Computing Methods. Volume 1 by I. S. Berezin and N. P. Zhidkov (Auth.)

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By I. S. Berezin and N. P. Zhidkov (Auth.)

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In this case a value of a function φ(χ) is calculated which is sufficiently precise for the computation. This will assume the values f(x0), /(#i)> · · · > f(%n) at the given points x0, xlf . . , xni and will approximate the function f(x) at the remaining points of the range (a, b) over which f(x) is defined, with a certain degree of accuracy. In the solution of the problem, operations will be performed on function φ(χ) instead of function f(x). The problem of obtaining such a function φ(χ) is called the problem of interpolation.

Performing these calculations for every possible case we shall get the following picture. Error 0 1 2 3 4 5 6 7 8 9 Number of combinations 11 10 9 8 7 6 5 4 3 2 10 1 1 I t will be a symmetrical picture in the case of negative errors. We see t h a t the number of combinations is very inconsiderable when the error is close to maximum. This will be more noticeable if three or more terms are added. Here it is difficult to calculate the number of combinations directly a n d we shall arrive a t it by a roundabout way.

Here n will be greater than or equal to m. Then, so t h a t the solution of this problem should be unambiguous, it is necessary t h a t m = n. Suppose m = n and the determinant φ0(ζ0) Δ = φ^) •

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