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We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations are enough to reduce round-off errors to the order of the unit round-off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs. [ABSTRACT FROM AUTHOR]

This paper presents a deep analysis of a time-dependent Schrödinger equation with fractional time derivative. After the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations; it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations. The main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations (possibly with high frequency as it is typical with Schrödinger equations) and a persisting memory (as a consequence of the fractional order); moreover, an accurate spatial discretization gives rise to systems of large to very large size, involving a noteworthy computational complexity. By means of a theoretical analysis the exact solution is split into two or three terms (depending on the order of the fractional derivative), thus to face the numerical computation by different and suitably selected methods: direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step-by-step strategy, in conjunction with matrix function, for the oscillating term. In both cases, Krylov subspace methods are employed for the computation of matrix functions and convergence results are presented. [ABSTRACT FROM AUTHOR]

Abstract: The shift-and-invert method is very efficient in eigenvalue computations, in particular when interior eigenvalues are sought. This method involves solving linear systems of the form . The shift is variable, hence when a direct method is used to solve the linear system, the LU factorization of needs to be computed for every shift change. We present two strategies that reduce the number of floating point operations performed in the LU factorization when the shift changes. Both methods perform first a preprocessing step that aims at eliminating parts of the matrix that are not affected by the diagonal change. This leads to about 43% and 50% flops savings respectively for the dense matrices. [Copyright &y& Elsevier]

Abstract: The inertia of a complex matrix A, is defined to be an integer triple, , where is the number of eigenvalues of A with positive real parts, is the number of eigenvalues with negative real parts and is the number of eigenvalues with zero real parts. In this paper we show that the inertia can be computed by Gereshgorin theorem and block shift-and-invert algorithm for equation or and this algorithm is compared by function eig.m and sptarn.m in Matlab. [Copyright &y& Elsevier]

The shift-and-invert Arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and/or the associated eigenvectors of a large unsymmetric matrix pair, but there is no guarantee for the approximate eigenvectors, Ritz vectors, obtained by this method to converge even though the subspace is good enough. In order to correct this problem, a refined shift-and-invert Arnoldi method is proposed that uses certain refined Ritz vectors to approximate the desired eigenvectors. The refined Ritz vectors can be computed cheaply and reliably by small-sized singular value decompositions. It is shown that the refined method converges. A refined shift-and-invert Arnoldi algorithm is developed, and several numerical examples are reported. Comparisons are drawn on the refined algorithm and the shift-and-invert Arnoldi algorithm, indicating that the former is considerably more efficient than the latter. [Copyright &y& Elsevier]

In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalue problem Ax = λBx, and analyze how the accuracy in the inner iterations affects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the inner threshold parameter as the convergence rate. We also discuss the total amount of work and asymptotic equivalence between this stopping criterion and a more standard one. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that four our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.