GaussHermiteQuadrature

Gauss-Hermite Quadrature (for approximating integrals w.r.t. a Gaussian density)

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Gauss-Hermite Quadrature

The basic idea in Gauss-Hermite Quadrature is that we can evaluate an integral of the form of the product of some function $f(x)$ and a `weighting function' $p(x)$ as an appropriately weighted sum of function evaluations at a specified set of points. In other words:

$$\int_{-\infty}^\infty f(x) p(x) dx \quad \approx \quad \sum_{i=1}^N w_i f(r_i),$$

where $p(x) = \exp(-x^2)$, and the weights $w_i$ and evaluation points $r_i$ come from the theory of Hermite polynomials (which are orthogonal polynomials w.r.t. weighting function $p(x)$ ).

In THIS repository, however, the weighting function used is a standard normal Gaussian density, $p(x) = 1/\sqrt{2\pi} \exp(-x^2 / 2)$, making it easy to evaluate $f(x)$ times a normal density instead of the standard weighting function.

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Using this repository

• Set the order of the Hermite polynomial $n$. This must be an integer >= 1. Higher $n$ will give higher accuracy, though it requires more evalauations of $f$.

n = 10; % order for Gauss-Hermite polynomial

• Call compGaussHermiteQuadCoeffs to obtain the evaluation points $r_i$ and the weights $w_i$

[rr,ww] = compGaussHermiteQuadCoeffs(n); % get points and weights

• Then, for any function of interest $f$, evaluate the integral of $f$ times a standard normal pdf over the reals via:

fIntegral = sum( f(rr) .* ww);

• To evaluate the integral of $f$ times a Gaussian with mean $\mu$ and variance $\sigma^2$, instead use:

fIntegral = sum( f(rr * sig + mu) .* ww));

• See the script testGaussHermiteQuadrature for an illustrated example, along with an evaluation of accuracy