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Walter F. Mascarenhas

Publications and source records attributed to Walter F. Mascarenhas.

At least 19 recordsLinked to original sources
arXiv2021

Root Finding With Interval Arithmetic

We consider the solution of nonlinear equations in one real variable, the problem usually called by root finding. Although this is an old problem, we believe that some aspects of its solution using interval arithmetic are not well understood, and we present our views on this subject. We argue that problems with just one variable are much simpler than problems with more variables, and we should use specific methods for them. We provide an implementation of our ideas in C++, and make this code available under the Mozilla Public License 2.0.

math.NA↗
arXiv2021

Solving systems of inequalities in two variables with floating point arithmetic

From a theoretical point of view, finding the solution set of a system of inequalities in only two variables is easy. However, if we want to get rigorous bounds on this set with floating point arithmetic, in all possible cases, then things are not so simple due to rounding errors. In this article we describe in detail an efficient data structure to represent this solution set and an efficient and robust algorithm to build it using floating point arithmetic. The data structure and the algorithm were developed as a building block for the rigorous solution of relevant practical problems. They were implemented in \texttt{C++} and the code was carefully tested. This code is available as supplementary material to the arxiv version of this article, and it is distributed under the Mozilla Public License 2.0.

cs.DS↗
arXiv2021

Computing the exact sign of sums of products with floating point arithmetic

IIn computational geometry, the construction of essential primitives like convex hulls, Voronoi diagrams and Delaunay triangulations require the evaluation of the signs of determinants, which are sums of products. The same signs are needed for the exact solution of linear programming problems and systems of linear inequalities. Computing these signs exactly with inexact floating point arithmetic is challenging, and we present yet another algorithm for this task. Our algorithm is efficient and uses only of floating point arithmetic, which is much faster than exact arithmetic. We prove that the algorithm is correct and provide efficient and tested \texttt{C++} code for it.

cs.CG↗
arXiv2021

On the differentiability of interval functions

Two articles published by Information Science discuss the derivatives of interval functions, in the sense of Svetoslav Markov. The authors of these articles tried to characterize for which functions and points such derivatives exist. Unfortunately, their characterization is inaccurate. This article describes this inaccuracy and explains how it can be corrected.

math.GM↗
arXiv2018

On the rate of convergence of Berrut's interpolant at equally spaced nodes

We extend the work by Mastroianni and Szabados regarding the barycentric interpolant introduced by J.-P. Berrut in 1988, for equally spaced nodes. We prove fully their first conjecture and present a proof of a weaker version of their second conjecture. More importantly than proving these conjectures, we present a sharp description of the asymptotic error incurred by the interpolants when the derivative of the interpolated function is absolutely continuous, which is a class of functions broad enough to cover most functions usually found in practice. We also contribute to the solution of the broad problem they raised regarding the order of approximation of these interpolants, by showing that they have order of approximation of order 1/n for functions with derivatives of bounded variation.

math.NA↗
arXiv2018

A simple canonical form for nonlinear programming problems and its use

We argue that reducing nonlinear programming problems to a simple canonical form is an effective way to analyze them, specially when the problem is degenerate and the usual linear independence hypothesis does not hold. To illustrate this fact we solve an open problem about constraint qualifications using this simple canonical form.

math.OC↗
arXiv2018

Moore: Interval Arithmetic in C++20

This article presents the Moore library for interval arithmetic in C++20. It gives examples of how the library can be used, and explains the basic principles underlying its design.

cs.MS↗
arXiv2018

Fast and accurate normalization of vectors and quaternions

We present fast and accurate ways to normalize two and three dimensional vectors and quaternions and compute their length. Our approach is an adaptation of ideas used in the linear algebra library LAPACK, and we believe that the computational geometry and computer aided design communities are not aware of the possibility of speeding up these fundamental operations in the robust way proposed here.

cs.CG↗
arXiv2017

Floating point numbers are real numbers

Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which continuous mathematics leads to sharp, simple and new results about the evaluation of sums, square roots and dot products in floating point arithmetic.

math.NA↗
arXiv2016

Moore: Interval Arithmetic in Modern C++

We present the library Moore, which implements Interval Arithmetic in modern C++. This library is based on a new feature in the C++ language called concepts, which reduces the problems caused by template meta programming, and leads to a new approach for implementing interval arithmetic libraries in C++.

cs.MS↗
arXiv2016

The stability of extended Floater-Hormann interpolants

We present a new analysis of the stability of extended Floater-Hormann interpolants, in which both noisy data and rounding errors are considered. Contrary to what is claimed in the current literature, we show that the Lebesgue constant of these interpolants can grow exponentially with the parameters that define them, and we emphasize the importance of using the proper interpretation of the Lebesgue constant in order to estimate correctly the effects of noise and rounding errors. We also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants. Our experiments show that extended interpolants mentioned in the literature satisfy this condition and, therefore, the formula used to implement them is not backward stable. Finally, we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation.

math.NA↗
arXiv2016

A rational Rodrigues' formula to interpolate rotations

We propose a rational version of the classic Rodrigues' rotation formula, which leads to a more accurate and efficient modelling of rotations and their derivatives in finite precision arithmetic. We explain how the rational Rodrigues' formula can be used to describe the kinematics of rigid bodies, in a practical example in which we model the rotation of a cell phone using the data obtained from its gyroscope.

math.NA↗
arXiv2014

The divergence of the barycentric Pade approximants

We explain that, like the usual Padé approximants, the barycentric Padé approximants proposed recently by Brezinski and Redivo-Zaglia can diverge. More precisely, we show that for every polynomial P there exists a power series S, with arbitrarily small coefficients, such that the sequence of barycentric Padé approximants of P + S do not converge uniformly in any subset of the complex plane with a non-empty interior.

math.NA↗
arXiv2014

The stability of barycentric interpolation at the Chebyshev points of the second kind

We present a new analysis of the stability of the first and second barycentric formulae for interpolation at the Chebyshev points of the second kind. Our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice. % OLD: We also explain that the first barycentric formula has accuracy problems which are not properly taken into account in the current literature. We also extend our current understanding regarding the accuracy problems of the first barycentric formula.

math.NA↗
arXiv2013

The divergence of the BFGS and Gauss Newton Methods

We present examples of divergence for the BFGS and Gauss Newton methods. These examples have objective functions with bounded level sets and other properties concerning the examples published recently in this journal, like unit steps and convexity along the search lines. As these other examples, the iterates, function values and gradients in the new examples fit into the general formulation in our previous work {\it On the divergence of line search methods, Comput. Appl. Math. vol.26 no.1 (2007)}, which also presents an example of divergence for Newton's method.

math.OC↗