## Numerical integration and differentiation

### Quick description

Given a function , compute (approximately) the definite integral for some subset or to compute (approximately) the a derivatives , or etc., using only a finite (or small) number of function evaluations , .

The most efficient numerical integration methods have been developed for scalar problems ( ), 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., ) are a particular challenge, as are integrands with singularities or localized "spikes".

### Prerequisites

Calculus, interpolation.

### Example 1: 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 2: If you have a singularity that has an simple form, use product integration methods

For example, for computing integrals of the form where is smooth on , determine a polynomial interpolant of , and compute 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 with . By using an interpolant of on and computing the resulting integral exactly, the only errors incurred are due to the interpolation of , not the fact that the integral has a "spike".

### More examples

Some examples for multidimensional integration would be nice to see here.

### To find the nth derivitive of f(x):

Use any value δ, which can be made small enough to approximate the derivitive to an arbitrary number of decimal places.
f'(x)≃(f(x+δ)-f(x))/δ
f''(x) ((f(x+2δ)-f(x+δ))-(f(x+δ)-f(x)))/δ=f(x+2δ)-2f(x+δ)+f(x)
f'''(x) (f(x+3δ)-f(x+2δ)+f(x+δ)-f(x))/δ
The pattern continues. The nth derivitive is
....................n
...................⎲
f(x+nδ)/δ+ ⎳ n(-1)^k f(x+kδ)/δ
...................k=1

The dots are there because the computer won't show spaces properly.

### Angela, I think you missed

Angela, I think you missed some coefficients, correct expression should be, And I'm sorry but its really difficult to understand the formula you write for derivative. I think the expression you wanted to write is the following- 