16,063 research outputs found
Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics
In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr-Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr-Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire's problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators
Real eigenvalues of non-Gaussian random matrices and their products
We study the properties of the eigenvalues of real random matrices and their
products. It is known that when the matrix elements are Gaussian-distributed
independent random variables, the fraction of real eigenvalues tends to unity
as the number of matrices in the product increases. Here we present numerical
evidence that this phenomenon is robust with respect to the probability
distribution of matrix elements, and is therefore a general property that
merits detailed investigation. Since the elements of the product matrix are no
longer distributed as those of the single matrix nor they remain independent
random variables, we study the role of these two factors in detail. We study
numerically the properties of the Hadamard (or Schur) product of matrices and
also the product of matrices whose entries are independent but have the same
marginal distribution as that of normal products of matrices, and find that
under repeated multiplication, the probability of all eigenvalues to be real
increases in both cases, but saturates to a constant below unity showing that
the correlations amongst the matrix elements are responsible for the approach
to one. To investigate the role of the non-normal nature of the probability
distributions, we present a thorough analytical treatment of the
single matrix for several standard distributions. Within the class of smooth
distributions with zero mean and finite variance, our results indicate that the
Gaussian distribution has the maximum probability of real eigenvalues, but the
Cauchy distribution characterised by infinite variance is found to have a
larger probability of real eigenvalues than the normal. We also find that for
the two-dimensional single matrices, the probability of real eigenvalues lies
in the range [5/8,7/8].Comment: To appear in J. Phys. A: Math, Theo
Characterizing and approximating eigenvalue sets of symmetric interval matrices
We consider the eigenvalue problem for the case where the input matrix is
symmetric and its entries perturb in some given intervals. We present a
characterization of some of the exact boundary points, which allows us to
introduce an inner approximation algorithm, that in many case estimates exact
bounds. To our knowledge, this is the first algorithm that is able to guaran-
tee exactness. We illustrate our approach by several examples and numerical
experiments
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