A Novel Divisional Bisection Method for the Symmetric Tridiagonal Eigenvalue Problem

Abstract

The embarrassingly parallel nature of the Bisection Algorithm makes it easy and efficient to program on a parallel computer, but with an expensive time cost when all symmetric tridiagonal eigenvalues are wanted. In addition, few methods can calculate a single eigenvalue in parallel for now, especially in a specific order. This paper solves the issue with a new approach that can parallelize the Bisection iteration. Some pseudocodes and numerical results are presented. It shows our algorithm reduces the time cost by more than 35–70% compared to the Bisection algorithm while maintaining its accuracy and flexibility.

Exploración comparativa de los métodos numéricos de Newton-Raphson y bisección para la resolución de ecuaciones no lineales

“…Este método proporcionó una aproximación menos precisa de la raíz después de 10 iteraciones. Aunque es un método más simple y robusto que no requiere el cálculo de la derivada, puede converger más lentamente hacia la raíz (Chu et al, 2022), especialmente si el intervalo inicial es grande o si la función tiene comportamientos oscilantes o no lineales en el intervalo. En la tabla 3 se observa la resolución de la ecuación exponencial (problema 2) por el método de Newton-Raphson.…”

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

“…a i and b i denote the ith component on the diagonal and sub-diagonal of A, respectively. According to [17], we have…”

“…We calculated the distances between λ 2001 of Φ and the last two sub-eigenvalues of Φ 1:η (η ∈ [2,2000]). Because by the Interlacing Property from [17], the close sub-eigenvalues to λ 2001 must be the last ones. The result is shown in Figure 2.…”