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Unformatted text preview: 01 3.8774e02 4.8645e03
7 1.8454e01 3.8546e02
8 1.8424e01
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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 Spring '14
 JosephE.Flaherty

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