arXiv subjects

George Labahn

Publications and source records attributed to George Labahn.

23 records · Page 2Linked to original sources

arXiv2016

On rational functions without Froissart doublets

In this paper we consider the problem of working with rational functions in a numeric environment. A particular problem when modeling with such functions is the existence of Froissart doublets, where a zero is close to a pole. We discuss three different parameters which allow one to monitor the absence of Froissart doublets for a given general rational function. These include the euclidean condition number of an underlying Sylvester-type matrix, a parameter for determing coprimeness of two numerical polynomials and bounds on the spherical derivative. We show that our parameters sharpen those found in a previous paper by two of the autours.

math.NA↗

arXiv2016

A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix

Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity $O^{\sim}\left(n^ωd\right)$ where $d$ is the degree of $\mathbf{F}$. Here soft-$O$ notation is Big-$O$ with log factors removed and $ω$ is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal entries of its Hermite normal form, having as cost $O^{\sim}\left(n^ωs\right)$ operations with $s$ the average of the column degrees of $\mathbf{F}$.

cs.SC↗

arXiv2016

Existence Problem of Telescopers: Beyond the Bivariate Case

In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, which has been solved recently. The existence criteria we present is needed for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.

cs.SC↗

arXiv2014

A Bayesian model for recognizing handwritten mathematical expressions

Recognizing handwritten mathematics is a challenging classification problem, requiring simultaneous identification of all the symbols comprising an input as well as the complex two-dimensional relationships between symbols and subexpressions. Because of the ambiguity present in handwritten input, it is often unrealistic to hope for consistently perfect recognition accuracy. We present a system which captures all recognizable interpretations of the input and organizes them in a parse forest from which individual parse trees may be extracted and reported. If the top-ranked interpretation is incorrect, the user may request alternates and select the recognition result they desire. The tree extraction step uses a novel probabilistic tree scoring strategy in which a Bayesian network is constructed based on the structure of the input, and each joint variable assignment corresponds to a different parse tree. Parse trees are then reported in order of decreasing probability. Two accuracy evaluations demonstrate that the resulting recognition system is more accurate than previous versions (which used non-probabilistic methods) and other academic math recognizers.

cs.AI↗

arXiv2014

Fast and deterministic computation of the determinant of a polynomial matrix

Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm is $\bigO \left(n^ωs\right)$ field operations where $s$ is the average column degree or the average row degree of $\mathbf{F}$. Here $\bigO$ notation is Big-$O$ with log factors omitted and $ω$ is the exponent of matrix multiplication.

cs.SC↗