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where f tj tl tm ] is the second divided di erence at the points tj , tl , tm
f tj tl tm ] = f tj ttl ] ; f tl tm] :
j ; tm
Thus, the second divided di erence is a divided di erence of rst divided di erences. The
second-degree interpolating polynomial is P2(t) = P1 (t) + f t0 t1 t2](t ; t0 )(t ; t1): (5.2.7b) Continuing, we construct the k th-degree polynomial as Pk (t) = Pk;1(t) + f t0 t1 : : : tk ](t ; t0 )(t ; t1) : : : (t ; tk ; 1)
7 (5.2.8a) where
f t0 t1 : : : tk ] = f t0 t1 : : : tkt 1]; t f t1 t2 : : : tk ]
is the k th divided di erence at the points t0 , t1 , : : : , tk .
The intermediate polynomials can be eliminated from (5.2.8a) to write the Newton
divided-di erence polynomial in the more explicit form
Pk (t) = f t0 t1 : : : ti] (t ; tj )
i=0 j =0 where the zeroth divided di erence is f tj ] = f (tj ).
The Newton divided-di erence representation is the traditional interpolating polynomial to be used when developing multistep formulas. It had some advantages for hand
computation and the divided di erences furnish approximations of solution derivatives
that may, e.g., be used for error estimation. Before specializing the approximation (5.2.9)
to our ODE application, let us note that the interpolation problem has a unique solution
as indicated by the following theorem. The di erent bases just simplify the interpolation
problem for speci c applications. Theorem 5.2.1. There is one and only one polynomial of degree k that interpolates a
function f (t) at k + 1 distinct points. Proof. Suppose that there are two polynomials Pk (t) and Qk (t) of degree k that interpolate f (t) at the points t0 t1 : : : tk . Subtract the two polynomials and de ne R(t) = Pk (t) ; Qk (t):
Now R(t) is also a polynomial of degree k that satis es R(tj ) = Pk (tj ) ; Qk (tj ) = 0 j = 0 1 : : : k: This, however, is impossible since a polynomial of degree k can only have k roots. Thus,
the interpolating polynomial is unique.
We may suspect that divided di erences are related to derivat...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14
- The Land