What question did this study set out to answer?

This research aims to approximate eigenvalues of perturbed tridiagonal systems using polynomial methods.

March 25, 2026Open Access

Approximating Eigenvalues of a Class of Perturbed Tridiagonal Systems

CCChristos ChorianopoulosUniversity of West Attica IFIoannis Th. FamelisUniversity of West Attica

Key Points

This research aims to approximate eigenvalues of perturbed tridiagonal systems using polynomial methods.
Studied rank-one updates of symmetric tridiagonal Toeplitz matrices.
Derived formulas for polynomial approximations up to eighth-order and quartic closed formulas.
Examined symmetries in coefficients of the derived polynomials.
Conducted numerical testing comparing results with MATLAB’s eig function.
Achieved high accuracy in eigenvalue approximations, close to machine precision.
Demonstrated low computational time when handling large matrices compared to MATLAB’s tools.
Observed significant differences in error for various polynomial degrees.

Abstract

We study a class of perturbed tridiagonal problems in the form of a rank-one update of a symmetric tridiagonal Toeplitz matrix. We derive computable formulas for up to eighth-order polynomial approximations or closed formulas for quartic approximation. Moreover, we study some symmetries that characterise the coefficients of these polynomials. Numerical testing suggested that the error is close to machine accuracy in the former case and surprising low in the latter, whereas for big matrices the computational time is clearly lower compared to the MATLAB’s eig function.

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