2,235 research outputs found
Explicit inverse of nonsingular Jacobi matrices
We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm–Liouville boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout a discrete Schrödinger operator and their solutions can be computed using recent advances in the study of linear difference equations. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provide the entries of the inverse matrixPeer ReviewedPostprint (author's final draft
On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and
the associated Green's matrices for whole-line and half-line self-adjoint
Hamiltonian finite difference systems with separated boundary conditions.Comment: 30 pages, to appear in J. Comput. Appl. Mat
Classification and Casimir Invariants of Lie-Poisson Brackets
We classify Lie-Poisson brackets that are formed from Lie algebra extensions.
The problem is relevant because many physical systems owe their Hamiltonian
structure to such brackets. A classification involves reducing all brackets to
a set of normal forms, and is achieved partially through the use of Lie algebra
cohomology. For extensions of order less than five, the number of normal forms
is small and they involve no free parameters. We derive a general method of
finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir
invariants of all low-order brackets are explicitly computed. We treat in
detail a four field model of compressible reduced magnetohydrodynamics.Comment: 59 pages, Elsevier macros. To be published in Physica
Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions
The aim of this paper is to show some examples of matrix-valued orthogonal
functions on the real line which are simultaneously eigenfunctions of a
second-order differential operator of Schr\"{o}dinger type and an integral
operator of Fourier type. As a consequence we derive integral representations
of these functions as well as other useful structural formulas. Some of these
functions are plotted to show the relationship with the Hermite or wave
functions
Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector
We consider algebraic iterative reconstruction methods with applications in
image reconstruction. In particular, we are concerned with methods based on an
unmatched projector/backprojector pair; i.e., the backprojector is not the
exact adjoint or transpose of the forward projector. Such situations are common
in large-scale computed tomography, and we consider the common situation where
the method does not converge due to the nonsymmetry of the iteration matrix. We
propose a modified algorithm that incorporates a small shift parameter, and we
give the conditions that guarantee convergence of this method to a fixed point
of a slightly perturbed problem. We also give perturbation bounds for this
fixed point. Moreover, we discuss how to use Krylov subspace methods to
efficiently estimate the leftmost eigenvalue of a certain matrix to select a
proper shift parameter. The modified algorithm is illustrated with test
problems from computed tomography
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