Perturbation theory for the matrix Mittag-Leffler function

Abstract

This paper establishes perturbation theories for the matrix Mittag-Leffler function E_α,β(A), where α > 0, β ∈ ℝ and A ∈ ℂ^{n × n}. We present upper bounds on ||E_α,β(A + Δ) – E_α,β(A)||, where Δ ∈ ℂ^{n × n}. Results of numerical experiments are reported in order to show how much larger the presented bounds are compared to ||E_α,β(A + Δ) – E_α,β(A)||. When α = β = 1, the function reduces to the matrix exponential. We compare the presented bound when α = β = 1 with an existing perturbation bound for the matrix exponential.