00. g, Bulirsch [1]). by More interesting is the nature of the approximation to for finite i and r. 2. 7) to order p+r, with a known value of p~1. 17) (£ is taken to be the class = ... 18) where Kpr is homogeneous of order 0 in its r+1 arguments (i. , it depends only on their ratios). 19) Proof. iej J=p f,p ,L 1 p+rep+r+Rp+r(np)' np Due to the linearity, we may interpolate the three terms separately.

1. (qJ;2~)(~)= {0'1 [(V+1) -6 -3 ~ n -- n -3~ (v) (V-1) (V-2)] - +3~ - - -~ - n n n V=O, , v = 2(1)n-1, with a shifted third difference for v = 1 and n, satisfy the Lipschitz-constants of the qJn2 are of order O(n 3 ). Thus if we are able to show that (! 33) for k=2,j=1,2, then we can apply the difference correction procedure once more. 33) to hold. Therefore, the solution of 2 1 2 1 1 1 1 Fn 'n = - qJn1 'n + ""2 qJn2 'n n n satisfies (; = d n Z + O(n - 3). In the case f(y)=gy, we have (n(vjn)=(1+gjnr and we obtain for (; g2 V {(1+-;;-g)V +V (1+-;;-g)V-1 2n2' 'n (;) 1+-ng)" +~n (1+-ng)"-1 --n 1+-ng)"-2J -2ng2 1 = ( 9 ( 2 ' v=O(1)n-1, v=n.

### Analysis of Discretization Methods for Ordinary Differential Equations by Hans J. Stetter

by Mark

4.2