The Hermitian Lanczos method for Hermitian matrices has a wellknown connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of R. This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomi...

This study proposes a new method for the acceleration of the projection method to compute a few eigenvalues with the largest real parts of a large nonsymmetric matrix. In the eld of the solution of the linear system, an acceleration using the least squares polynomial which minimizes its norm on the boundary of the convex hull formed with the unwanted eigenvalues are proposed. We simplify this m...

The PageRank algorithm plays an important role in modern search engine technology. It involves using the classical power method to compute the principle eigenvector of the Google matrix representing the web link graph. However, when the largest eigenvalue is not well separated from the second one, the power method may perform poorly. This happens when the damping factor is sufficiently close to...

In order reduction of high order linear time invariant systems based on two-sided Krylov subspace methods, the Lanczos algorithm is commonly used to find the bases for input and output Krylov subspaces and to calculate the reduced order model by projection. However, this method can be numerically unstable even for systems with moderate number of states and can only find a limited number of bior...

In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem Ax = λBx are needed. If exact linear solves with A − σB are available, implicitly restarted Arnoldi with purification is a common approach for problems where B is positive semidefinite. In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of ...

In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem Ax = λBx are needed. If exact linear solves with A−σB are available, implicitly restarted Arnoldi with purification is a common approach for problems where B is positive semidefinite. In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of th...

The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize p(A)b over polynomials p of degree n. The difference is that p is normalized at z 0 for GMRES and at z x for Arnoldi. Analogous "ideal GMRES" and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes p(/l)II instead. Investigation of these true ...

Guided and leaky modes of planar dielectric waveguides are eigensolutions of a singular Sturm-Liouville problem. This paper describes how this problem can be transformed into a quartic eigenvalue problem, which in turn can be converted into a generalized eigenvalue problem. Thus standard iterative methods, such as Arnoldi methods, can be used to compute the spectrum. We show how the shifts in t...

This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi-Tikhonov method. This new rule is essentially based on the discrepancy principle, although no initial knowledge of the norm of the error that affects the right-hand side is assumed; an increasingly more accurate approximation of this q...