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Unformatted text preview: ::: 12 J Y
J j =1 (x ; )
j (9.3.3) where P (x) is an interpolating polynomial having the form of (9.2.3a) (Section 5.2 or 5],
Chapter 5). Recall, that the divided di erence f 1 2 : : : ] is de ned recursively as f l l +1 ::: l + k (
= J f( )
l l+1 f l+k ;f l
l+k ; l ::: ::: l+k;1 if k = 0
if k > 0 : (9.3.4) Recall (Lemma 5.2.1) that divided di erences and derivatives are related in that there
exists a point 2 ( 1 ), 1 < 2 < < , such that
J J f
when f (x) 2 C ( 1
J J l l +1 () = f J !( )
J ::: l (9.3.5) k ). We can specialize this result to the problem at hand. Lemma 9.3.1. Suppose y 2 C +1(a b), then the error (9.2.3c) in the interpolation
J (9.3.3) satis es jR(x)j = O(h ) x 2 (x J 1 j; j x) j = 1 2 ::: J j (9.3.6a) where h =x ;x
j j (9.3.6b) 1 j; and j j denotes a vector norm.
Proof. Using (9.2.3c) R=y 0 i 1 i 2 ::: iJ Y
J j =1 ( ; ):
j Since and , j = 1 2 : : : J , are on 0 1], the product appearing above has at most
unit magnitude. Applying (9.3.5) in this case implies
j y 0 i 1 i 2...
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