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Numerical Integration

The problem is to compute the numerical value of the definite integral

I(f)=∫abf(x)dxI(f) = \int_a^b f(x) \ud x
(1)

where f:[a,b]→Rf:[a,b] \to \re is a given function. The solution to this problem usually takes the following form: Form a partition or grid

a≤x0≤x1≤…≤xn≤ba \le x_0 \le x_1 \le \ldots \le x_n \le b
(2)

and approximate the integral by a formula of the type

In(f)=∑j=0nwjf(xj)I_n(f) = \sum_{j=0}^n w_j f(x_j)
(3)

where the {wj}\{ w_j \} are some weights. We may be interested in the following type of questions.

  1. Given the nodes {xj}\{ x_j \} how to find the weights to obtain a good approximation ?

  2. Can we find both the nodes and weights so that the formula is as accurate as possible ?

To measure the accuracy, we may establish a result like

∣I(f)−In(f)∣=O(1np),for some p>0|I(f) - I_n(f)| = \order{\frac{1}{n^p}}, \qquad \textrm{for some $p > 0$}
(4)

Then the approximations converge fast if pp is very large and this is algebraic convergence. A still better approximation is one which would converge exponentially

∣I(f)−In(f)∣=O(e−αn),for some α>0|I(f) - I_n(f)| = \order{\ee^{-\alpha n}}, \qquad \textrm{for some $\alpha > 0$}
(5)

1Integration via function approximation

Let {fn}\{ f_n \} be a family of approximations to ff such that

lim⁡n→∞∥f−fn∥∞=0\lim_{n \to \infty} \norm{f - f_n}_\infty = 0
(6)

i.e., they converge pointwise sense. Then define

In(f)=∫abfn(x)dx=I(fn)I_n(f) = \int_a^b f_n(x) \ud x = I(f_n)
(7)

Then the error in the integral is

En(f)=I(f)−In(f)=∫ab[f(x)−fn(x)]dxE_n(f) = I(f) - I_n(f) = \int_a^b [f(x) - f_n(x)] \ud x
(8)

so that

∣En(f)∣≤∫ab∣f(x)−fn(x)∣dx≤(b−a)∥f−fn∥∞|E_n(f)| \le \int_a^b |f(x) - f_n(x)| \ud x \le (b-a) \norm{f - f_n}_\infty
(9)

We thus automatically obtain convergence of the integral approximations.

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