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We saw when discussing numerical differentiation that when we have a sequence of numbers which are approximations to a number A whose errors decrease by a given factor r from term to term, we can extrapolate the sequence to get a more rapidly converging one. The general extrapolation rule is: to get rid of errors that decrease by a factor of z, take the current result times z minus the previous result and divide this difference by z -1. Here for example, if we look at the answers provided by the trapezoid rule
for values 16d, 8d, 4d, 2d, d, say, we get a sequence of approximate answers whose
errors can be expected to decrease by a factor of roughly 4 each time. If we denote
these answers as
and look at the sequence ,
(for j = 2 to 5) the terms in A that go down as a factor of 4 will cancel out
in each B and we will be left with the higher order terms only. Exercise 25.3 Verify this statement. Exercises:25.4 Use the results of Exercise 2 which give the A's to compute the B's
to E's. Compare their accuracy on an integral whose value you know. (for example
sin (x) from x = 0 to 1) 25.5 Can you find a function for which this approach to integrating is lousy? What might you do to improve it? |