F. Patrício

A priori estimates for the zeros of interval polynomials

The aim of this paper is to study the zeros of interval polynomials. The characterization of such zeros is given as a function of the interval coefficients and estimates for such zeros are established. Interval polynomials of degree two, three and four are considered.

Cubic spline functions and initial value problems

A numerical process is presented which provides a cubic spline function approximation for the solution of initial value problems in ordinary differential equations. With interpolate cubic spline functions we can achieveO(h4) convergence.

A numerical method for solving initial-value-problems with spline functions

A numerical method, using spline functions of degree five, for obtaining approximate solutions to initial value problems is presented. It is shown that the method is stable and the convergence is analysed. Some numerical experiments are included.

On the interval Legendre polynomials

In this paper, the extention to interval theory of the classical Legendre polynomials is considered. The so-called interval Legendre polynomials are introduced and some properties are studied. Based on these polynomials an interval minimum square approximation is introduced when continuous and discrete data are taken.

Instability in linear multistep methods

Classes of multistep methods with k steps, order k+1 and depending on a certain number of free parameters, one of them representing the size of the real interval of stability are constructed. A criterion to select automatically multistep methods of such classes, which are fitted with the eigenvalues of the jacobian matrix of a differential system is also presented

Numerical oscillations on nonuniform grids

In this paper we study a class of numerical methods used to solve two-point boundary-value problems on nonuniform grids. Particular attention is devoted to numerical oscillations which are quantified for different methods. Numerical experiments are also included.

Monotonicity on nonuniform grids

In this paper we study a class of numerical methods used to solve two-point boundary value problems on nonuniform grids. Particular attention is devoted to positive solutions, i.e. conditions under which the solutions of the problem are positive. Applications to steady states of air pollution problems are also referred to.

Multiderivative variable stepsize variable formula methods

Abstract In this paper we present a study of consistency, stability and convergence properties of linear multiderivative multistep variable stepsize variable formula methods.

A class of hybrid formulae for the numerical integration of stiff systems

A class of hybrid formulae suitable for the numerical integration of stiff systems of first order ordinary differential equations is presented. These formulae are defined in a predictor-corrector mode and contain an off-step pointxn+v, which is distinct from the points defined by the step used. The parameterv is examined in an attempt to obtain better stability properties and higher order formulae of one step only. We obtain order five and eitherA-stability or stiff-stability according to the values ofv used. Some numerical results are presented.