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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_poly.cardinal_b_splines"></a><a class="link" href="cardinal_b_splines.html" title="Cardinal B-splines">Cardinal B-splines</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_poly.cardinal_b_splines.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.cardinal_b_splines.synopsis"></a></span><a class="link" href="cardinal_b_splines.html#math_toolkit.sf_poly.cardinal_b_splines.synopsis">Synopsis</a>
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</h5>
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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">special_functions</span><span class="special">/</span><span class="identifier">cardinal_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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">cardinal_b_spline</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">cardinal_b_spline_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">cardinal_b_spline_double_prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="identifier">Real</span> <span class="identifier">forward_cardinal_b_spline</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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Cardinal <span class="emphasis"><em>B</em></span>-splines are a family of compactly supported
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functions useful for the smooth interpolation of tables.
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</p>
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<p>
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The first <span class="emphasis"><em>B</em></span>-spline <span class="emphasis"><em>B</em></span><sub>0</sub> is simply
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a box function: It takes the value one inside the interval [-1/2, 1/2], and
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is zero elsewhere. <span class="emphasis"><em>B</em></span>-splines of higher smoothness are
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constructed by iterative convolution, namely, <span class="emphasis"><em>B</em></span><sub>1</sub> :=
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<span class="emphasis"><em>B</em></span><sub>0</sub> ∗ <span class="emphasis"><em>B</em></span><sub>0</sub>, and <span class="emphasis"><em>B</em></span><sub><span class="emphasis"><em>n</em></span>+1</sub> :=
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<span class="emphasis"><em>B</em></span><sub><span class="emphasis"><em>n</em></span> </sub> ∗ <span class="emphasis"><em>B</em></span><sub>0</sub>.
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For example, <span class="emphasis"><em>B</em></span><sub>1</sub>(<span class="emphasis"><em>x</em></span>) = 1 - |<span class="emphasis"><em>x</em></span>|
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for <span class="emphasis"><em>x</em></span> in [-1,1], and zero elsewhere, so it is a hat
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function.
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</p>
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<p>
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A basic usage is as follows:
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</p>
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<pre class="programlisting"><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">cardinal_b_spline</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">cardinal_b_spline_prime</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">cardinal_b_spline_double_prime</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
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<span class="comment">// B₀(x), the box function:</span>
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<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">cardinal_b_spline</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">x</span><span class="special">);</span>
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<span class="comment">// B₁(x), the hat function:</span>
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<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">cardinal_b_spline</span><span class="special"><</span><span class="number">1</span><span class="special">>(</span><span class="identifier">x</span><span class="special">);</span>
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<span class="comment">// First derivative of B₃:</span>
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<span class="identifier">yp</span> <span class="special">=</span> <span class="identifier">cardinal_b_spline_prime</span><span class="special"><</span><span class="number">3</span><span class="special">>(</span><span class="identifier">x</span><span class="special">);</span>
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<span class="comment">// Second derivative of B₃:</span>
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<span class="identifier">ypp</span> <span class="special">=</span> <span class="identifier">cardinal_b_spline_double_prime</span><span class="special"><</span><span class="number">3</span><span class="special">>(</span><span class="identifier">x</span><span class="special">);</span>
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</pre>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/central_b_splines.svg"></object></span> <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/central_b_spline_derivatives.svg"></object></span>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/central_b_spline_second_derivatives.svg"></object></span>
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.cardinal_b_splines.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.cardinal_b_splines.caveats"></a></span><a class="link" href="cardinal_b_splines.html#math_toolkit.sf_poly.cardinal_b_splines.caveats">Caveats</a>
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</h4>
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<p>
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Numerous notational conventions for <span class="emphasis"><em>B</em></span>-splines exist.
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Whereas Boost.Math (following Kress) zero indexes the <span class="emphasis"><em>B</em></span>-splines,
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other authors (such as Schoenberg and Massopust) use 1-based indexing. So
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(for example) Boost.Math's hat function <span class="emphasis"><em>B</em></span><sub>1</sub> is what Schoenberg
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calls <span class="emphasis"><em>M</em></span><sub>2</sub>. Mathematica, like Boost, uses the zero-indexing
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convention for its <code class="computeroutput"><span class="identifier">BSplineCurve</span></code>.
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</p>
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<p>
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Even the support of the splines is not agreed upon. Mathematica starts the
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support of the splines at zero and rescales the independent variable so that
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the support of every member is [0, 1]. Massopust as well as Unser puts the
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support of the <span class="emphasis"><em>B</em></span>-splines at [0, <span class="emphasis"><em>n</em></span>],
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whereas Kress centers them at zero. Schoenberg distinguishes between the
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two cases, called the splines starting at zero forward splines, and the ones
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symmetric about zero <span class="emphasis"><em>central</em></span>.
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</p>
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<p>
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The <span class="emphasis"><em>B</em></span>-splines of Boost.Math are central, with support
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support [-(<span class="emphasis"><em>n</em></span>+1)/2, (<span class="emphasis"><em>n</em></span>+1)/2]. If
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necessary, the forward splines can be evaluated by using <code class="computeroutput"><span class="identifier">forward_cardinal_b_spline</span></code>,
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whose support is [0, <span class="emphasis"><em>n</em></span>+1].
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.cardinal_b_splines.h2"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.cardinal_b_splines.implementation"></a></span><a class="link" href="cardinal_b_splines.html#math_toolkit.sf_poly.cardinal_b_splines.implementation">Implementation</a>
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</h4>
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<p>
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The implementation follows Maurice Cox' 1972 paper 'The Numerical Evaluation
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of B-splines', and uses the triangular array of Algorithm 6.1 of the reference
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rather than the rhombohedral array of Algorithm 6.2. Benchmarks revealed
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that the time to calculate the indexes of the rhombohedral array exceed the
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time to simply add zeroes together, except for <span class="emphasis"><em>n</em></span> >
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18. Since few people use <span class="emphasis"><em>B</em></span> splines of degree 18, the
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triangular array is used.
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.cardinal_b_splines.h3"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.cardinal_b_splines.performance"></a></span><a class="link" href="cardinal_b_splines.html#math_toolkit.sf_poly.cardinal_b_splines.performance">Performance</a>
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</h4>
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<p>
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Double precision timing on a consumer x86 laptop is shown below:
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</p>
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<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>
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<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
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<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>
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<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>
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<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>
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<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>
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<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">0.21</span><span class="special">,</span> <span class="number">0.33</span><span class="special">,</span> <span class="number">0.29</span>
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<span class="special">-----------------------------------------</span>
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<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
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<span class="special">-----------------------------------------</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">1</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">18.8</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">2</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">25.3</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">3</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">29.3</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">4</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">33.8</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">5</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">36.7</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">6</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">39.1</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">7</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">43.6</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">8</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">62.8</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">9</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">70.2</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">10</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">83.8</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">11</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">94.3</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">12</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">108</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">13</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">122</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">14</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">138</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">15</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">155</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">16</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">170</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">17</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">192</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">18</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">174</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">19</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">180</span> <span class="identifier">ns</span>
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<span class="identifier">CardinalBSpline</span><span class="special"><</span><span class="number">20</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="number">194</span> <span class="identifier">ns</span>
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<span class="identifier">UniformReal</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="number">11.5</span> <span class="identifier">ns</span>
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</pre>
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<p>
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A uniformly distributed random number within the support of the spline is
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generated for the argument, so subtracting 11.5 ns from these gives a good
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idea of the performance of the calls.
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.cardinal_b_splines.h4"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.cardinal_b_splines.accuracy"></a></span><a class="link" href="cardinal_b_splines.html#math_toolkit.sf_poly.cardinal_b_splines.accuracy">Accuracy</a>
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</h4>
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<p>
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Some representative ULP plots are shown below. The error grows linearly with
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<span class="emphasis"><em>n</em></span>, as expected from Cox equation 10.5.
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/b_spline_ulp_3.svg"></object></span> <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/b_spline_ulp_5.svg"></object></span>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/b_spline_ulp_9.svg"></object></span>
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</p>
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<h4>
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<a name="math_toolkit.sf_poly.cardinal_b_splines.h5"></a>
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<span class="phrase"><a name="math_toolkit.sf_poly.cardinal_b_splines.references"></a></span><a class="link" href="cardinal_b_splines.html#math_toolkit.sf_poly.cardinal_b_splines.references">References</a>
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</h4>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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I.J. Schoenberg, <span class="emphasis"><em>Cardinal Spline Interpolation</em></span>,
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SIAM Volume 12, 1973
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</li>
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<li class="listitem">
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Rainer Kress, <span class="emphasis"><em>Numerical Analysis</em></span>, Springer, 1998
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</li>
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<li class="listitem">
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Peter Massopust, <span class="emphasis"><em>On Some Generalizations of B-splines</em></span>,
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arxiv preprint, 2019
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</li>
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<li class="listitem">
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Michael Unser and Thierry Blu, <span class="emphasis"><em>Fractional Splines and Wavelets</em></span>,
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SIAM Review 2000, Volume 42, No. 1
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</li>
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<li class="listitem">
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Cox, Maurice G. <span class="emphasis"><em>The numerical evaluation of B-splines.</em></span>,
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IMA Journal of Applied Mathematics 10.2 (1972): 134-149.
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</li>
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</ul></div>
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</div>
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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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