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

Tracking p-adic precision

We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 sequence

Spherical functions on the space of pp-adic unitary hermitian matrices

We investigate the space $X$ of unitary hermitian matrices over \frp-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic, i.e., 2\notin \frp. In the latter half we assume odd residual characteristic, and give explicit formulas of typical spherical functions on $X$, where Hall-Littlewood symmetric polynomials of type $C_n$ appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show the Schwartz space \SKX is a free Hecke algebra \hec-module of rank $2^n$, where $2n$ is the size of matrices in $X$, and give the explicit Plancherel formula on \SKX.Comment: to appear in International Journal of Number Theor