Calculation of Densities of States and Spectral Functions by Chebyshev Recursion and Maximum Entropy

TL;DR

This paper introduces an efficient algorithm combining Chebyshev recursion and maximum entropy methods to accurately compute spectral properties of large sparse Hamiltonian matrices, with linear scaling and improved conditioning.

Contribution

It presents a novel approach that enhances spectral property calculations using Chebyshev moments and maximum entropy, improving accuracy and efficiency over traditional methods.

Findings

High energy resolution achieved without numerical instability.

Linear scaling of computational resources with system size.

Better conditioning of spectral inference from Chebyshev moments.

Abstract

We present an efficient algorithm for calculating spectral properties of large sparse Hamiltonian matrices such as densities of states and spectral functions. The combination of Chebyshev recursion and maximum entropy achieves high energy resolution without significant roundoff error, machine precision or numerical instability limitations. If controlled statistical or systematic errors are acceptable, cpu and memory requirements scale linearly in the number of states. The inference of spectral properties from moments is much better conditioned for Chebyshev moments than for power moments. We adapt concepts from the kernel polynomial approximation, a linear Chebyshev approximation with optimized Gibbs damping, to control the accuracy of Fourier integrals of positive non-analytic functions. We compare the performance of kernel polynomial and maximum entropy algorithms for an electronic structure example.