On the Characteristic Polynomial of Regular Linear Matrix Pencil

Linear matrix pencil, denoted by (A,B), plays an important role in control systems and numerical linear algebra. The problem of finding the eigenvalues of (A,B) is often solved numerically by using the well-known QZ method. Another approach for exploring the eigenvalues of (A,B) is by way of its characteristic polynomial, P(λ)=A − λB. There are other applications of working directly with the characteristic polynomial, for instance, using Routh-Hurwitz analysis to count the stable roots of P(λ) and transfer function representation of control systems governed by differential-algebraic equations. In this paper, we present an algorithm for algebraic construction of the characteristic polynomial of a regular linear pencil. The main theorem reveals a connection between the coefficients of P(λ) and a lexicographic combination of the rows between matrices A and B

Augmented Block Householder Arnoldi Method

AbstractComputing the eigenvalues and eigenvectors of a large sparse nonsymmetric matrix arises in many applications and can be a very computationally challenging problem. In this paper we propose the Augmented Block Householder Arnoldi (ABHA) method that combines the advantages of a block routine with an augmented Krylov routine. A public domain MATLAB code ahbeigs has been developed and numerical experiments indicate that the code is competitive with other publicly available codes