14.1 The method of weighted residuals can be used to find approximations of a given function. Let f( x) be a function that one would like to approximate with a polynomial of the order n in a certain domain, say 0 ? x ? 1. Let the polynomial be given by

Ideally

In terms of the weighted residuals equation this may also be expressed as

Now, suppose that w is also a polynomial of the order n:

This may also be written in an alternative format:

with

Likewise h( x) can be written as

with

Use of these expressions yields

Notice that both and are arrays with polynomial coefficients independent of x, while is an array with known functions of the coordinate x. Therefore, this equation may also be written as

The integral expression on the left-hand side is a matrix:

while the integral on the right-hand side is a column:

The equation must be satisfied for all hence, with the use of the above matrix-array notation it follows that: