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This paper presents an algorithm, Voted Kernel Regularization , that provides the flexibility of using potentially very complex kernel functions such as predictors based on much higher-degree polynomial kernels, while benefitting from…
Root-finding method is an iterative process that constructs a sequence converging to a solution of an equation. Householder's method is a higher-order method that requires higher order derivatives of the reciprocal of a function and has…
We extend coded distributed computing over finite fields to allow the number of workers to be larger than the field size. We give codes that work for fully general matrix multiplication and show that in this case we serendipitously have…
Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial…
We present a unified constructive digit-by-digit framework for exact root extraction using only integer arithmetic. The core contribution is a complete correctness theory for the fractional square root algorithm, proving that each computed…
In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial…
A Radial Basis Function Generated Finite-Differences (RBF-FD) inspired technique for evaluating definite integrals over the volume of the ball in three dimensions is described. Such methods are necessary in many areas of Applied…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…
Let F_q be a finite field with q elements with prime power q and let r>1 be an integer with $q\equiv 1 \pmod{r}$. In this paper, we present a refinement of the Cipolla-Lehmer type algorithm given by H. C. Williams, and subsequently improved…
The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In…
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…
New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the…
We discuss the babylonian method of extracting the root square of a number, from the point of view of modern mathematics. We also speculate that the babylonian mathematics was rich enough for a generalization of this method, despite the…
Let $R$ be a reduced irreducible root system, $h$ its Coxeter number and $m$ a positive integer smaller than $h$. Choose of base of $R$, whence a corresponding height function, and let $R(m)$ be the set of roots whose height is a multiple…
We consider a list decoding algorithm recently proposed by Pellikaan-Wu \cite{PW2005} for $q$-ary Reed-Muller codes $\mathcal{RM}_q(\ell, m, n)$ of length $n \leq q^m$ when $\ell \leq q$. A simple and easily accessible correctness proof is…
Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any $\eps > 0$, the author and Rudra (2006,08) presented an $n^{O(1/\eps)}$ time…
We consider the $q$th root number function for the symmetric group. Our aim is to develop an asymptotic formula for the multiplicities of the $q$th root number function as $q$ tends to $\infty$. We use character theory, number theory and…
We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the…
We introduce beyond-worst-case analysis into symbolic computation. This is an extensive field which almost entirely relies on worst-case bit complexity, and we start from a basic problem in the field: isolating the real roots of univariate…
This work presents a new evolutionary optimization algorithm in theoretical mathematics with important applications in scientific computing. The use of the evolutionary algorithm is justified by the difficulty of the study of the…