QuantLib: BSpline Class Reference

A free/open-source library for quantitative finance Reference manual - version 1.12

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BSpline Class Reference

B-spline basis functions. More...

`#include <ql/math/bspline.hpp>`

## Public Member Functions
| BSpline (Natural p, Natural n, const std::vector< Real > &knots)
Real | operator() (Natural i, Real x) const

## Detailed Description

B-spline basis functions.

Follows treatment and notation from:

Weisstein, Eric W. "B-Spline." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/B-Spline.html

\( (p+1) \)-th order B-spline (or p degree polynomial) basis functions \( N_{i,p}(x), i = 0,1,2 \ldots n \), with \( n+1 \) control points, or equivalently, an associated knot vector of size \( p+n+2 \) defined at the increasingly sorted points \( (x_0, x_1 \ldots x_{n+p+1}) \). A linear B-spline has \( p=1 \), quadratic B-spline has \( p=2 \), a cubic B-spline has \( p=3 \), etc.

The B-spline basis functions are defined recursively as follows:

\[ \begin{array}{rcl} N_{i,0}(x) &=& 1 \textrm{\ if\ } x_{i} \leq x < x_{i+1} \\ &=& 0 \textrm{\ otherwise} \\ N_{i,p}(x) &=& N_{i,p-1}(x) \frac{(x - x_{i})}{ (x_{i+p-1} - x_{i})} + N_{i+1,p-1}(x) \frac{(x_{i+p} - x)}{(x_{i+p} - x_{i+1})} \end{array} \]