求解大规模非线性特征值问题的方法和理论

A wide variety of applications require the solution of nonlinear eigenvalueproblems, most of them arsing in mathematics, physics, mechanics, computational geometry and so on. The main purpose of this research is, aiming at several important large-scale.nonlinear eigenvalue problems in these fields, to exploit their special.structures to develop several high performance numerical methods, and.give the theoretic analysis of these methods. Based on this goal we deeply.study a certain kind of positive definite quadratic eigenvalue problems.with wide application backgrounds, and propose three numercial methods for computing the largest or the smallest real eigenvalue: eigenvalue curve method, smallest singular value method and inexact Newton method. Comparing with former methods, the three methods all can make full use of the symmetry and ositive definition of coefficient matrices, and can also efficiently compute a good approximation to the largest or the smallest real eigenvalue even when the eigenvalue is surrounded by.国家自然科学基金资助项目结题报告.3 complex eigenvalues. Since the several smallest positive eigenvalues are.usually the factors of safety for applied external force in the positive.definite quadratic systems, our numerical methods are of great pratical value in engineering designing. Key words: quadratic eigenvalue problem, eigenvalue curve method, smallest singular value method, inexact Newton method

非线性特征值问题广泛出现于数学、物理、力学和计算机图形学等一系列领域中。针对这些领域中提出的若干重要的大规模非线性特征值问题，充分利用其特殊性，提出和发展几个高性能的数值计算方法，并给出这些方法的理论分析，将有着重要的理论价值和广阔的应用前景，而且对推动我国矩阵计算这一领域的研究垮入世界先进行列有不可低估的作用。