Compression of multivariate discrete measures and applications

We discuss two methods for the compression of multivariate discrete measures, with applications to node reduction in numerical cubature and least-squares approximation. The methods are implemented in the Matlab computing environment, in dimension two

Computing with Infinite and Infinitesimal Numbers in Matlab easily: The Grossone-Light Toolbox

The introduction of the novel numeral system introduced in [1] (based on the notion of grossone) opens new frontiers in numerical computing, allowing to easily perform computations involving infinite and infinitesimal numbers in a numerical way. This work is aimed at making this powerful numeral system easily usable and accessible to the large community of Matlab users. We have implemented (using pure Matlab code) the GrossoneLight Matlab Toolbox, a collection of classes, functions and examples that make grossone-based computing straightforward. The toolbox is called "light" because it introduces some limitations to the more general numeral system discussed in [1]. In particular, only numbers made of integer powers of grossone can be represented, together with bound on the minimum and maximum number of such powers. However, even in presence of such limitations, the implemented numeral system is powerful enough to solve basic numerical linear algebra problems. Following the Matlab object-oriented abstraction paradigm, available in latest Matlab releases, we have been able to implement two classes: the GrossNumber class and the GrossArray class. The first class allows to represent a number made of integer grossone powers, where the coecient used as multiplier for each power is a standard double-precision Matlab oating-point number. The GrossNumber class has been equipped with basic operations (addition, multiplication, etc) by operator overloading. This allows to operate on GrossNumber objects as any other Matlab scalar variable. The GrossArray class has been introduced to handle operations on arrays of GrossNumber objects more eciently. The speedup can be signicant, especially when the code is written in a vectorized fashion and a GPGPU (General Purpose Graphics Processing Units) is available on the machine running the toolbox

The Likelihood of Mixed Hitting Times

We present a method for computing the likelihood of a mixed hitting-time model that specifies durations as the first time a latent L\'evy process crosses a heterogeneous threshold. This likelihood is not generally known in closed form, but its Laplace transform is. Our approach to its computation relies on numerical methods for inverting Laplace transforms that exploit special properties of the first passage times of L\'evy processes. We use our method to implement a maximum likelihood estimator of the mixed hitting-time model in MATLAB. We illustrate the application of this estimator with an analysis of Kennan's (1985) strike data.Comment: 35 page

An extension of MATLAB to continuous functions and operators

An object-oriented MATLAB system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices. About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with our "chebfun" objects using almost exactly the usual MATLAB syntax. All functions live on [-1,1] and are represented by values at sufficiently many Chebyshev points for the polynomial interpolant to be accurate to close to machine precision. Each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation, differentiation, integration, zerofinding, or transforms. Applications in approximation theory and numerical analysis are explored, and possible extensions for more substantial problems of scientific computing are mentioned