35a which we repeat here for convenience pk1t k1 x 1i

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Unformatted text preview: l corrector formula. We'll illustrate the procedure for the third-order corrector (5.7.5b). Setting = 1 in (5.7.5b) and subtracting (5.7.5a) yields (1) (0) (0) 0 0 0 yn = yn + 38h f (tn yn ) ; 3yn;1 + 3yn;2 ; yn;3]: 50 From the above predicted interpolant, we have 0 0 0 0 yn(0) ; 3yn;1 + 3yn;2 ; yn;3 = 0: 0 (0) Let us de ne yn(1) = f (tn yn ) and add this to both sides of this relation to obtain 0 0 (0) 0 0 0 yn(1) = yn(0) + f (tn yn ) ; 3yn;1 + 3yn;2 ; yn;3: The previous relations may be used for all iterates and not just for the rst correction thus, we obtain (5.7.8a) with 2 3 3=8 6 7 c=6 1 7 405 (5.7.8b) 0 ( ( 0 0 0 g(yn ;1)) = h f (tn yn ;1)) ; 3yn;1 + 3yn;2 ; yn;3]: (5.7.8c) The form of the predictor-corrector pair given by (5.7.6c) and (5.7.8) avoids saving yn;1, yn;2, and yn;3. Functional iteration is assumed. The formulas would have to be modi ed for Newton iteration. Our development does not reveal the generality of (5.7.6c) and (5.7.8). Gear 10], Chapters 7 and 9, provides additional details. The nal converged iterates are passed to the next step thus, if convergence occurs after corrector iterations, ( yn = yn ) . The representation (5.7.6c) and (5.7.8) may not be best when automatic step and order changes are involved. Formulas involving backward di erences may, for example, be best for these cases. Having the general representation (5.7.6c) and (5.7.8), we can switch to another one by a linear transformation. Thus, let an = Tyn (5.7.9a) and use (5.7.6b) and (5.7.8) to get a new predictor-corrector pair a(0) = Aan;1 n (5.7.9b) a(n ) = a(n ;1) + lP (a(n )) (5.7.9c) 51 where A = TBT;1 l = Tc P (an) = g(T;1an ): (5.7.9d) Example 5.7.3. Let us develop the backward-di erence form of the Adams predictorcorrector pair (5.7.5a,b). Let 2 3 yn 0 6 hyn an = 6 rhy0 4 n 0 r2hyn 7 7: 5 Using the de nition of the backward-di erence operators (5.2.15) 2 yn 6 hyn 6 00 4 rhyn 0 r2hyn 32 32 10 00 yn 7 6 0 1 0 0 7 6 hyn 7=6 76 0 0 5 4 0 1 ;1 0 5 4 hyn;1 0 0 1 ;2 1 hyn;2 3 7 7 = Tyn: 5 Nordsieck 15] suggested representing the sol...
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