By Annie Cuyt Lucwuytack (Eds.)

Whereas such a lot textbooks on Numerical research talk about linear suggestions for the answer of varied numerical difficulties, this ebook introduces and illustrates nonlinear tools. It provides numerous nonlinear options ensuing frequently from using Pad? approximants and rational interpolants.

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Z. Convergence of continued fraction ezpanaiona Domain of uniform convergence of the continued fraction ( l . l 5 a . 1. This theorem was originally proved by Stieltjes [23, 28 p. 1201 in 1894. We illustrate it with t he following example. A continued fraction expansion for the function [lo p. 6141 is 'I + 1/21 11 4- - Iz l 1 + 3/21 -- Iz l 1 +... 8. this continued fraction expansion converges for all z not on the negative real axis. The next result is due t o Van Vleck [20 p. 3941 and dates from 1904.

2. ) which is in fact the k t h convergent of Pn/Q,. ) : with RPj = 1 = Sin), R P ) = Cia) and Sl;' = 0. ) Remark t h a t in comparison with P n - l / Q n - l the expression Pn/Qn contains an e x t r a term in each of the involved convergents of B;. Also Bn is not taken into account in P n - l / Q n - l . ) we must b e able to proceed from one row in the table of subconvergents t o the next row. 14. , n - 1 and Proof We shall perform the proof only for Rp' - Rf-') because it is completely analogous for sin) .

We know that 4- l ~ l 5l 1 Ib + 511 ( D 2- 2 d n ) l b + 5 1 1 2 Dd, J b + 5 1 ) 2 En In order to bound I(C - r n ) / ( C - Cn)\ we shall now calculate an upper bound for I(Tn - zl)/Tn\. 9. Modifying factors and because Using this upper bound for J f k - 1 -t Bk-11 we can also prove it to be an upper bound for \ & 2 + & 2 \ . Repeating this procedure as long as l a k l I d, and J B k - i l 5 e n , finally assures for k - I = n IFn NOWsince dn _< la1/2 = I(b +z + Pnl 5 E n l ) I /~2 ~ and 2dn _< 2 D 2 / 3 we have Next we shall compute an upper bound for Ihn/(h, + x1)1 which is the second factor in I(C - rn)/(C- Cn)l.