Calculating a function of a matrix with a real spectrum

TL;DR

This paper presents a new method for approximating the function of a matrix with a real spectrum by using block representations, interpolating polynomials, and enhanced precision to improve computational stability.

Contribution

It introduces a block matrix approach combined with interpolating polynomials and increased precision to efficiently compute matrix functions with real spectra.

Findings

The proposed method improves numerical stability when calculating functions of matrices with close eigenvalues.

Using block diagonalization reduces computational complexity in matrix function calculations.

Enhanced decimal precision helps mitigate errors in polynomial interpolation for matrix functions.

Abstract

Let $T$ be a square matrix with a real spectrum, and let $f$ be an analytic function. The problem of the approximate calculation of $f(T)$ is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that $T$ is triangular and its diagonal entries $t_{ii}$ are arranged in increasing order. To avoid calculations using the differences $t_{ii}-t_{jj}$ with close (including equal) $t_{ii}$ and $t_{jj}$, it is proposed to represent $T$ in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the $f(T)$ entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform scalar operations (such as the building of interpolating polynomials) with an enlarged number of decimal digits.