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<a class="el" href="namespace_quant_lib.html">QuantLib</a>::<a class="el" href="class_quant_lib_1_1_b_spline.html">BSpline</a></div>
<h1>BSpline Class Reference</h1><!-- doxytag: class="QuantLib::BSpline" --><code>#include <ql/math/bspline.hpp></code>
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<a href="class_quant_lib_1_1_b_spline-members.html">List of all members.</a><hr><a name="_details"></a><h2>Detailed Description</h2>
B-spline basis functions.
<p>
Follows treatment and notation from:<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>
<img class="formulaInl" alt="$ (p+1) $" src="form_102.png">-th order B-spline (or p degree polynomial) basis functions <img class="formulaInl" alt="$ N_{i,p}(x), i = 0,1,2 \ldots n $" src="form_103.png">, with <img class="formulaInl" alt="$ n+1 $" src="form_104.png"> control points, or equivalently, an associated knot vector of size <img class="formulaInl" alt="$ p+n+2 $" src="form_105.png"> defined at the increasingly sorted points <img class="formulaInl" alt="$ (x_0, x_1 \ldots x_{n+p+1}) $" src="form_106.png">. A linear B-spline has <img class="formulaInl" alt="$ p=1 $" src="form_107.png">, quadratic B-spline has <img class="formulaInl" alt="$ p=2 $" src="form_108.png">, a cubic B-spline has <img class="formulaInl" alt="$ p=3 $" src="form_109.png">, etc.<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_110.png">
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<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="98d1f469c2fccf68567d10f6919a28f3"></a><!-- doxytag: member="QuantLib::BSpline::BSpline" ref="98d1f469c2fccf68567d10f6919a28f3" 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#g7e529c39c477ba1f5a22264d93e8457a">Natural</a> p, <a class="el" href="group__types.html#g7e529c39c477ba1f5a22264d93e8457a">Natural</a> n, const std::vector< <a class="el" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &knots)</td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="f4b5a32b1cb2ea56aea90bf5db3122b2"></a><!-- doxytag: member="QuantLib::BSpline::operator()" ref="f4b5a32b1cb2ea56aea90bf5db3122b2" args="(Natural i, Real x) const " -->
<a class="el" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>operator()</b> (<a class="el" href="group__types.html#g7e529c39c477ba1f5a22264d93e8457a">Natural</a> i, <a class="el" href="group__types.html#g4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> x) const </td></tr>
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