13 in the form i i q h iq hrq r 1 1 h 3 i1 iq

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Unformatted text preview: h i q j) j = i+1 q 1 (hj ) = z (t hj ) j = +1 +2 R ; 1 ( j) = ( h zth ii +1 ::: i + ;1 q and R ; i i ::: i + q thus, the required interpolation conditions i q (h) = z (t hj ) R are satis ed for all choices of j iq (h). = +1 +2 i i ::: + ;1 i q If we additionally select iq (hi ) = 1 iq (hi+q ) = 0 then i i q (hi ) = Rq R Since iq (h) 1 ; ( i) = ( h i i+1 q (hi+q ) = Rq 1 (hi+q ) = z (t hi+q ): i) zth R ; is a linear function of then h ; i; iq (h) = i+q : i+q h h (4.1.4b) h h Combining (4.1.4a) and (4.1.4b) i q (h) = R (; h i i+q )Rq h ; 1 ( )+( i; ) h i; h h i+1 q 1 (h) hR ; i+q h : (4.1.5) i i We'll denote the extrapolated solutions q (0) simply as q . These functions may be simply generated in a tableau as indicated in Figure 4.1.2. Using (4.1.5), the entries of the tableau are R ( ( ( ( )= 1) = 2) = 3) = ... 0 z t h0 R0 zth R0 zth 0 R0 zth R R0 R1 1 1 R1 2 2 R1 ... 3 0 R2 1 R2 ... 0 R3 ... Figure 4.1.2: Tableau for Richardson's extrapolation. 4 i q R = i+1 i i (0) = hi Rq 1 ; hi+q Rq q hi ; hi+q ; R 1 ; = i+1 i q 1 ; Rq 1 : i ( hh+q ) ; 1 i i+1 q 1+ R R ; ; (4.1.6) ; i If this extrapolation technique is used with a rst-order method, then the values of q increase in accuracy as either or increase, provided that ( ) is smooth. In this case, i the global error ( ) ; q is ( q+1), provided that i ! 0 as ! 1 4]. Finally, we note that it is n...
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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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