Large, sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on Chebyshev polynomials for an interval in the complex plane. Knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends. Adaptive Chebyshev algorithms, in which t...

Abstract The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive definite. The e...

Large, sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on Chebyshev polynomials for an interval in the complex plane. Knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends. Adaptive Chebyshev algorithms , in which ...

We define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem (MEP). The pseudospectrum for the MEP is defined and characterized. We show that the distance from a right definite MEP to the closest non right definite MEP is related to the smallest unbounded pseudospectrum. Some numerical results are given.

We offer criteria for the existence of positive solutions for two-point right focal eigenvalue problems (−1) n−p y Δ n (t) = λ f (t, y(σ n−1 (t)), y Δ (σ n−2 (t)),..., y Δ p−1 (σ n−p (t))), t ∈ [0,1] ∩ T, y Δ i (0) = 0, 0 ≤ i ≤ p − 1, y Δ i (σ(1)) = 0, p ≤ i ≤ n − 1, where λ > 0, n ≥ 2,1 ≤ p ≤ n − 1 are fixed and T is a time scale.

abstract This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of...

abstract This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of...

The detection of a Hopf bifurcation in a large scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer et. al. (SINUM, 34, (1997) pp. 1-21) proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenv...

In this paper, we study large m asymptotics of the l1 minimal m-partition problem for Dirichlet eigenvalue. For any smooth domain Ω ⊂ ℝn such that ∣Ω∣ = 1, prove limit $${\rm{lim}}_{m \to \infty}l_m^1\left(\Omega \right) {c_0}$$ exists, and constant c0 is independent shape Ω. Here, $$l_m^1\left({\rm{\Omega}} \right)$$ denotes value normalized sum first Laplacian eigenvalues