Numerical integration and differentiation

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Numerical integration and differentiation

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[QUICK DESCRIPTION]
Given a function $f\colon {\R}^m\to {\R}$, compute (approximately) the definite integral
[maths] \int_A f(x)\, dx, [/maths]
for some subset $A\subseteq {\R}^m$ or to compute (approximately) the a derivatives $\nabla f({\bold x})$, or $\partial^2 f/\partial x_i\partial x_j({\bold x})$ etc., using only a finite (or small) number of function evaluations $f({\bold x}_i)$, $i=1, 2, \ldots, N$.

The most efficient numerical integration methods have been developed for scalar problems ($m=1$), including methods such as Gauss quadrature and Romberg integration. 
But there are good methods for many regions in higher dimensions.  Regions with high dimensionality (e.g., $m\geq 6$) are a particular challenge, as are integrands with singularities or localized "spikes".

[PREREQUISITES]
Calculus, interpolation.

[EXAMPLE int-int|Integrate (or differentiate) the interpolant]

If you have an effective and accurate method of interpolation, you can use that to construct an integration method as the integral of the interpolant.

[EXAMPLE product-int|If you have a singularity that has an simple form, use product integration methods]
For example, for computing integrals of the form 
[maths] \int_0^1 x^\alpha\,f(x)\,dx, [/maths]
where $f$ is smooth on $[0,1]$, determine a polynomial interpolant $p$ of $f$, and compute
[maths] \int_0^1 x^\alpha\,p(x)\,dx. [/maths]

This technique is also useful for dealing with smooth functions with localized "spikes" that would otherwise require very many integration points, even with adaptive methods: for example, consider computing
[maths] \int_a^b k^{1/2}\,e^{-kx^2}\,f(x)\,dx [/maths]
with $k\gg 1$.  By using an interpolant of $f$ on $[a,b]$ and computing the resulting integral exactly, the only errors incurred are due to the interpolation of $f$, not the fact that the integral has a "spike".

[GENERAL DISCUSSION]

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Numerical integration and differentiation

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