# Some methods have error expansions that only contain

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Unformatted text preview: successive column of the extrapolation tableau. If the trapezoidal rule is solved exactly at each step, then it has an expansion in powers of 2 ( = 2). Extrapolation with rational functions is also possible ( 5], Section II.9). The basic idea is to approximate ( ) by a rational function ( ) and then evaluate (0). Bulirsch i and Stoer 1] derived a scheme where q ( ) is de ned as the rational approximation that interpolates ( ) at = i i+1 i+q for = i i+1 i+q . The values i i of q = q (0) can be obtained from the following recursion h zth Rh R zth R h h h ::: R h h h h >h > ::: > h R i R;1 R i q = i+1 q 1+ R ; =0 i R0 =( i) (4.1.8a) zth i+1 i q 1 ; Rq 1 i i i i i ( hh+q )2 1 ; Rq+11 ; Rm 1 Rq+11 ; Rq+22 ] ; 1 i R ; 7 ; ; ; ; ; (4.1.8b) when the method has an error expansion in powers of 2. The computation of cording to (4.1.8b) is equivalent to interpolating ( ) by a function of the form h i j (h) = R ( i q R ac- zth a0 +a1 h2 +:::+aj hj b0 +b1 h2 +:::+bj hj a0 +a1 h2 +:::+aj;1 hj;1 b0 +b1 h2 +:::+bj;1 hj;1 if is even if is odd j j : (4.1.8c) There are several enhancements to the basic extrapolation approach. Some of these follow. 1. A variable-order extrapolation algorithm could choose the order (i.e., the number of columns in the tableau) adaptively. Error estimates c...
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