Abstract
In this paper, we present highly accurate numerical results for the lowest four energy eigenvalues of the quartic, sextic and octic anharmonic oscillators over a wide range of the anharmonicity parameter λ. Also, we provide illustrative graphs describing the dependence of the eigenvalues on λ. Our computation is carried out by using higher-order finite-difference approximation, involving the nine-and-ten-point differentiation formulas. In addition, we apply Richardson’s extrapolation method in our calculation for the purpose of achieving a maximum numerical precision. The main advantage of utilizing the finite-difference approach lies in its simplicity and capability to transform the time-independent Schrödinger equation into an eigenvalue matrix equation. This allows the use of numerical matrix algebra for obtaining several eigenvalues and eigenvectors simultaneously without consuming much of the computer time. The method is illustrated in a simple pedagogical way through which the close relation between differential and algebraic eigenvalue problems are clearly seen. The findings of our computations via MATLAB are tested on a number of accurate results derived by different methods.