By J. B. Tatum
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Additional info for Celestial Mechanics Vol.1
I therefore just provide the method only, without proof. The essence of the method is to set up a table of differences as illustrated below. The first two columns are x and y. The entries in the remaining columns are the differences between the two entries in the column immediately to the left. 5 , etc. 5 x8 y8 Let us suppose that we want to find y for a value of x that is a fraction θ of the way from x4 to x5. 3 Here the Bn are the Besselian interpolation coefficients, and the successive terms in parentheses in the expansion are the sums of the numbers in the boxes in the table.
32 We write down this equation for the four adjacent tabulated points and solve for the coefficients, and hence we can evaluate the function for any value of x that we like in the interval between x1 and x4 . Unfortunately, this might well involve more computational effort than evaluating the original function itself. 6A The problem has been solved in a convenient fashion in terms of finite difference calculus, the logical development of which would involve an additional substantial chapter beyond the intended scope of this book.
Suppose a function y(x) is tabulated at x = x1 and x = x2 , the interval x2 − x1 being δx. 1 where y1 = y(x1) and y2 = y(x2). Here it is assumed that is a fraction between 0 and 1; if θ is outside this range (that is negative, or greater than 1) we are extrapolating, not interpolating, and that is always a dangerous thing to do. Let us now look at the situation where linear interpolation is not good enough. Suppose that a function y(x) is tabulated for four points x1 , x2 , x3 , x4 of the argument x, the corresponding values of the function being y1 , y2 , y3 , y4 .