ARNOLDI TYPE ALGORITHMS FOR LARGE UNSYMMETRICMULTIPLE EIGENVALUE PROBLEMS

As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix A involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for A symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.

IMPLICIT DETERMINANT METHOD FOR SOLVING AN HERMITIAN EIGENVALUE OPTIMIZATION PROBLEM

Implicit determinant method is an effective method for some linear eigenvalue optimization problems since it solves linear systems of equations rather than eigenpairs.In this paper,we generalize the implicit determinant method to solve an Hermitian eigenvalue optimization problem for smooth case and non-smooth case.We prove that the implicit determinant method converges locally and quadratically.Numerical experiments confirm our theoretical results and illustrate the efficiency of implicit determinant method.

Natural Frequency of Planetary Transmission with Thin-Walled Ring Gear on Elastic Supports

A lumped parameter-rigid elastic coupled dynamic model of two-stage planetary gears for a hybrid car is established through the inter-stage coupled method,in which the supports of the ring gear of planet set Ⅱ are represented as an elastic foundation with radial and tangential uniform distributed stiffness,and the ring gear of planet set Ⅱ is modeled as an elastic continuum body. The natural frequencies based on the eigenvalue problem of dynamic model of planetary transmission are solved and the associated vibration modes are discussed. The rules are revealed which are the influences of the ring gear elastic supports stiffness and rim thickness on natural frequencies of planetary transmission. The theoretical analysis indicates that the vibration modes of planetary transmission with thin-walled ring gear on elastic supports are classified into seven types: Ⅰ/Ⅱ stage coupled rotational mode,Ⅰ stage translational mode,Ⅰ stage planet mode,Ⅱ stage translational mode,Ⅱ stage degenerate planet mode,Ⅱ stage distinct planet mode and purely ring gear mode. For each vibration mode, its properties are summarized. The numerical solutions show that the elastic supports stiffness and rim thickness of the ring gear of planet set Ⅱ have different influences on natural frequencies.

Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.

Multiplicity Results for a Fourth-Order Boundary Value Problem

This paper deals with multiplicity results for nonlinear elastic equations of the type wheree∈L￣2(0,1),g:[0,1]×R×R→R is a bounded contimuous function.and the pair(α,β)satisfiesand

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

Sensitivity Analysis of Semi-simple Eigenvalues of Regular Quadratic Eigenvalue Problems

This paper discusses the sensitivity analysis of semisimple eigenvalues and associated eigen-matrix triples of regular quadratic eigenvalue problems analytically dependent on several parameters. The directional derivatives of semisimple eigenvalues are obtained. The average of semisimple eigenvalues and corresponding eigen-matrix triple are proved to be analytic, and their partial derivatives are given. On these grounds, the sensitivities of the semisimple eigenvalues and corresponding eigenvector matrices are defined.

Proof of a conjecture on a discretized elliptic equation with cubic nonlinearity

We sharpen and prove a conjecture suggested by Chen and Xie, which states that in Galerkineigenfunction discretization for -Δu = u3 , when the finite-dimensional subspace is taken as the eigensubspace corresponding to an N-fold eigenvalue of -Δ, the discretized problem has at least 3N-1 distinct nonzero solutions. We also present a related result on the multiplicities of eigenvalues of -Δ.