By Annie Cuyt Lucwuytack (Eds.)

Whereas so much textbooks on Numerical research speak about linear options for the answer of varied numerical difficulties, this booklet introduces and illustrates nonlinear equipment. It offers numerous nonlinear concepts ensuing quite often from using Pad? approximants and rational interpolants.

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**Extra resources for Nonlinear Methods in Numerical Analysis**

**Example text**

Take 1 1 + -12z 2 + z y + -g21 2 + 61 3 + -&+ --5y2+ 2 2 1 1 + 241 4 + 61 3y + -412 y 2 + -xy3 + -y4 + . . 6 24 1 1 1 1 = 1 + z(l + + 6-z2 + . ) + y ( l f -I/2 + -I/2 + . ) 2 6 1 1 1 + q ( 1 + 12 + -9 + -61x 2 + -sy 1- -r/2 + . ,"' with i 5 j. 4. A multivariate Viscouatou algorithm and 1 a22 = 2 p) = 1 2 36 d y ) = 16. /) - - - - +1 i p +1 O y Y 2 gl(s) -- - 2 + ... 2 +... X and we finally get for (ez+Y - e2 - eY + l ) / z y the branched continued fraction I 1 ~ 11 + ( + and for Iq+ ,*+ . ) + (,*+ I++ xY14 I1+ ..

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. 9. The convergence is uniform in every closed and bounded subset G of the complex plane that contains no poles of f (4. ) con- l} t o a function f ( z ) meromorphic in t h a t cut complex plane. Z. Let us also illustrate the previous theorem with an example.

But the algorithm appears to be niinierically stable in a lot of cases [2]. I. 2. 2. Forward algorithms. The following theorem can be found in [14]. 11. The nth convergent of the continued fraction i=l is the first unknown XI,, li of the tridiagonal system of linear equations ... bl -1 0 a2 b2 - 1 0 a3 b3 ... 0 *. -1 0 ... ) 0 b, (see also problem (5)). Consequently, algorithms for the solution of a linear tridiagonal system, and especially for the computation of t h e first unknown, are also algorithms for the calculation of C,.