Factorization of Multivariate Positive Laurent Polynomials

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting

p-Adic Stability In Linear Algebra

Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants , characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.Comment: ISSAC 2015, Jul 2015, Bath, United Kingdom. 201

Accurate Computations with Collocation and Wronskian Matrices of Jacobi Polynomials

In this paper an accurate method to construct the bidiagonal factorization of collocation and Wronskian matrices of Jacobi polynomials is obtained and used to compute with high relative accuracy their eigenvalues, singular values and inverses. The particular cases of collocation and Wronskian matrices of Legendre polynomials, Gegenbauer polynomials, Chebyshev polynomials of the first and second kind and rational Jacobi polynomials are considered. Numerical examples are included. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

Accurate computations with Wronskian matrices of Bessel and Laguerre polynomials

This paper provides an accurate method to obtain the bidiagonal factorization of Wronskian matrices of Bessel polynomials and of Laguerre polynomials. This method can be used to compute with high relative accuracy their singular values, the inverse of these matrices, as well as the solution of some related systems of linear equations. Numerical examples illustrating the theoretical results are included. © 2022 The Author