Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r−1, we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 3r fo...

We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first-order linear differential operators on the real line. The applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations, for example reaction-diffusion or solitary wave equations. The...

In this paper, we study numerical approximations of eigenvalues when using projection method for spectral approximations of completely continuous operators. We improve the theory depending on the ascent of T − μ and provide a new approach for error estimate, which depends only on the ascent of Th − μh. Applying this estimator to the integral operator eigenvalue problems, we obtain asymptoticall...

We study the iterative solution of dense linear systems that arise from boundary element discretizations of the electrostatic integral equation in magnetoencephalography (MEG). We show that modern iterative methods can be used to decrease the total computation time by avoiding the time-consuming computation of the LU decomposition of the coefficient matrix. More importantly, the modern iterativ...

This handout provides the essential elements needed to understand finite random matrix theory. A cursory observation should reveal that the tools for infinite random matrix theory are quite different from the tools for finite random matrix theory. Nonetheless, there are significantly more published applications that use finite random matrix theory as opposed to infinite random matrix theory. Ou...

In the present paper we consider the Volterra integration operator V on the Wiener algebra W (D) of analytic functions on the unit discD of the complex plane C. A complex number is called an extended eigenvalue of V if there exists a nonzero operator A satisfying the equation AV = V A. We prove that the set of all extended eigenvalues of V is precisely the set C\{0}, and describe in terms of Du...