A novel nonsymmetric K−-Lanczos algorithm for the generalized nonsymmetric K−-eigenvalue problems Original Research Article

In this article, we present a novel algorithm, named nonsymmetric K−-Lanczos algorithm, for computing a few extreme eigenvalues of the generalized eigenvalue problem Mx = λLx, where the matrices M and L have the so-called K±-structures. We demonstrate a K−-tridiagonalization procedure preserves the K±-structures. An error bound for the extreme K−-Ritz value obtained from this new algorithm is presented. When compared with the class nonsymmetric Lanczos approach, this method has the same order of computational complexity and can be viewed as a special 2 × 2-block nonsymmetric Lanczos algorithm. Numerical experiments with randomly generated K−-matrices show that our algorithm converges faster and more accurate than the nonsymmetric Lanczos algorithm.