An approximation of matrix exponential by a truncated Laguerre series

2023/12/12 by E. D. Khoroshikh, Khoroshikh, E. D., V. G. Kurbatov +1
Computer Science · Mathematics · #33C45 #65F60 #97N50 #Dynamical Systems (math.DS) #FOS: Mathematics #Fractional Differential Equations Solutions #Functional Analysis (math.FA) #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2312.07291

openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Laguerre functions l_n,τ^α, n=0,1,…, are constructed from generalized Laguerre polynomials. The functions l_n,τ^α depend on two parameters: scale τ>0 and order of generalization α>-1, and form an orthogonal basis in L₂[0,∞). Let the spectrum of a square matrix A lie in the open left half-plane. Then the matrix exponential H_A(t)=e^At, t>0, belongs to L₂[0,∞). Hence the matrix exponential H_A can be expanded in a series H_A=∑_n=0^∞ S_n,τ,α,A l_n,τ^α. An estimate of the norm ‖ H_A-∑_n=0^N S_n,τ,α,A l_n,τ^α‖_L2[0,∞) is proposed. Finding the minimum of this estimate over τ and α is discussed. Numerical examples show that the optimal α is often almost 0, which essentially simplifies the problem.

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