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Abstract. A direct numerical method is proposed for the determination of all isolated zeros of a system of multivariate polynomial equations. By "polynomial combination", the system is reduced to a special form which may be interpreted as a multiplication table for power products modulo the system. The zeros are then formed from an ordinary eigenvalue problem for the matrix of the multiplication table. Degenerate situations may be handled by perturbing them into general form and reaching the zeros of the unperturbed system via a homotopy method.

Summary. Recently, a number of closely related techniques for error estimation and iterative improvement in discretization algorithms have been proposed. In this article, we expose the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.

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The order p which is obtainable with a stable k -step method in the numerical solution of y′ = f ( x , y ) is limited to p = k + 1 by the theorems of Dahlquist. In the present paper the customary schemes are modified by including the value of the derivative at one “nonstep point;” as usual, this value is gained from an explicit predictor. It is shown that the order of these generalized predictor-corrector methods is not subject to the above restrictions; stable k -step schemes with p = 2 k + 2 have been constructed for k ≤ 4. Furthermore it is proved that methods of order p actually converge like h p uniformly in a given interval of integration. Numerical examples give some first evidence of the power of the new methods.

This is an introductory survey of the defect correction approach which may serve as a unifying frame of reference for the subsequent papers on special subjects.

Assume that the solution r/(h) of a finite algorithm depending upon a parameter h> 0 converges for h-->0 to the solution y of a certain infinitesimal problem. We consider asymptotic expansions o/ the discretization error e(h) :=~(h)--y:where 00. Except in the case of the EulerMaclaurin sum formula representing the expansion (t.1) for the approximation of definite integrals by trapezoidal sums, the existence of an asymptotic expansion and its sequence of exponents {p,} had only been conjectured in applications of Richardson-extrapolation to functional equations. Quite recently, GRAGG treated the case of initial value problems for first order differential equations (see [7]).In w 2 of this paper, we will --under suitable conditions --prove the existence of such expansions (usually with p~=p +v--1) for a very general class of diseretization algorithms for non-linear functional equations in Banach-spaces. In the proof, the sequence {e~} will be recursively constructed. If the expansion of the local discretization error (see (2.2)) contains only even powers of h, this fact is preserved ~ in the expansion of e(h). In these cases, Richardson-extrapolation is particularly effective in improving the numerical results.In w 3, we will apply our abstract theorem to several important functional equations and their discretizations : Initial and boundary value problems for both ordinary and partial differential equations, integral equations and integrodifferential equations. For all these infinitesimal functional equations our theorem will provide hypotheses under which the application of Richardson-extrapolation is justified for a given discretization algorithm.In w 4, we will actually compute the first terms of the expansion (I