281 research outputs found
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
Spectral Approximation for Quasiperiodic Jacobi Operators
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals
and in more general studies of structures exhibiting aperiodic order. The
spectra of these self-adjoint operators can be quite exotic, such as Cantor
sets, and their fine properties yield insight into associated dynamical
systems. Quasiperiodic operators can be approximated by periodic ones, the
spectra of which can be computed via two finite dimensional eigenvalue
problems. Since long periods are necessary to get detailed approximations, both
computational efficiency and numerical accuracy become a concern. We describe a
simple method for numerically computing the spectrum of a period- Jacobi
operator in operations, and use it to investigate the spectra of
Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse
potentials
On simple bounds for eigenvalues of symmetric tridiagonal matrices
How much can be said about the location of the eigenvalues of a symmetric tridiagonal matrix just by looking at its diagonal entries? We use classical results on the eigenvalues of symmetric matrices to show that the
diagonal entries are bounds for some of the eigenvalues regardless of the size of the off-diagonal entries. Numerical examples are given to illustrate that our arithmetic-free technique delivers useful information on the location of the eigenvalues.FEDER Funds through “Programa Operacional Factores de Competitividade - COMPETE
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
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