Interpolation and Quadrature

To facilitate the development of explicit and implicit methods, it is necessary to briefly consider the origins of interpolation and quadrature formulas (i.e., numerical approximation to integration). There are essentially two methods for performing the differencing operation (as a means to approximate differentiation) one is the forward difference, and the other is the backward difference. Only the backward difference is of use in the development of eiqjlicit and implicit methods. 

The same procedure can be applied for higher order differences. 

From this definition, the second and third order differences are 

Having defined the necessary backward difference relations, we present the Newton backward interpolation formula, written generally as 

If we keep two terms in the RHS of Eq. 7.78, the resulting formula is equivalent to a linear equation passing through two points (t , y ) and (t i y i). Similarly, if three terms are retained, the resulting formula corresponds to a parabolic curve passing through three data points (t ,y ), U -i, y -i) and (t 2, y -2)- Note that these latter points are prior to (t , y ). 

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Interpolation and Quadrature 247 spaced times). The forward difference in finite difference terms is... 

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