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Numerical methods for the solution of ordinary differential equations Newbery, Arthur Christopher Rolls

Abstract

Families of three- and four-point corrector formulae are derived, which differ from standard formulae in that they express yո in terms of more than one previously computed ordinate. It is shown that the standard formulae are special cases of the more general formulae derived here. By theoretical argument and by numerical experiments it is shown that the standard formulae are often inferior to others which are developed in this thesis. The three-point family, with its associated truncation error, is given in (7) and (9) of Chapter 2 on page 12. The four-point family is given in (41) on page 25. With the help of Rutishauser's method each family is examined for stability. In the four-point case a procedure is described, whereby the magnitude of the coefficient in the error term can be minimized subject to the restriction that the formula shall remain stable. Also a theorem is proved, which states that no stable four-point formula can have a truncation error of degree higher than fifth in the step-size h.

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