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Unformatted text preview: ot necessary to halve the step size after each pass. Other step size sequences
can be used to reduce computation. For example
R i yt R q yt Oh f h 2 h0 3 h0 = h0 = 4 h0 = i 6 8 h0 = h0 = ::: g : This should be done carefully since some sequences lead to a loss of stability ( 5], Section
II.9).
Example 4.1.2. Consider the solution of
y 0 =; 0 y 1 <t y (0) = 1 using the forward Euler method with step sizes i = 2 i, = 0 1
8. The results at
= 1 are reported in Table 4.1.1. The entries in the rst column ( = 0) are computed
by Euler's method with step size i. Subsequent columns are obtained using (4.1.6). Let
us verify the entry in the rst row and second column of the upper table by using (4.1.6)
with = 0 and = 1 thus,
1
0
0
1
= 0 + h00 ; 0 = 0 25 + 0 25 ; 0 0 = 0 5
1
20;10
( h1 ) ; 1
; h i ::: t q h i q R R R R : : : : :: : i
Columns converge at increasing powers of thus, errors ( (1) ; q ) decrease as ( )
in the rst ( = 0) column, as ( 2) in the second ( = 1) column, etc. This is con rmed
in the bottom table, which consists of the errors divided by q+1.
i
Increasing or increases the number of calculations. If the basic integration scheme
is Euler's method, computations increase at the rate of 2q+i when the basic step 0 = 1
is repeatedly halved. If worstcase roundo occurs at each step, the answe...
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 Spring '14
 JosephE.Flaherty

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