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

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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...
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