David Roe

Tracking p-adic precision

We present a new method to propagate

p-Adic Stability In Linear Algebra

p

Near-Ultraviolet Absorption Spectrum of 5,10-Dihydrophenazine

-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.

Characteristic Polynomials of p-adic Matrices

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.

Division and Slope Factorization of p-Adic Polynomials

In the course of other work it became necessary to synthesize and determine the near-uv absorption spectrum of 5,10 dihydrophenazine. Since, to our knowledge the spectrum of this compound has not been reported previously, we wish to present our findings here.

ZpL: a p-adic Precision Package

We analyze the precision of the characteristic polynomial XM of an nxn p-adic matrix M using differential precision methods developed previously. When M is an integral matrix whose entries are all given at the same precision O(pN), we give a criterion (checkable within Õ(nω) operations in Fp) for the existence of a coefficient of XM with more accuracy than O(pN). In general, we provide two algorithms for determining the optimal precision of the coefficients of XM and of Ms eigenvalues. We provide evidence showing that classical algorithms do not reach this optimal precision in general.

Photoreduction of phenazine in acidic methanol

We study two important operations on polynomials defined over complete discrete valuation fields: Euclidean division and factorization. In particular, we design a simple and efficient algorithm for computing slope factorizations, based on Newton iteration. One of its main features is that we avoid working with fractional exponents. We pay particular attention to stability, and analyze the behavior of the algorithm using several precision models.

Chemiluminescence from the Reduction of Rubrene Radical Cations

We present a new package ZpL for the mathematical software system SageMath. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in a previous paper by the same authors. The underlying algorithms are mostly based on automatic differentiation techniques. We introduce them, study their complexity and discuss our design choices. We illustrate the benefits of our package (in comparison with previous implementations) with a large sample of examples coming from linear algebra, commutative algebra and differential equations.