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By W. W. Rouse Ball

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Shampine 1984a, 1985b). 5). , all the eigenvalues of / have negative real parts), then the local errors may damp out with increasing x\ otherwise, the errors will be magnified with increasing x (cf. Lambert 1980). Bulirsch and Stoer (1966) constructed asymptotic upper and lower bounds on the global errors emanating from extrapolation methods to IVPs. Shampine (1985d) generalized this idea for any one-step method endowed with an asymptotically correct local error estimator. 3: Geometrical Representation of Local and Global Error 30 NUMERICAL INTEGRATION ALGORITHMS estimation of the global error (cf.

3) to be exact whenever the theoretical solution y (x) is a polynomial of degree < 2. 2) is implicit, the solution yn+i has to be generated iteratively. , yn+i = v ^ J . 8) with Newton Iteration (IMPTR-1). 3. 6) is entirely analogous to the stability constraint. 3. 236, 0) TMPTR-1 ^1 1+1/2Z r-oo m Interval of Absolute Stability μ(ζ) 1-1/2Z 44 THEORY OF ONE-STEP METHODS I Xh I < 2 for simple iteration to converge in general. 3 have not taken this into consideration. 7) r 0 < je < 1 adopting uniform meshsizes {h =2 , r = 2(2)12}.

2a) where ·«, is the maximum norm defined as 3O- = max \yr\ . 3), which we adopt simply because of its ease of computation. 4), as it is incapable of differentiating between the test problems y = Xy or y = -λν. 2b) where Vix) (which can be negative) is the one-sided Lipschitz "constant" of / with respect to y. 2a), but the converse is not true in general. , Ince (1956), Henrici (1962), Pontryagin (1962), and Coddington and Levinson (1955). 1. 2). 4) has a unique solution in R. Since the Lipschitz constant L is not readily available, it suffices to check if fy (the partial derivative of fix, y) with respect to y) is defined and continuous in R.

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