Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems

In this article, the quasi-Laguerre iteration is established in the spirit of Laguerre's iteration for solving polynomial f with all real zeros. The new algorithm, which maintains the monotonicity and global convergence of the Laguerre iteration, no longer needs to evaluate f". The ultimate convergence rate is + 1. When applied to approximate the eigenvalues of a symmetric tridiagonal matrix, the algorithm substantially improves the speed of Laguerre's iteration.