By John Todd

Numerical research.

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These must be used with care. The new series may converge slower than the original- this often happens if the old series is fairly rapidly convergent and one is being greedy. , by round-off errors in the calculators. In Chapters 9, 10 we will discuss another type of acceleration process: the h 2 - extrapolation of L. F. Richardson. 1. \1<1. \. 75 and we find =2- 67 2- n. 25 = 2. However, if the relation (1) is approximate, it may be expected that the value of X given by (2) is a better approximation to the limit than those used.

W e cannot p u r s u e this question further at this stage: All we can d o is t o advise t h e user of packaged p r o g r a m s t o b e careful. 3 . NEWTON'S PROCESS T h e classical result of N e w t o n , that if x^ is an approximation t o a r o o t of / ( x ) = 0, then is a better o n e , which is motivated in t h e diagram, is well suited for t h e a p p r o x i m a t e determination of t h e roots of / ( x ) = 0, o n c e they h a v e b e e n separated. W e outline a proof that t h e sequence of approximations has quadratic convergence.

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