Wave Propagation in Linear Media | re-examined

5.2 Convergence acceleration
(1)
,1 =0
(2)
,1 =0
(3)
,1 =0
(4)
,1 =0
(5)
,1 =0
.
(0)
0
(1)
0
(2)
0
(3)
0
(4)
0
= S0
= S1
= S2
= S3
= S4
.
(0)
1
(1)
1
(2)
1
(3)
1
.
(0)
2 = S e(2) 0
(1)
2 = S e(2) 1
(2)
2 = S e(2) 2
.
(0)
3
(1)
3
.
(0)
4 = S e(4) 0
Figure 5.1: The -array. Only the values (n)
2k in the even columns are valid estimates for the sequence
limit S.
then the -algorithm gives the exact result (0)
2p
of the original sequence are required.
.
. ..
= S [113], which means that 2p + 1 elements
In any case, a sequence transformation may be seen as a functional relation between a member
of the transformed sequence and a certain subset of the original sequence. Since our goal is
to obtain a better estimate for the limit of the sequence, we normally desire only one result
from the function and not a transformed sequence. Thus we can write
Tn = e S (k)
n = F (Sn;Sn+1 :::Sn+k) ; (5.13)
with k +1 being the number of sequence members involved in the transformation and n
the starting index. If the original sequence comprised N members, then we would typically
compute S e(k) N,k or S e(N) 0 as the (hopefully) best approximation to the exact limit S.
131