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Abstract: For the numerical solution of stochastic differential equations an economical Runge–Kutta scheme of second order in the weak sense is proposed. Numerical stability is studied and some examples are presented to support the theoretical results.

Abstract: For the numerical solution of stochastic differential equations an economical Runge–Kutta scheme of second order in the weak sense is proposed. Numerical stability is studied and some examples are presented to support the theoretical results.

Recently, by Costabile, Gualtieri and Serra (1999), an iterative method was presented for the computation of zeros of C1 functions. This method combines the assured convergence of the bisection-like algorithms with a superlinear convergence speed which characterizes Newton-like methods. The order of the method and the cost per iteration is exactly equivalent to the Newton method. In this paper we present a new iterative method for the computation of the zeros of C1 functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order 1+ ≐2.41 for C3 functions. Compared with the methods of order 1+ presented by Traub (1964), our methods ensure global convergence. Then we consider a generalization of this procedure which gives a class of methods of order (n+ )/2, where n is the degree of the approximating polynomial, with one-point iteration functions with memory. Finally a number of numerical tests are performed. The numerical results seem to show that, at least on a set of problems, the new methods work better than the methods proposed, and, therefore, than both the Newton and Alefeld and Potra (1992) methods.

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval [a,b]?R that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of Cm-functions on a 2-dimensional simplex; a method to generalize the expansion on an N-dimensional simplex is also discussed. This new expansion is applied to find new cubature formulas for 2-dimensional simplex.

Recently, by Costabile, Gualtieri and Serra (1999), an iterative method was presented for the computation of zeros of C1 functions. This method combines the assured convergence of the bisection-like algorithms with a superlinear convergence speed which characterizes Newton-like methods. The order of the method and the cost per iteration is exactly equivalent to the Newton method. In this paper we present a new iterative method for the computation of the zeros of C1 functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order 1+$\$\backslash sqrt\{2\}\$$ ≐2.41 for C3 functions. Compared with the methods of order 1+$\$\backslash sqrt\{2\}\$$ presented by Traub (1964), our methods ensure global convergence. Then we consider a generalization of this procedure which gives a class of methods of order (n+$\$\backslash sqrt\{n^\{2\}+4\}\$$)/2, where n is the degree of the approximating polynomial, with one-point iteration functions with memory. Finally a number of numerical tests are performed. The numerical results seem to show that, at least on a set of problems, the new methods work better than the methods proposed, and, therefore, than both the Newton and Alefeld and Potra (1992) methods.

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval [a,b]⊂R that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of Cm-functions on a 2-dimensional simplex; a method to generalize the expansion on an N-dimensional simplex is also discussed. This new expansion is applied to find new cubature formulas for 2-dimensional simplex.

For the numerical solution of the initial value problem a parallel, global integration method is derived and studied. It is a collocation method. If f(x,y)≡f(x) the method coincides with the Filippi's modified Clenshaw–Curtis quadrature [11]. Two numerical algorithms are considered and implemented, one of which is the application of the new method to Picard iterations, so it is a waveform relaxation technique [3]. Numerical experiments are favourably compared with the ones given by the known GAM [2], GBS [14] and Sarafyan [18] methods.

In this paper we generalize an expansion in Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Starting from this expansion we obtain useful kernels, which are substantially different from Sard's for a wide class of linear functionals that includes the truncation error for cubature formulas.

Abstract: In this paper a detailed study of the convergence and stability of a numerical method for the differential problem

Sommario: Si dimostra che per il grado di precisione,N, della formula di quadratura di tipo Tschebyscheff:

Given a real functionf eC2k[0,1],k = 1 and the corresponding Bernstein polynomials {Bn(f)}n we derive an asymptotic expansion formula forBn(f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanBn(f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.

This paper develops a general theory for a class of Runge-Kutta methods which are based, in addition to the stages of the current step, also on the stages of the previous step. Such methods have been introduced previously for the case of one and two stages. We show that for any numbers of stages methods of orderp withs+1 ≤p ≤ 2s can be constructed. The paper terminates with a study of step size change and stability.

Abstract: A six order formula of an explicit method for the numerical initial value problem, proposed in [1], [3], is determined. This formula which for f(x,y)≡f(x) is the Gauss quadrature formula with three points, requires only three stages by comparison with seven stages required by an explicit six order Runge-Kutta method. Convergence and stability are considered and numerical example are also given.

Sommario: Per l’integrazione numerica dell’equazione differenziale:

Riassunto: Per la risoluzione numerica dell'equazione differenziale ordinaria con valore iniziale

Sommario: Per l'integrazione numerica del problema con valore iniziale

Sommario: Si studia la stabilità dei metodi pseudo Runge-Kutta. Si confrontano gli intervalli di stabilità con gli analoghi metodi di Runge-Kutta e a differenze finite.

Riassunto: Per l'integrazione numerica dell'equazione differenziale ordinaria

A family of two-step fourth order methods, which requires two function evaluations per step, is derived fory″=f(x,y). We then show the existence of a sub-family of these methods which when applied toy″=−k2y,k real, areP-stable.