The Rayleigh-Ritz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ri...

This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (μ + τ, x̃) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value...

We consider the two-sided Arnoldi method and propose a two-sided Krylov–Schurtype restarting method. We discuss the restart for standard Rayleigh–Ritz extraction as well as harmonic Rayleigh–Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer–Fike theorem and Saad’s theorem....

Ritz Values and Arnoldi Convergence for Nonsymmetric Matrices by Russell Carden The restarted Arnoldi method, useful for determining a few desired eigenvalues of a matrix, employs shifts to refine eigenvalue estimates. In the symmetric case, using selected Ritz values as shifts produces convergence due to interlacing. For nonsymmetric matrices the behavior of Ritz values is insufficiently under...

This paper presents an error analysis of the Lanczos algorithm in finite-precision arithmetic for solving the standard nonsymmetric eigenvalue problem, if no breakdown occurs. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. The theory developed illustra...

The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to precond...

The Rayleigh-Ritz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in t...

Generalized block Lanczos methods for large unsymmetric eigenproblems are presented, which contain the block Arnoldi method, and the block Arnoldi algorithms are developed. The convergence of this class of methods is analyzed when the matrix A is diagonalizable. Upper bounds for the distances between normalized eigenvectors and a block Krylov subspace are derived, and a priori theoretical error...

The Rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices. The Rayleigh-Ritz method finds the stationary values of the Rayleigh quotient, called Ritz values, on a given trial subspace as optimal, in some sense, approximations to eigenvalues. In the present paper, we derive upper bounds for proximity of the Ritz valu...