Newton forward difference polynomial

When the function yix) is known, we have the flexibility of choosing the discrete points. With this flexibility, we can choose points such that the accuracy of the numerical integration can be enhanced. 

If the function y(x) is described by a collection of discrete points, then it is probably best to fit a polynomial to these points. The Lagrangian polynomial can be used, which can be fitted exactly. Alternately, the method of least square could be used. For the latter method, the polynomial may not pass through the discrete points. The Lagrangian polynomial, described in Chapter 8, can be used to exactly fit the discrete data for unequally spaced data points. For equally spaced data, the Newton forward difference polynomial will be very useful for the integration procedure. The following section will deal with equally spaced data. 

Assume that we have a set of equally spaced data at equally spaced points, Xq, Xj,. .., x , x +i,. .. and let yj be the values of y at the point Xj. The forward difference is defined as follows  

One can apply this forward difference operator to Ay to obtain A y that is, 

The same procedure can be applied to higher order differences. 

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The Newton forward difference polynomial (Finlayson 1980) is defined as... 

Having found the Nth degree Newton forward difference polynomial to approximate the integrand y(jc) (y P x)X the integral of Eq. E.1 can be readily integrated analytically after the integrand y(jc) has been replaced by P ix) as... 

The forward and the backward difference approximations give the derivatives of the Newton interpolation polynomial at the edges of the interpolation range. However, the central difference is derived from the Newton interpolation at the center of the range of interpolation. Accuracy of an interpolation formula based on equispaced points is highest at the center of the interpolation range. Therefore, the central difference interpolation formula is always more accurate than the forward or backward difference approximations. 

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