The order conditions 3315 are related to the previous

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Unformatted text preview: ask if there is an optimal way of selecting the collocation points. Appropriate strategies would select them so that accuracy and/or stability are maximized. Let's handle accuracy rst. The following theorems discuss relevant accuracy issues. Theorem 3.3.5. (Alekseev and Grobner) Let x, y, and z satisfy x (t z( )) = f (t x(t z( ))) x( 0 y (t) = f (t y(t)) 0 z (t) = f (t z(t)) + g(t z(t)) 0 with fy (t y ) 2 C 0, t > 0. Then, Z t z( )) = z( ) y(0) = y0 z(0) = y0 (3.3.17a) (3.3.17b) (3.3.17c) @x(t z( )) g( z( ))d : (3.3.17d) @z 0 Remark 2. Formula (3.3.17d) is often called the nonlinear variation of parameters. Remark 3. The parameter identi es the time that the initial conditions are applied in (3.3.17a). A prime, as usual, denotes t di erentiation. Remark 4. Observe that y (t) = x(t 0 y0 ). z(t) ; y(t) = 31 Proof. cf. Hairer et al. 16], Section I.14, and Problem 1. Theorem 3.3.5 makes it easy for us to associate the collocation error with a quadrature error as indicated below. Theorem 3.3.6. Consider the quadrature rule Z tn tn;1 F (t)dt = h Z 0 1 F (tn 1 + h)d = h ; s X i=1 bi F (tn...
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