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Interpolation and Extrapolation part 6

Occasionally you may wish to know not the value of the interpolating polynomial that passes through a (small!) number of points, but the coefficients of that polynomial. A valid use of the coefficients might be, for example, to compute simultaneous interpolated values | 120 Chapter 3. Interpolation and Extrapolation 3.5 Coefficients ofthe Interpolating Polynomial Occasionally you may wish to know not the value of the interpolating polynomial that passes through a small number of points but the coefficients of that polynomial. A valid use of the coefficients might be for example to compute simultaneous interpolated values of the function and of several of its derivatives see 5.3 or to convolve a segment of the tabulated function with some other function where the moments of that other function i.e. its convolution with powers of x are known analytically. However please be certain that the coefficients are what you need. Generally the coefficients of the interpolating polynomial can be determined much less accurately than its value at a desired abscissa. Therefore it is not a good idea to determine the coefficients only for use in calculating interpolating values. Values thus calculated will not pass exactly through the tabulated points for example while values computed by the routines in 3.1 3.3 will pass exactly through such points. Also you should not mistake the interpolating polynomial and its coefficients for its cousin the best fit polynomial through a data set. Fitting is a smoothing process since the number of fitted coefficients is typically much less than the number of data points. Therefore fitted coefficients can be accurately and stably determined even in the presence of statistical errors in the tabulated values. See 14.8. Interpolation where the number of coefficients and number of tabulated points are equal takes the tabulated values as perfect. If they in fact contain statistical errors these can be magnified into oscillations of the interpolating polynomial in between the tabulated points. As before we take the tabulated points to be yi y xj . If the interpolating polynomial is written as y co c1x c2x2 cN xN 3.5.1 then the cfs are required to satisfy the linear equation -1 xo 2 xo N 3 x0 r 3 co yo 1 . X1 . 2 x i .