By Ki Hang Kim
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U,(x)] (1) or (2) it being assumed for instance, that each u,(x) is continuous and that L u,(x) is convergent for x near a in the first case, and for x between a and b in the second. It will appear that the concept of uniform convergence is the appropriate one in the present context. This concept is particularly natural in the numerical study of convergence. We discuss, for two examples, the dependence of the no( E) on the argument x. Consider first the series 1 co Since n~o (n + x + 1)(n +x +2)' 1 1 -------------=-----(n+x+1)(n+x+2) (n+x+1) 1 (n+x+2) we have n-1 1 L= 1+x ,=0 1 n+1+x so that the sum is (1 + X)-1 and the remainder after n terms is (n + 1 + X)-1.
60 Chapter 5 Hence the significant part of the sum for Bn (f, x) occurs for r near nx but, since f is continuous, f(rln) is then near f(x). Thus Bn(f,x)=';=f(x). We have noted the apparent advantage of the Bn(f, x), that they can be written down as a mean of the values of f at the points rln, r = 0, 1, 2, ... , n. 5. This order result is true at any point Xo when f(x) has a second derivative at Xo. This means that in order to get a 6 decimal approximation to sin x we have to take the Bernstein polynomial of order about a million.
If we take p= q= in our machine we fail since p2/4 is out of range. We can counter this by calculating the radical as P SORT [1 - (q/p )/p]. This matter has been discussed at length by G. E. Forsythe. The Solution of Equations 2. BAD 47 EXAMPLES The following examples (due to A. M. Ostrowski, G. E. Forsythe and 1. H. Wilkinson) show some of the troubles which can arise with equations of higher degrees and indicates that the approximate solution of polynomial equations, which would appear to be a comparatively simple problem, is far from being so.