quintic_hermite.h2"></a>
<span class="phrase"><a name="math_toolkit.quintic_hermite.complexity_and_performance"></a></span><a class="link" href="quintic_hermite.html#math_toolkit.quintic_hermite.complexity_and_performance">Complexity
and Performance</a>
</h4>
<p>
The following google benchmark demonstrates the cost of the call operator for
this interpolator:
</p>
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="special">(</span><span class="number">16</span> <span class="identifier">X</span> <span class="number">4300</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span> <span class="identifier">s</span><span class="special">)</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">1024</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">11264</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">0.92</span><span class="special">,</span> <span class="number">0.64</span><span class="special">,</span> <span class="number">0.35</span>
<span class="special">--------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
<span class="special">--------------------------------------------------</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8</span> <span class="number">8.14</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16</span> <span class="number">8.69</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32</span> <span class="number">9.42</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">64</span> <span class="number">9.90</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">128</span> <span class="number">10.4</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">10.9</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">11.6</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">12.3</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">12.8</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">13.6</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">14.6</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">15.5</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">17.4</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">18.5</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">18.8</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">19.8</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">20.5</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">21.6</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">22.5</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">24.2</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">26.6</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">26.7</span> <span class="identifier">ns</span>
<span class="identifier">QuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">1.14</span> <span class="identifier">lgN</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">5.22</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">5.21</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">5.21</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">5.32</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">5.33</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">5.50</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">5.74</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">7.74</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">10.6</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">10.7</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">10.6</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">10.5</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">10.6</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">7.57</span> <span class="special">(</span><span class="number">1</span><span class="special">)</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">5.27</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">5.26</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">5.26</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">5.28</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">5.30</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">5.41</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">5.89</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">5.97</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">5.96</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">5.92</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">5.94</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">5.96</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">5.93</span> <span class="identifier">ns</span>
<span class="identifier">CardinalQuinticHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">5.64</span> <span class="special">(</span><span class="number">1</span><span class="special">)</span>
</pre>
</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="pchip.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="bilinear_uniform.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
|
