Improving the computation of the τVI Painlevé function using the quadrature method for the fredholm determinant

Abstract

The Painlevé transcendent functions are important tools in theoretical physics, they appear in a variety of physical systems going from quantum integrable systems to random matrix theory. The accessory parameter problem for ODEs, which has connections to black hole scattering problem, can be solved by using the connection between the Painlevé VI transcendent with isomonodromic deformations of a linear ordinary differential equation. In this case, the isomonodromic V I function plays a major role, and finding its roots is equivalent to solving the accessory parameter problem. The V I function can be expressed as a function of a Fredholm determinant. In this dissertation, we will discuss the two main different methods of calculation of the V I in the Fredholm determinant form. We will also present how to construct codes for both methods and analyze them in order to understand which one is the most numerically efficient to find the roots of the V I function.