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<!-- doxytag: class="QuantLib::BSpline" -->
<p>B-spline basis functions.
<a href="class_quant_lib_1_1_b_spline.html#details">More...</a></p>
<p><code>#include <ql/math/bspline.hpp></code></p>
<p><a href="class_quant_lib_1_1_b_spline-members.html">List of all members.</a></p>
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Public Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a98d1f469c2fccf68567d10f6919a28f3"></a><!-- doxytag: member="QuantLib::BSpline::BSpline" ref="a98d1f469c2fccf68567d10f6919a28f3" args="(Natural p, Natural n, const std::vector< Real > &knots)" -->
 </td><td class="memItemRight" valign="bottom"><b>BSpline</b> (<a class="el" href="group__types.html#ga7e529c39c477ba1f5a22264d93e8457a">Natural</a> p, <a class="el" href="group__types.html#ga7e529c39c477ba1f5a22264d93e8457a">Natural</a> n, const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &knots)</td></tr>
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<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>operator()</b> (<a class="el" href="group__types.html#ga7e529c39c477ba1f5a22264d93e8457a">Natural</a> i, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> x) const </td></tr>
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<hr/><a name="details" id="details"></a><h2>Detailed Description</h2>
<div class="textblock"><p>B-spline basis functions. </p>
<p>Follows treatment and notation from:</p>
<p>Weisstein, Eric W. "B-Spline." From MathWorld--A Wolfram Web Resource. <<a href="http://mathworld.wolfram.com/B-Spline.html">http://mathworld.wolfram.com/B-Spline.html</a>></p>
<p><img class="formulaInl" alt="$ (p+1) $" src="form_170.png"/>-th order B-spline (or p degree polynomial) basis functions <img class="formulaInl" alt="$ N_{i,p}(x), i = 0,1,2 \ldots n $" src="form_171.png"/>, with <img class="formulaInl" alt="$ n+1 $" src="form_172.png"/> control points, or equivalently, an associated knot vector of size <img class="formulaInl" alt="$ p+n+2 $" src="form_173.png"/> defined at the increasingly sorted points <img class="formulaInl" alt="$ (x_0, x_1 \ldots x_{n+p+1}) $" src="form_174.png"/>. A linear B-spline has <img class="formulaInl" alt="$ p=1 $" src="form_175.png"/>, quadratic B-spline has <img class="formulaInl" alt="$ p=2 $" src="form_176.png"/>, a cubic B-spline has <img class="formulaInl" alt="$ p=3 $" src="form_177.png"/>, etc.</p>
<p>The B-spline basis functions are defined recursively as follows:</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \begin{array}{rcl} N_{i,0}(x) &=& 1 \textrm{\ if\ } x_{i} \leq x < x_{i+1} \\ &=& 0 \textrm{\ otherwise} \\ N_{i,p}(x) &=& N_{i,p-1}(x) \frac{(x - x_{i})}{ (x_{i+p-1} - x_{i})} + N_{i+1,p-1}(x) \frac{(x_{i+p} - x)}{(x_{i+p} - x_{i+1})} \end{array} \]" src="form_178.png"/>
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