A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [t] we prove that a linear multistep method of design order q ~p ~ t which satisfies the weak stability root condition, applied to the differential equation y" (t) = / (t, y (t)) where f is Lipschitz continuous in its second argument, will exhibit actual convergence of order o(h p-l) if y has a (p--t)th derivative y(p-1) that is a Riemann integral and order o(h t') if y(p-1} is the integral of a function of bounded variation. This result applies for a function y taking on values in any real vector space, finite or infinite dimensional.
Abstract.It is well known that the calculation of an accurate approximate derivative f'(x) of a nontabular function fix) on a finite-precision computer by theis a delicate task. If ft is too large, truncation errors cause poor answers, while if ft is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of d(ft) to /', a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.
Abstract.This paper develops fourth order discretizations to the two-point boundary value problem y(2kt)=f(t,y(t),y(1\t)), o^0) -"o^(1)(°) = 60' al ?W + "l^1^1) = 5 1-These discretizations have the desirable properties that they are tridiagonal and of "positive type".
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