We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order. In numerical experiments we compare our integrators with some other numerical schemes, including the stan...

Deterministic branch and bound methods for the solution of general nonlinear programs have become increasingly popular during the last decade or two, with increasing computer speed, algorithmic improvements, and multiprocessors. There are presently several commercial packages. Although such packages are based on exhaustive search, not all of them rigorously take account of roundoff error, singu...

The asymptotic behaviour of a class of least squares problems when subjected to structured perturbations is considered. It is permitted that the number of rows (observations) in the design matrix can be unbounded while the number of degrees of freedom (variables) is fixed. It is shown that for certain classes of random data the solution sensitivity depends asymptotically on the condition number...

Spectral functions of symmetric matrices – those depending on matrices only through their eigenvalues – appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. We then round off t...

A new criterion for selective reorthogonalization in the Gram-Schmidt procedure is given. We establish its comportment in presence of rounding errors when the criterion is used with modified Gram-Schmidt algorithm and show counter-example matrices which prove that standard criteria are not always valid. Experimentally, our criterion is fine also for the classical Gram-Schmidt algorithm with reo...

methodology, and indeed a part of their radiance comes from Davidson's reinterpretation of Foucault's works of his "middle period", best contained in Les mots et les choses and L'Arche'ologie du savoir. Davidson shows the similarity and difference between Canguilhem's and Foucault's projects better than any other Anglophone commentator, and he relates all these theoretical insights back to the ...

Approximate, non-singular kernels are often used in moment-method formulations coping with thin-wire structures. Their use has important consequences, one of which is the appearance of oscillations in the computed currents when the number of sub-domain basis functions is sufficiently large. These oscillations are not due to round-off errors. In this paper, a smoothing procedure is used in conju...

In this paper, Laplace homotopy perturbation method, which is combined form of the Laplace transform and the homotopy perturbation method, is employed to obtain a quick and accurate solution to the fractional Black Scholes equation with boundary condition for a European option pricing problem. The Black-Scholes formula is used as a model for valuing European or American call and put options on ...

A variant of the MINRES method, often referred to as the MR-II method, has in the last few years become a popular iterative scheme for computing approximate solutions of large linear discrete ill-posed problems with a symmetric matrix. It is important to terminate the iterations sufficiently early in order to avoid severe amplification of measurement and round-off errors. We present a new L-cur...

This paper gives a broad overview of a complete framework for assessing the predictive uncertainty of scientific computing applications. The framework is complete in the sense that it treats both types of uncertainty (aleatory and epistemic) and incorporates uncertainty due to the form of the model and any numerical approximations used. Aleatory (or random) uncertainties in model inputs are tre...