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Splines and Gaussian quadrature

A spline is a piecewise polynomial that is smooth across the joints between pieces; cubic splines are most common and avoid Runge's phenomenon. Gaussian quadrature is an integration method that chooses points and weights optimally, so that $n$ points give an exact answer for polynomials up to degree $2n-1$ .

\int_a^b f\,dx \approx \sum_{i=1}^n w_i f(x_i)

Gaussian quadrature with

n

optimally chosen points and weights

Symbols

$x_i$	quadrature points (not evenly spaced)
$w_i$	the weight of each point

Example

With $n = 2$ points, Gaussian quadrature gives an exact answer for all polynomials up to degree $2\cdot 2 - 1 = 3$ .

Gaussian quadrature needs far fewer points than trapezoid/Simpson for the same accuracy, but the points are not evenly spaced.

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Part of Numerical Methods: Interpolation and integration.