Formulas involving backward di erences may for

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Unformatted text preview: ution in terms of scaled derivatives at t = tn , i.e., as a n = yn hy0 n h2 y00 : : : hk;1 y(k;1)]T : 2! n (k ; 1)! n (5.7.10) Step size variation is simpli ed with this representation. The transformation between representations may be obtained from the Newton backward-di erence polynomial (5.3.3a) Pl;1(t) = l ;1 X 0 (;1)i ; i+ 1 riyn: i=0 (5.7.11) (The relationship between the order k and the degree of the Newton polynomial l ; 1 will become clear shortly.) Since Pl;1(t) is an approximation of y0(t), di erentiating (5.7.11) and setting = 1 gives a relation between the Nordsieck and backward-di erence representations. A similar linear relationship between the backward-di erence and standard representation (5.7.6b) can be used to relate the Nordsieck and the standard variables. Example 5.7.4. We'll construct a relationship between the Nordsieck and standard representations for the three-step (fourth-order) Adams method (5.7.5b). The Nordsieck vector for this method is 2 3 0 00 000 an = yn hyn h yn h yn ]T : 2 6 52 A second-degree polynomial su ces to get the necessary derivatives thus, setting l = 3 in (5.7.11), we have 0 0 01 P2 = yn + ( ; 1)ryn + 2 ( ; 1) r2yn: Di erentiating while using (5.7.7b) 2 01 0 h dPdt(t) = dPd2( ) = ryn + 2 (2 ; 1)r2yn: 00 Setting = 1 (t = tn ) gives an approximation of yn as 0 00 0 hyn = ryn + 1 r2 yn: 2 Di erentiating again 2 2 P P 0 h2 d dt22(t) = d d 2( ) = r2yn: 2 Setting = 1 000 0 h2 yn = r2yn: Summarizing the results in matrix form 2 32 32 3 yn 10 0 0 yn 6 hyn 7 6 0 1 0 0 7 6 hyn 7 6 2 000 7 = 6 76 0 0 7: 4 h yn=2 5 4 0 0 1=2 1=4 5 4 rhyn 5 000 0 h3 yn =6 0 0 0 1=6 r2hyn Multiplying by the transformation of Example 5.7.3 gives the desired relationship between the Nordsieck and traditional forms as 32 32 3 2 yn 10 00 yn 6 hyn 7 6 0 1 0 0 7 6 hyn 7 : 6 2 000 7 = 6 76 0 0 7 4 h yn=2 5 4 0 3=4 ;1 1=4 5 4 hyn;1 5 000 0 h3yn =6 0 1=6 ;1=3 1=6 hyn;2 Example 5.7.5. Storage was at a premium when the above transformations were developed. Now, with a third-degree interpolating polynomial and l...
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