# 6788e 01 13212e 01 24129e 02 22263e 03 10452e 04 1

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 01 -3.8774e-02 4.8645e-03 7 1.8454e-01 -3.8546e-02 8 1.8424e-01 Table 4.1.1: Solution of Example 4.1.2 by Richardson's extrapolation. The upper table i presents solution, the middle table presents errors in q , and the lower table presents q+1 . errors divided by i q i q i q i R h i i = 1=ri, for 1 < r 2. The worst case rounding errors in R0 will be proportional i i r , since this is approximately the number of steps. The error in R could be as large h 1 r i+1 + r i+1 + ri r ;1 = 6 r i+1 ( r + 1=r ): r ;1 to as The error in i q could be as large as 2 q i+q ( + 1 )( + 1 ) ( + 1 ) 2;1 q;1 ;1 Values of near 2 will make the expression i+q large whereas values of near unity will make the denominators in the above expression small. Bulirsch and Stoer 2] used sequences with = 3 2 or = 4 3 so that i+q grows more slowly than with = 2. If the extrapolation is started with a method of order ( 1) and the method has an expansion in of the form R r r r =r r =r r r r ::: =r r : r r = r = q r r p p h q X ( ) = ( )+ zth yt i=0 c p+i + O(hp+q+1): i (t)h i In this case, the approximation q has an error of order ( p+q+1). Some methods have error expansions that only contain even powers of , e.g., R Oh h ( ) = ( )+ zth yt q X i i (t)h c i=1 (q+1) +( Oh ) where is typically 2 or 4. The extrapolation algorithm can be modi ed to take advantage of this by utilizing polynomials in to obtain h i Rq i = Rq+11 + i+1 i q 1 ; Rq 1 : i ( hh+q ) ; 1 i R ; ; (4.1.7) ; Now, the order of the approximation is increased by in each...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online