The lagrange basis k y t tj li t j 1 j 6i ti tj t

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Unformatted text preview: t0)(t ; t1 ) : : : (t ; ti;1 )(t ; ti+1) : : : (t ; tk ) (ti ; t0 )(ti ; t1 ) : : : (ti ; ti;1)(ti ; ti+1 ) : : : (ti ; tk ) i = 0 1 ::: k (5.2.3a) leads to the more direct solution Pk (t) = k X i=0 f (ti)Li(t): (5.2.3b) A disadvantage of the Lagrange basis is that the computation of the basis elements Li (t), i = 0 1 : : : k, must be repeated when changing the polynomial degree by adding or deleting an interpolation point. As discussed in Chapter 4, the Aitken-Neville recursion remedies this by obtaining Pk (t) as a combination of polynomials of degree k ; 1. Thus, we de ne the family of interpolants +1 Pji(t) = (t ; ti+j )Pji;1(t) + (ti ; t)Pji;1 (t)ti ; ti + j i = 0 1 ::: k ;j j = 0 1 : : : k (5.2.4a) where Pji(t) is a polynomial of degree j satisfying the interpolation conditions Pji(tm ) = f (tm ) m = i i + 1 : : : i + j: (5.2.4b) The desired interpolating polynomial is Pk0(t). The Aitken-Neville algorithm simpli es degree changes, but its complexity is slightly larger than necessary ( 6], Chapter 3). The Newton form of the interplating polynomial has a lower complexity and, like the Aitken-Neville procedure, is hierarchical. Again we'll de ne a sequence of polynomials of increasing degree with P0(t) = f (t0): (5.2.5) The degree one polynomial is obtained by adding a correction to P0(t) in the form P1 (t) = P0(t) + a1 (t ; t0 ): First, we easily verify that P1(t0 ) = P0(t0 ) = f (t0): 6 The coe cient a1 is determined so that P1(t1 ) = f (t1) thus, using (5.2.5), and f (t1 ) = f (t0) + a1 (t1 ; t0 ) ;f a1 = f (t1 ) ; t (t0 ) : t 1 0 The notation is simpli ed by de ning the rst divided di erence at the points tj and tl as ;f f tj tl ] = f (tjt) ; t (tl ) : (5.2.6a) j l Then P1(t) = P0 (t) + f t0 t1 ](t ; t0): (5.2.6b) In a similar manner, the second-degree polynomial is written in the form P2(t) = P1(t) + a2(t ; t0 )(t ; t1): By construction, P2 (t) satis es the interpolation requirements at t0 and t1 . Satisfaction of the interplation condition P2(t2 ) = f (t2) determines a2 as a2 = f t0 t1 t2...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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