The same result can be obtained by approximating by a

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Unformatted text preview: mial ) that interpolates ( ) at = 0 and 0 2. Thus, let 1( zth= zth zth Oh yt zth= zth R th zth h h h= ( ) = 0( ) + 1( ) R1 t h t t h: The interpolation conditions require ( z t h0 and ( ) = 0( ) + 1( ) t t h0 2) = 0 ( ) + 1 (2) 2 z t h0 = t t h0 : z(t,h) z(t,h 0 ) R 1 (t,h) z(t,h 0 /2) R 1 (t,0) h 0/2 h0 h Figure 4.1.1: Interpretation of Richardson's extrapolation for Example 4.1.1. Thus, 0 2) ; ( =2 ( z t h0 = z t h0 ) 1 =2 ( h0 2) ; ( z t h0 = z t h0 )] : As shown in Figure 4.1.1, the higher-order solution is obtained as ( 0) = R1 t 0 2) ; ( =2 ( z t h0 = ) z t h0 : The general extrapolation procedure consists of generating a sequence of solutions ( j ), = 0 1 , with 0 1 q and interpolating these solutions by an 0 degree polynomial q ( ). The desired higher-order approximation is the value of 0 the interpolating polynomial at = 0, i.e., q ( 0). The meaning of the superscript 0 will become clear shortly. The Aitken-Neville algorithm provides a simple way of generating the necessary approximations in the form a recurrence relation where higher-order solutions are obtained i from lower-order ones. Suppressing the dependence, let q ( ) be the unique polynomial of degree that satis es the interpolation conditions zth j ::: q h q th R >h > ::: > h th h R t t R h q i q (h) = z (t hj ) R Consider R i q (h) j = ii +1 ::: i + q: (4.1.3) in the form i i q (h) = iq (h)Rq R 1 ; ( ) + (1 ; h 3 i+1 iq (h))Rq 1 (h) ; (4.1.4a) where iq (h) is a linear polynomial in . By assumption,...
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