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# These methods are of order s which is less than the

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Unformatted text preview: -c) and 2 3 f (tn 1 + c1h) 6 f (tn 1 + c2 h) 7 7: f =6 6 7 ... 4 5 f (tn 1 + csh) ; (3.3.23) ; ; Let ^ Y = T 1Y ; ^ = T 1l l ^ A = T 1 AT ; ; ^ = T 1f : f ; (3.3.24) Butcher 9] chose the collocation points ci = i, i = 1 2 : : : s, where i is the i th root of the s th-degree Laguerre polynomial Ls(t) and is chosen so that the numerical method has favorable stability properties. butcher also selected T to have elements Tij = Li 1 ( j ): ; Then 2 6 6 ^ =6 A6 6 4 ... 36 3 7 7 7: 7 7 5 (3.3.25) ^ Thus, A is lower bidiagonal with the single eigenvalue . The linearized system (3.3.9) is easily solved in the transformed variables. (A similar transformation also works with Radau methods 17].) Butcher 9] and Burrage 5] show that it is possible to nd Astable SIRK methods for s 8. These methods are also L-stable with the exception of the seven-stage method. Problems 1. Verify that (3.3.17d) is correct when f (t y) = ay with a a constant. 2. Consider the method yn = yn 1 + h (1 ; )f (tn 1 yn 1) + f (tn yn)] ; ; ; with 2 0 1]. The method corresponds to the Euler method with = 0, the trapezoidal rule with = 1=2, and the backward Euler method and when = 1. 2.1. Writ...
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