Very-high-precision solutions of a class of Schr{\"o}dinger equations

TL;DR

This paper presents a high-precision numerical method for solving a class of Schrödinger equations, achieving thousands to a million decimal places with linear memory scaling but super-quadratic time complexity.

Contribution

The authors develop a method that efficiently computes eigenvalues of Schrödinger equations to extremely high precision, optimizing memory and algebraic operation scaling.

Findings

Memory requirement scales linearly with precision P.

Time complexity increases faster than P^2 due to high-precision multiplication costs.

Method enables extremely accurate eigenvalue computations for Schrödinger problems.

Abstract

We investigate a method to solve a class of Schr{\"o}dinger equation eigenvalue problems numerically to very high precision $P$ (from thousands to a million of decimals). The memory requirement, and the number of high precision algebraic operations, of the method scale essentially linearly with $P$ when only eigenvalues are computed. However, since the algorithms for multiplying high precision numbers scale at a rate between $P^{1.6}$ and $P\,\log P\,\log\log P$, the time requirement of our method increases somewhat faster than $P^2$.