Bisection eigenvalue method for Hermitian matrices with quasiseparable representation and a related inverse problem
This paper builds off of the bisection method for computing eigenvalues of a matrix $A$. This is an iterative method that can find any eigenvalue of a real, symmetric, tridiagonal matrix based on the Sturm sequence property: for any given real number $\lambda$, the number of sign changes in the sequence of the characteristic polynomials $\gamma_0(\lambda)\equiv 1, \gamma_1(\lambda), \gamma_2(\lambda), \dots, \gamma_N(\lambda)$ of the principal leading submatrices of $A$ is the number of eigenvalues that are less than $\lambda$. The corresponding eigenvectors can also be computed. The authors study the bisection method for Hermitian matrices with quasiseparable representation. This extends their work [ETNA, Electron. Trans. Numer. Anal. 44, 342--366 (2015; Zbl 1332.65049)] for quasiseparable matrices of order one: here, the authors present an algorithm for using this method for Hermitian matrices whose quasiseparable generators of any order are known. This can be modified for a general matrix $A_0$ having a given quasiseparable representation, where the algorithm is applied to the Hermitian matrix $A=A_0^*A_0$. In order to check the accuracy and perform numerical tests of the developed eigenvalue algorithm, one requires a set of matrices with prescribed eigenvalues from which one can obtain the quasiseparable generators without building the entire matrix. The authors present an algorithm for solving this inverse problem. Various numerical experiments are performed with respect to errors as well as speed of the algorithm. For the entire collection see [Zbl 1411.47002].
scientific article; zbMATH DE number 7137669