The complex-step method is a clever way of obtaining a numerical approximation to the first derivative of a function, avoiding the round-off error that plagues standard finite difference approximations. An extension of the method allows second derivatives to be calculated with reduced round-off error. This article provides an overview of the method, discusses its practical implementation, with ...

Abstract. The determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy, namely the Wronskian of two solutions of the Schrödinger equation which are regular respectively at the origin and at infinity. We show in this paper how to evaluate that Wronskian exactly, except for numerical rounding errors. The procedure is il...

We study the Euler–Frobenius numbers, a generalization of the Eulerian numbers, and the probability distribution obtained by normalizing them. This distribution can be obtained by rounding a sum of independent uniform random variables; this is more or less implicit in various results and we try to explain this and various connections to other areas of mathematics, such as spline theory. The mea...

Various performance indices are used for the design of serial manipulators. One method of optimization relies on the condition number of the Jacobian matrix. The minimization of the condition number leads, under certain conditions, to isotropic configurations, for which the roundoff-error amplification is lowest. In this paper, the isotropy conditions, introduced elsewhere, are the motivation b...

We present a case study of a formal veri cation of a numerical program that computes the discretization of a simple partial di erential equation. Bounding the rounding error was tricky as the usual idea, that is to bound the absolute value of the error at each step, fails. Our idea is to nd out a precise analytical expression that cancels with itself at the next step, and to formally prove the ...

The joint probability density function, evaluated at the observed data, is commonly used as the likelihood function to compute maximum likelihood estimates. For some models, however, there exist paths in the parameter space along which this densityapproximation likelihood goes to infinity and maximum likelihood estimation breaks down. In applications, all observed data are discrete due to the r...

The omitted variables problem is one of regression analysis’ most serious problems. The standard approach to the omitted variables problem is to find instruments, or proxies, for the omitted variables, but this approach makes strong assumptions that are rarely met in practice. This paper introduces best projection reiterative truncated projected least squares BP-RTPLS , the third generation of ...

If rounded data are used in estimating moments and regression coffiecients, the estimates are typically more or less biased. The purpose of the paper is to study the bias inducing effect of rounding, which is also seen when population moments intstead of their estimates are considered. Under appropriate conditions this effect can be approximately specified by versions of Sheppard’s correction f...

In the past few decades, discontinuous piecewise affine maps have found considerable interest in the theory of dynamical systems. For an overview, we refer the reader to [1, 7, 12, 13, 17, 18], for particular instances to [29, 16, 25] (polygonal dual billiards), [15] (polygonal exchange transformations), [10, 31, 11, 8] (digital filters) and [19, 21, 22] (propagation of round-off errors in line...