Computing the Eigenvectors of Nonsymmetric Tridiagonal Matrices

The computation of the eigenvalue decomposition of matrices is one of the most investigated problems in numerical linear algebra. In particular, real nonsymmetric tridiagonal eigenvalue problems arise in a variety of applications. In this paper the problem of computing an eigenvector corresponding to a known eigenvalue of a real nonsymmetric tridiagonal matrix is considered, developing an algorithm that combines part of a $$QR$$ sweep and part of a $$QL$$ sweep, both with the shift equal to the known eigenvalue. The numerical tests show the reliability of the proposed method.