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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.cardinal_quintic_b"></a><a class="link" href="cardinal_quintic_b.html" title="Cardinal Quintic B-spline interpolation">Cardinal Quintic B-spline
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interpolation</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.cardinal_quintic_b.h0"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_quintic_b.synopsis"></a></span><a class="link" href="cardinal_quintic_b.html#math_toolkit.cardinal_quintic_b.synopsis">Synopsis</a>
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</h4>
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<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_quintic_b_spline</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cardinal_quintic_b_spline</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="comment">// If you don't know the value of the derivative at the endpoints, leave them as NaNs and the routine will estimate them.</span>
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<span class="comment">// y[0] = y(a), y[n - 1] = y(b), step_size = (b - a)/(n -1).</span>
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<span class="identifier">cardinal_quintic_b_spline</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">y</span><span class="special">,</span>
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<span class="identifier">size_t</span> <span class="identifier">n</span><span class="special">,</span>
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<span class="identifier">Real</span> <span class="identifier">t0</span> <span class="comment">/* initial time, left endpoint */</span><span class="special">,</span>
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<span class="identifier">Real</span> <span class="identifier">h</span> <span class="comment">/*spacing, stepsize*/</span><span class="special">,</span>
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<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">left_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()},</span>
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<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">right_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()})</span>
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<span class="identifier">cardinal_quintic_b_spline</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="identifier">Real</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">y</span><span class="special">,</span>
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<span class="identifier">Real</span> <span class="identifier">t0</span> <span class="comment">/* initial time, left endpoint */</span><span class="special">,</span>
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<span class="identifier">Real</span> <span class="identifier">h</span> <span class="comment">/*spacing, stepsize*/</span><span class="special">,</span>
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<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">left_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()},</span>
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<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">right_endpoint_derivatives</span> <span class="special">=</span> <span class="special">{</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limit</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()})</span>
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<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>
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<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>
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<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>
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<span class="special">};</span>
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<span class="special">}}}</span>
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</pre>
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<h4>
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<a name="math_toolkit.cardinal_quintic_b.h1"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_quintic_b.cardinal_quintic_b_spline_interp"></a></span><a class="link" href="cardinal_quintic_b.html#math_toolkit.cardinal_quintic_b.cardinal_quintic_b_spline_interp">Cardinal
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Quintic B-Spline Interpolation</a>
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</h4>
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<p>
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The cardinal quintic B-spline interpolator is very nearly the same as the cubic
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B-spline interpolator, with the modification that the basis functions are constructed
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by convolving a box function with itself five times, rather than three times
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as is done with the cubic B-spline.
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</p>
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<p>
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The basis functions of the quintic B-spline interpolator are more smooth than
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the cubic <span class="emphasis"><em>B</em></span>-spline interpolator, and hence this is very
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useful for computing second derivatives. For example, the second derivative
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of the cubic spline interpolator is a piecewise linear function, whereas the
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second derivative of the quintic <span class="emphasis"><em>B</em></span>-spline is a cubic spline.
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The graph of the second derivative of the quintic <span class="emphasis"><em>B</em></span>-spline
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is therefore more visually appealing, though whether it is in fact more accurate
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depends on the smoothness of your data.
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</p>
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<p>
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And example usage is as follows:
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</p>
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<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_quintic_b_spline</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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_quintic_b_spline</span><span class="special">;</span>
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<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">512</span><span class="special">);</span>
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<span class="comment">// fill v with data . . .</span>
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<span class="keyword">double</span> <span class="identifier">t0</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="comment">// initial time</span>
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<span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">0.125</span><span class="special">;</span> <span class="comment">// spacing</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">left_endpoint_derivatives</span><span class="special">{</span><span class="identifier">first_derivative_at_t0</span><span class="special">,</span> <span class="identifier">second_derivative_at_t0</span><span class="special">};</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">right_endpoint_derivatives</span><span class="special">{</span><span class="identifier">first_derivative_at_tf</span><span class="special">,</span> <span class="identifier">second_derivative_at_tf</span><span class="special">};</span>
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<span class="keyword">auto</span> <span class="identifier">qs</span> <span class="special">=</span> <span class="identifier">cardinal_quintic_b_spline</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="identifier">t0</span><span class="special">,</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">left_endpoint_derivatives</span><span class="special">,</span> <span class="identifier">right_endpoint_derivatives</span><span class="special">);</span>
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<span class="comment">// Evaluate the interpolant at a point:</span>
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<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">qs</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
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<span class="comment">// Evaluate the derivative of the interpolant:</span>
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<span class="keyword">double</span> <span class="identifier">yp</span> <span class="special">=</span> <span class="identifier">qs</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
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<span class="comment">// Evaluate the second derivative of the interpolant:</span>
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<span class="keyword">double</span> <span class="identifier">ypp</span> <span class="special">=</span> <span class="identifier">qs</span><span class="special">.</span><span class="identifier">double_prime</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
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</pre>
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<p>
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This routine will estimate the endpoint derivatives if they are not provided.
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<span class="emphasis"><em>Try to avoid this if possible.</em></span> The endpoint derivatives
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must be evaluated by finite differences and this is not robust again perturbations
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in the data. So if you have some way of knowing the endpoint derivatives, make
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sure to provide them.
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</p>
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<h4>
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<a name="math_toolkit.cardinal_quintic_b.h2"></a>
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<span class="phrase"><a name="math_toolkit.cardinal_quintic_b.references"></a></span><a class="link" href="cardinal_quintic_b.html#math_toolkit.cardinal_quintic_b.references">References</a>
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</h4>
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<p>
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Cox, Maurice G. <span class="emphasis"><em>Numerical methods for the interpolation and approximation
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of data by spline functions.</em></span> Diss. City, University of London, 1975.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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