Brad J.C. Baxter

A New Error Estimate of the Fast Gauss Transform

The fast Gauss transform of L. Greengard and J. Strain [SIAM J. Sci. Statist. Comput., 12 (1991), pp. 79--94] reduces the computational complexity of the evaluation of the sum of N Gaussians at M points in d-dimensional space from

On the Foundations of Computational Mathematics

{\cal O}(MN)

A COVARIANCE MATRIX INVERSION PROBLEM ARISING FROM THE CONSTRUCTION OF PHYLOGENETIC TREES

to

Rapid evaluation of radial basis functions

{\cal O}(M+N)

Invertible and non-invertible information sets in linear rational expectations models

floating-point operations. In this note, we provide numerical evidence that the error estimate of Lemma 2.1 in [SIAM J. Sci. Statist. Comput., 12 (1991), pp. 79--94] is erroneous and then proceed to calculate a replacement error estimate for the fast Gauss transform, incorporating an improved upper bound for Hermite functions.

Invertible and Non-Invertible Information Sets in Dynamic Stochastic General Equilibrium

Publisher Summary This chapter discusses the interaction of computational numerical with pure mathematics.. As far as computer scientists are concerned, their genuine “interface of interaction” with numerical thinking is in two distinct areas: complexity theory and high-performance computing. While complexity theory has always been a glamourous activity in theoretical computer science, it has only recently emerged as a focus of concerted activity in numerical circles, occasionally leading to a measure of acrimony. It is to be hoped that, eventually, computer scientists will find complexity issues involving real-number computations to be challenging, worthwhile, and central to the understanding of theoretical computation. Likewise, the ongoing development of parallel computer architectures and the computational grid is likely to lead to considerably better numerical/computational interaction at the more practical, engineering-oriented end.

On Spherical Averages of Radial Basis Functions

We describe an efficient algorithm for the inversion of covariance matrices that arise in the context of phylogenetic tree construction. Phylogenetic trees describe the evolutionary relationships between species, and their construction is computationally demanding. Many approaches involve the symmetric matrix of evolutionary distances between species. Regarding these distances as random variables, the corresponding set of variances and covariances form a rank-4 tensor, and the innerproduct defined by its inverse can be used to assign statistical scores to candidate trees. We describe a natural set of assumptions for the phylogenetic tree under construction, and show how under these assumptions the covariance tensor for a tree with n leaves can be inverted in O(n 2 ) operations. In addition to presenting the inversion algorithm, we hope this article will open algebraic and computational problems from the field of phylogeny to a wider audience.