Interpolation of Equally Spaced Points

If the collocation points are chosen at equidistant intervals within the interval of integration, then the collocation method is equivalent to polynomial interpolation of equally spaced points and to the finite difference method. This is not at all surprising, as the development of interpolating polynomials and finite differences were all based on expanding the function in Taylor series (see Chap. 3). It is not necessary, however, to choose the collocation points at equidistant intervals. In fact, it is more advantageous to locate the collocation points at the roots of appropriate orthogonal polynomials, as the following discussion shows. 

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