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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.cubic_b"></a><a class="link" href="cubic_b.html" title="Cubic B-spline interpolation">Cubic B-spline interpolation</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.cubic_b.h0"></a>
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<span class="phrase"><a name="math_toolkit.cubic_b.synopsis"></a></span><a class="link" href="cubic_b.html#math_toolkit.cubic_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">cubic_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">class</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cubic_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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">BidiIterator</span><span class="special">></span>
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<span class="identifier">cubic_b_spline</span><span class="special">(</span><span class="identifier">BidiIterator</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">BidiIterator</span> <span class="identifier">b</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>
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<span class="identifier">Real</span> <span class="identifier">left_endpoint_derivative</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>
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<span class="identifier">Real</span> <span class="identifier">right_endpoint_derivative</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>
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<span class="identifier">cubic_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">f</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">length</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>
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<span class="identifier">Real</span> <span class="identifier">left_endpoint_derivative</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>
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<span class="identifier">Real</span> <span class="identifier">right_endpoint_derivative</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>
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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">x</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">x</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> <span class="comment">// namespaces</span>
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</pre>
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<h4>
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<a name="math_toolkit.cubic_b.h1"></a>
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<span class="phrase"><a name="math_toolkit.cubic_b.cubic_b_spline_interpolation"></a></span><a class="link" href="cubic_b.html#math_toolkit.cubic_b.cubic_b_spline_interpolation">Cubic
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B-Spline Interpolation</a>
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</h4>
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<p>
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The cubic B-spline class provided by boost allows fast and accurate interpolation
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of a function which is known at equally spaced points. The cubic B-spline interpolation
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is numerically stable as it uses compactly supported basis functions constructed
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via iterative convolution. This is to be contrasted to traditional cubic spline
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interpolation is ill-conditioned as the global support of cubic polynomials
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causes small changes far from the evaluation point exert a large influence
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on the calculated value.
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</p>
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<p>
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There are many use cases for interpolating a function at equally spaced points.
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One particularly important example is solving ODE's whose coefficients depend
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on data determined from experiment or numerical simulation. Since most ODE
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steppers are adaptive, they must be able to sample the coefficients at arbitrary
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points; not just at the points we know the values of our function.
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</p>
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<p>
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The first two arguments to the constructor are either:
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</p>
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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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A pair of bidirectional iterators into the data, or
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</li>
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<li class="listitem">
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A pointer to the data, and a length of the data array.
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</li>
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</ul></div>
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<p>
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These are then followed by:
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</p>
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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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The start of the functions domain
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</li>
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<li class="listitem">
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The step size
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</li>
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</ul></div>
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<p>
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Optionally, you may provide two additional arguments to the constructor, namely
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the derivative of the function at the left endpoint, and the derivative at
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the right endpoint. If you do not provide these arguments, they will be estimated
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using one-sided finite-difference formulas. An example of a valid call to the
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constructor is
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</p>
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<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">f</span><span class="special">{</span><span class="number">0.01</span><span class="special">,</span> <span class="special">-</span><span class="number">0.02</span><span class="special">,</span> <span class="number">0.3</span><span class="special">,</span> <span class="number">0.8</span><span class="special">,</span> <span class="number">1.9</span><span class="special">,</span> <span class="special">-</span><span class="number">8.78</span><span class="special">,</span> <span class="special">-</span><span class="number">22.6</span><span class="special">};</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>
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<span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cubic_b_spline</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">spline</span><span class="special">(</span><span class="identifier">f</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">f</span><span class="special">.</span><span class="identifier">end</span><span class="special">(),</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">h</span><span class="special">);</span>
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</pre>
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<p>
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The endpoints are estimated using a one-sided finite-difference formula. If
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you know the derivative at the endpoint, you may pass it to the constructor
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via
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</p>
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<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cubic_b_spline</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">spline</span><span class="special">(</span><span class="identifier">f</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">f</span><span class="special">.</span><span class="identifier">end</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">a_prime</span><span class="special">,</span> <span class="identifier">b_prime</span><span class="special">);</span>
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</pre>
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<p>
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To evaluate the interpolant at a point, we simply use
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">spline</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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and to evaluate the derivative of the interpolant we use
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">yp</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="identifier">x</span><span class="special">);</span>
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</pre>
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<p>
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Be aware that the accuracy guarantees on the derivative of the spline are an
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order lower than the guarantees on the original function, see <a href="http://www.springer.com/us/book/9780387984087" target="_top">Numerical
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Analysis, Graduate Texts in Mathematics, 181, Rainer Kress</a> for details.
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</p>
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<p>
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Finally, note that this is an interpolator, not an extrapolator. Therefore,
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you should strenuously avoid evaluating the spline outside the endpoints. However,
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it is not an error if you do, as often you cannot control where (say) an ODE
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stepper will evaluate your function. As such the interpolant tries to do something
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reasonable when the passed a value outside the endpoints. For evaluation within
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one stepsize of the interval, you can assume something somewhat reasonable
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was returned. As you move further away from the endpoints, the interpolant
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decays to its average on the interval.
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</p>
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<h4>
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<a name="math_toolkit.cubic_b.h2"></a>
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<span class="phrase"><a name="math_toolkit.cubic_b.complexity_and_performance"></a></span><a class="link" href="cubic_b.html#math_toolkit.cubic_b.complexity_and_performance">Complexity
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and Performance</a>
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</h4>
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<p>
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The call to the constructor requires 𝑶(<span class="emphasis"><em>n</em></span>) operations, where
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<span class="emphasis"><em>n</em></span> is the number of points to interpolate. Each call the
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the interpolant is 𝑶(1) (constant time). On the author's Intel Xeon E3-1230,
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this takes 21ns as long as the vector is small enough to fit in cache.
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</p>
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<h4>
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<a name="math_toolkit.cubic_b.h3"></a>
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<span class="phrase"><a name="math_toolkit.cubic_b.accuracy"></a></span><a class="link" href="cubic_b.html#math_toolkit.cubic_b.accuracy">Accuracy</a>
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</h4>
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<p>
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Let <span class="emphasis"><em>h</em></span> be the stepsize. If <span class="emphasis"><em>f</em></span> is four-times
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continuously differentiable, then the interpolant is <span class="emphasis"><em>𝑶(h<sup>4</sup>)</em></span>
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accurate and the derivative is <span class="emphasis"><em>𝑶(h<sup>3</sup>)</em></span> accurate.
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</p>
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<h4>
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<a name="math_toolkit.cubic_b.h4"></a>
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<span class="phrase"><a name="math_toolkit.cubic_b.testing"></a></span><a class="link" href="cubic_b.html#math_toolkit.cubic_b.testing">Testing</a>
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</h4>
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<p>
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Since the interpolant obeys <span class="emphasis"><em>s(x<sub>j</sub>) = f(x<sub>j</sub>)</em></span> at all interpolation
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points, the tests generate random data and evaluate the interpolant at the
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interpolation points, validating that equality with the data holds.
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</p>
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<p>
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In addition, constant, linear, and quadratic functions are interpolated to
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ensure that the interpolant behaves as expected.
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</p>
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<h4>
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<a name="math_toolkit.cubic_b.h5"></a>
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<span class="phrase"><a name="math_toolkit.cubic_b.example"></a></span><a class="link" href="cubic_b.html#math_toolkit.cubic_b.example">Example</a>
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</h4>
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<p>
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This example demonstrates how to use the cubic b spline interpolator for regularly
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spaced data.
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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">cubic_b_spline</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
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<span class="special">{</span>
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<span class="comment">// We begin with an array of samples:</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">500</span><span class="special">);</span>
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<span class="comment">// And decide on a stepsize:</span>
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<span class="keyword">double</span> <span class="identifier">step</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span>
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<span class="comment">// Initialize the vector with a function we'd like to interpolate:</span>
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<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"><</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
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<span class="special">{</span>
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<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">sin</span><span class="special">(</span><span class="identifier">i</span><span class="special">*</span><span class="identifier">step</span><span class="special">);</span>
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<span class="special">}</span>
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<span class="comment">// We could define an arbitrary start time, but for now we'll just use 0:</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cubic_b_spline</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">spline</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="number">0</span> <span class="comment">/* start time */</span><span class="special">,</span> <span class="identifier">step</span><span class="special">);</span>
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<span class="comment">// Now we can evaluate the spline wherever we please.</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">gen</span><span class="special">;</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">random</span><span class="special">::</span><span class="identifier">uniform_real_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">absissa</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()*</span><span class="identifier">step</span><span class="special">);</span>
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<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"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">absissa</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"sin("</span> <span class="special"><<</span> <span class="identifier">x</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", spline interpolation gives "</span> <span class="special"><<</span> <span class="identifier">spline</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cos("</span> <span class="special"><<</span> <span class="identifier">x</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">cos</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", spline derivative interpolation gives "</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="identifier">x</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="comment">// The next example is less trivial:</span>
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<span class="comment">// We will try to figure out when the population of the United States crossed 100 million.</span>
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<span class="comment">// Since the census is taken every 10 years, the data is equally spaced, so we can use the cubic b spline.</span>
|
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<span class="comment">// Data taken from https://en.wikipedia.org/wiki/United_States_Census</span>
|
|
<span class="comment">// An eye</span>
|
|
<span class="comment">// We'll start at the year 1860:</span>
|
|
<span class="keyword">double</span> <span class="identifier">t0</span> <span class="special">=</span> <span class="number">1860</span><span class="special">;</span>
|
|
<span class="keyword">double</span> <span class="identifier">time_step</span> <span class="special">=</span> <span class="number">10</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">population</span><span class="special">{</span><span class="number">31443321</span><span class="special">,</span> <span class="comment">/* 1860 */</span>
|
|
<span class="number">39818449</span><span class="special">,</span> <span class="comment">/* 1870 */</span>
|
|
<span class="number">50189209</span><span class="special">,</span> <span class="comment">/* 1880 */</span>
|
|
<span class="number">62947714</span><span class="special">,</span> <span class="comment">/* 1890 */</span>
|
|
<span class="number">76212168</span><span class="special">,</span> <span class="comment">/* 1900 */</span>
|
|
<span class="number">92228496</span><span class="special">,</span> <span class="comment">/* 1910 */</span>
|
|
<span class="number">106021537</span><span class="special">,</span> <span class="comment">/* 1920 */</span>
|
|
<span class="number">122775046</span><span class="special">,</span> <span class="comment">/* 1930 */</span>
|
|
<span class="number">132164569</span><span class="special">,</span> <span class="comment">/* 1940 */</span>
|
|
<span class="number">150697361</span><span class="special">,</span> <span class="comment">/* 1950 */</span>
|
|
<span class="number">179323175</span><span class="special">};/*</span> <span class="number">1960</span> <span class="special">*/</span>
|
|
|
|
<span class="comment">// An eyeball estimate indicates that the population crossed 100 million around 1915.</span>
|
|
<span class="comment">// Let's see what interpolation says:</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cubic_b_spline</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">p</span><span class="special">(</span><span class="identifier">population</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">population</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="identifier">t0</span><span class="special">,</span> <span class="number">10</span><span class="special">);</span>
|
|
|
|
<span class="comment">// Now create a function which has a zero at p = 100,000,000:</span>
|
|
<span class="keyword">auto</span> <span class="identifier">f</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="keyword">return</span> <span class="identifier">p</span><span class="special">(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">-</span> <span class="number">100000000</span><span class="special">;</span> <span class="special">};</span>
|
|
|
|
<span class="comment">// Boost includes a bisection algorithm, which is robust, though not as fast as some others </span>
|
|
<span class="comment">// we provide, but lets try that first. We need a termination condition for it, which</span>
|
|
<span class="comment">// takes the two endpoints of the range and returns either true (stop) or false (keep going),</span>
|
|
<span class="comment">// we could use a predefined one such as boost::math::tools::eps_tolerance<double>, but that</span>
|
|
<span class="comment">// won't stop until we have full double precision which is overkill, since we just need the </span>
|
|
<span class="comment">// endpoint to yield the same month. While we're at it, we'll keep track of the number of</span>
|
|
<span class="comment">// iterations required too, though this is strictly optional:</span>
|
|
|
|
<span class="keyword">auto</span> <span class="identifier">termination</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">left</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">right</span><span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">double</span> <span class="identifier">left_month</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round</span><span class="special">((</span><span class="identifier">left</span> <span class="special">-</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">left</span><span class="special">))</span> <span class="special">*</span> <span class="number">12</span> <span class="special">+</span> <span class="number">1</span><span class="special">);</span>
|
|
<span class="keyword">double</span> <span class="identifier">right_month</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round</span><span class="special">((</span><span class="identifier">right</span> <span class="special">-</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">right</span><span class="special">))</span> <span class="special">*</span> <span class="number">12</span> <span class="special">+</span> <span class="number">1</span><span class="special">);</span>
|
|
<span class="keyword">return</span> <span class="special">(</span><span class="identifier">left_month</span> <span class="special">==</span> <span class="identifier">right_month</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">left</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">right</span><span class="special">));</span>
|
|
<span class="special">};</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">iterations</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
|
|
<span class="keyword">auto</span> <span class="identifier">result</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">tools</span><span class="special">::</span><span class="identifier">bisect</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">1910.0</span><span class="special">,</span> <span class="number">1920.0</span><span class="special">,</span> <span class="identifier">termination</span><span class="special">,</span> <span class="identifier">iterations</span><span class="special">);</span>
|
|
<span class="keyword">auto</span> <span class="identifier">time</span> <span class="special">=</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span> <span class="comment">// termination condition ensures that both endpoints yield the same result</span>
|
|
<span class="keyword">auto</span> <span class="identifier">month</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round</span><span class="special">((</span><span class="identifier">time</span> <span class="special">-</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">time</span><span class="special">))*</span><span class="number">12</span> <span class="special">+</span> <span class="number">1</span><span class="special">);</span>
|
|
<span class="keyword">auto</span> <span class="identifier">year</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">time</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The population of the United States surpassed 100 million on the "</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">month</span> <span class="special"><<</span> <span class="string">"th month of "</span> <span class="special"><<</span> <span class="identifier">year</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Found in "</span> <span class="special"><<</span> <span class="identifier">iterations</span> <span class="special"><<</span> <span class="string">" iterations"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
|
|
<span class="comment">// Since the cubic B spline offers the first derivative, we could equally have used Newton iterations,</span>
|
|
<span class="comment">// this takes "number of bits correct" as a termination condition - 20 should be plenty for what we need,</span>
|
|
<span class="comment">// and once again, we track how many iterations are taken:</span>
|
|
|
|
<span class="keyword">auto</span> <span class="identifier">f_n</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="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span><span class="identifier">p</span><span class="special">(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">-</span> <span class="number">100000000</span><span class="special">,</span> <span class="identifier">p</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="identifier">t</span><span class="special">));</span> <span class="special">};</span>
|
|
<span class="identifier">iterations</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
|
|
<span class="identifier">time</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">tools</span><span class="special">::</span><span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">f_n</span><span class="special">,</span> <span class="number">1910.0</span><span class="special">,</span> <span class="number">1900.0</span><span class="special">,</span> <span class="number">2000.0</span><span class="special">,</span> <span class="number">20</span><span class="special">,</span> <span class="identifier">iterations</span><span class="special">);</span>
|
|
<span class="identifier">month</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round</span><span class="special">((</span><span class="identifier">time</span> <span class="special">-</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">time</span><span class="special">))*</span><span class="number">12</span> <span class="special">+</span> <span class="number">1</span><span class="special">);</span>
|
|
<span class="identifier">year</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">floor</span><span class="special">(</span><span class="identifier">time</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The population of the United States surpassed 100 million on the "</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">month</span> <span class="special"><<</span> <span class="string">"th month of "</span> <span class="special"><<</span> <span class="identifier">year</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Found in "</span> <span class="special"><<</span> <span class="identifier">iterations</span> <span class="special"><<</span> <span class="string">" iterations"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
|
|
<span class="special">}</span>
|
|
</pre>
|
|
<pre class="programlisting"><span class="identifier">Program</span> <span class="identifier">output</span> <span class="identifier">is</span><span class="special">:</span>
|
|
</pre>
|
|
<pre class="programlisting">sin(4.07362) = -0.802829, spline interpolation gives - 0.802829
|
|
cos(4.07362) = -0.596209, spline derivative interpolation gives - 0.596209
|
|
sin(0.677385) = 0.626758, spline interpolation gives 0.626758
|
|
cos(0.677385) = 0.779214, spline derivative interpolation gives 0.779214
|
|
sin(4.52896) = -0.983224, spline interpolation gives - 0.983224
|
|
cos(4.52896) = -0.182402, spline derivative interpolation gives - 0.182402
|
|
sin(4.17504) = -0.85907, spline interpolation gives - 0.85907
|
|
cos(4.17504) = -0.511858, spline derivative interpolation gives - 0.511858
|
|
sin(0.634934) = 0.593124, spline interpolation gives 0.593124
|
|
cos(0.634934) = 0.805111, spline derivative interpolation gives 0.805111
|
|
sin(4.84434) = -0.991307, spline interpolation gives - 0.991307
|
|
cos(4.84434) = 0.131567, spline derivative interpolation gives 0.131567
|
|
sin(4.56688) = -0.989432, spline interpolation gives - 0.989432
|
|
cos(4.56688) = -0.144997, spline derivative interpolation gives - 0.144997
|
|
sin(1.10517) = 0.893541, spline interpolation gives 0.893541
|
|
cos(1.10517) = 0.448982, spline derivative interpolation gives 0.448982
|
|
sin(3.1618) = -0.0202022, spline interpolation gives - 0.0202022
|
|
cos(3.1618) = -0.999796, spline derivative interpolation gives - 0.999796
|
|
sin(1.54084) = 0.999551, spline interpolation gives 0.999551
|
|
cos(1.54084) = 0.0299566, spline derivative interpolation gives 0.0299566
|
|
The population of the United States surpassed 100 million on the 11th month of 1915
|
|
Found in 12 iterations
|
|
The population of the United States surpassed 100 million on the 11th month of 1915
|
|
Found in 3 iterations
|
|
</pre>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014, 2017 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, 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></td>
|
|
</tr></table>
|
|
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