广义特征值问题的Collatz包含定理

THE COLLATZ INCLUSION THEOREM EXTENDED FOR GENERALIZED EIGENVALUE PROBLEMS

工程中已发展了许多矩阵特征值问题的近似求解方法，由Ｄｕｎｃｌｅｙ法给出固有频率基频的下界，Ｒａｙｌｅｉｇｈ－Ｒｉｔｚ近似法建立的方程，给出基频的上界，以及通常的矩阵迭代法给出的矩阵的固有频率程序中是以某一元素迭代前后比值确定的，这样实际上很难说是上界或下界．Ｃｏｌｌａｔｚ包含定理仅适用于对称标准特征值问题，可以给出特征值上、下界． 采用矩阵 Ｃｈｏｌｅｓｋｙ三角分解的原理，把 Ｃｏｌｌａｔｚ包含定理推广到适用于具有对称矩阵的一般结构系统的广义征值问题，对于分解刚度矩阵或质量矩阵可给出基频，或最高固有频率．为了验证理论的正确性，给出了算例．

There are a lot of developed approaches to attract fundamental eigenvalue of matrix in engineering, Duncley method gives lower bound and the Rayleigh-Ritz method gives upper bound. As for he matrix iteration method is difficult to say which gives lower or upper bound, due to the eigenvalue is obtained by taking the ratio of before with after iteration of an arbitrary element. The Collatz inclusion theorem studied by many authors, it can be used to find the lower and upper bounds both, but the theorem can be only used to standard eigenvalue problem. In this paper, the Collatz inclusion theorem is extended to generalized eigenvalue problems. When the mass matrix or stiffness matrix is positive definite symmetric matrix, therefore the generalized eigenvalue problem can be reduced to standard eigenvalue problem by using Cholesky decomposition. The fundamental natural frequency or the highest natural frequency can be obtained from decomposition of mass matrix or stiffness matrix respectively. To verify the theory, some examples are presented.