Some numerical methods approximate the Jacobian using finite differences as part of their process. The finite difference method of approximation resembles the limit definition of the derivative. Since a computer cannot handle the mathematical concept of limits, a sufficiently small number has to be chosen as an increment. However, when the increments are chosen outside of certain acceptable ranges, we need to account for error stemming from issues regarding representability. In this thesis we will look at the mathematical limitations of the increments. We will also go over how they are chosen in 3 different solvers, and suggest ways to improve upon them.

{2022}

Relating the Arnoldi approximation to the best Chebyshev approximation of the characteristic polynomial - A study using a complex valued variant of the Remez algorithm

The Remez algorithm is a beautiful algorithm that finds the best approximation to a function by finding points satisfying the alternation condition. In 1987 Ping Tang published his doctoral thesis detailing a version of the Remez algorithm modified for functions of complex values with complex coefficients. He later followed the thesis with an article presenting an algorithm for finding the coefficients of a complex valued function. The Arnoldi approximation is a well-known iterative method for approximating the m eigenvalues of a matrix by a smaller set of n eigenvalues. The accuracy of the Arnoldi approximation is however dependent on an initial choice of a vector. The best choice of this initial vector leads to the so called ideal Arnoldi approximation. In 1994 Anne Greenbum and Lloyd Trefethen published an article discussing the possibility of finding the ideal Arnoldi approximation using Chebyshev approximation. The purpose of this thesis is to experiment with Ping Tangs algorithm to approximate the characteristic equation of a matrix on a circle set and compare the results to that of the ideal Arnoldi approximation to see if the ideal Arnoldi is a best approximation in a Chebyshev sense.