Wave Propagation in Linear Media | re-examined

6.6 Controlling the accuracy of the extrapolation result
17.5
15
12.5
10
7.5
2.5
Figure 6.12: Relative extrapolation error , log e S (k)
0 ,S
S
Precision[x] ( ).
50 100 150 200 250
for varying k ( ) and the respective results of
number of sequence members included in the extrapolation. One might argue that altering the
setting for WorkingPrecision should have a similar e ect. The following example explores
this possibility.
Example 6.6.3 The integral (6.12)
I =
Z 1
a
x
x2 p
sin x2 , a2dx =
, a2 2
shall be evaluated for a = 50 with di erent values for WorkingPrecision and with WynnDegree->
Infinity. As g. 6.13 shows, a high precision of the internal calculations naturally allows for a higher
precision of the result, but this is only true if enough sequence members are taken into account. Before
the respective saturation is reached, the setting of the internal precision has almost no e ect on the
precision of the result. But there is an even more important aspect to this parameter: If the meta
algorithm was to change the WorkingPrecision, the entire sequence of partial sums would have tobe
re-computed for each iteration, which isvery time-consuming. If, on the other hand, the number of
sequence elements treated with SequenceLimit is altered, then only a few new members need to be
added in each cycle. For these reasons, the value of the internal precision remains xed throughout
the computation. However, it must be set such that the required precision can be reached at all.
159