Volume 65, pp. 479-499, 2026.

Computing the exponential of tridiagonal Toeplitz matrices with applications to the numerical solution of the heat equation

Mehdi Tatari and Majed Hamadi

Abstract

The computation of the exponential of a tridiagonal matrix and its applications have long attracted considerable interest. One application considered in this work arises from the method of lines for solving the heat equation, where the partial differential equation is transformed into a system of ordinary differential equations (ODEs). The solution of this system depends on the exponential of a tridiagonal Toeplitz matrix. Strang and MacNamara [SIAM Review, 56 (2014), pp. 525–546] proposed an approximate method for computing the exponential of a symmetric tridiagonal Toeplitz matrix arising in the solution of ODEs. Their approach is based on approximating the entries of the matrix exponential by modified Bessel functions of the first kind evaluated at specific values. Consequently, the matrix exponential can be represented as the difference between a Toeplitz matrix and a Hankel matrix. In this paper, we extend this idea to the general class of tridiagonal Toeplitz matrices and improve the stability of the method by approximating the matrix exponential with a banded matrix, thereby making the computational complexity independent of the matrix size. In addition, we present an error analysis for the proposed methods and derive bounds for the entries of the exponential of tridiagonal Toeplitz matrices. As the main contribution of this work, the proposed approach is employed to solve the heat equation, and its uniform stability is established. Furthermore, by means of a splitting technique, the method is extended to two-dimensional problems. Numerical experiments demonstrate the effectiveness and efficiency of the proposed methods and the sharpness of the derived bounds.

Full Text (PDF) [957 KB], BibTeX , DOI: 10.1553/etna_vol65s479

Key words

tridiagonal Toeplitz matrix, matrix exponential, modified Bessel functions of the first kind, heat equation

AMS subject classifications

65F60, 65M06, 65M12

Links to the cited ETNA articles

[6]	Vol. 28 (2007-2008), pp. 16-39 Michele Benzi and Nader Razouk: Decay bounds and $O$($n$) algorithms for approximating functions of sparse matrices