In this paper, we propose an inexact inverse iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the lin...

When a matrix is represented as a product of "elementary" matrices, the matrix, its transpose, its inverse and inverse transpose are readily available for vector multiplication. By an "elementary matrix" is meant one formed from the identity matrix by replacing one column ; thus an elementary matrix can be compactly recorded by the subscript of the altered column and the values of the elements ...

We study the local convergence rates of several most widely used single-vector Newton-like methods for the solution of a degenerate eigenvalue of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. This problem has not been completely understood, since the Jacobian associated with Newton’s method is singular at the desired eigenpair, and the standard convergence theory is not applic...

A new Schulz-type method to compute the Moore-Penrose inverse of a matrix is proposed. Every iteration of the method involves four matrix multiplications. It is proved that this method converge with fourth-order. A wide set of numerical comparisons shows that the average number of matrix multiplications and the average CPU time of our method are considerably less than those of other methods.

The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix factorization–free” method) derives from the well–known inverse iteration procedure in such a way that ...

Very often, inverse iteration is used with shifts to accelerate convergence to an eigenvector. In this paper, it is shown that, if the eigenvalue belongs to a nonlinear elementary divisor, the vector sequences may diverge even when the shift sequences converge to the eigenvalue. The local behavior is discussed through a 2 X 2 example, and a sufficient condition for the convergence of the vector...

On the basis of least squares support vector machine regression (LSSVR), an adaptive and iterative support vector machine regression algorithm based on chunking incremental learning (CISVR) is presented in this paper. CISVR is an iterative algorithm and the samples are added to the working set in batches. The inverse of the matrix of coefficients from previous iteration is used to calculate the...

The equation (1) is solved by the finite element method. We use four corner eight nodes isoparametric elements. Algebraic system of equations for node field values is produced by Galerkin method. To find several modes simultaneously in arbitrary spectrum domain, we use subspace iteration method with frequency shift. If the permittivity and permeability of lossy filling depend on frequency, the ...

The standard Incomplete LU (ILU) preconditioners often fail for general sparse indeenite matrices because they give rise tòunstable' factors L and U. In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditioners based on minimizing kI?AMk F , where AM is the preconditioned matrix. An iterative descent-type method...