THE COLLATZ INCLUSION THEOREM EXTENDED FORGENERALIZED EIGENVALUE PROBLEMS
广义特征值问题的Collatz包含定理

There are a lot of developed approaches to attract fundamental eigenvalue of matrix inengineering, 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 theeigenvalue is obtained by taking the ratio of before with after iteration of an arbitrary element. TheCollatz inclusion theorem studied by many authors, it can be used to find the lower and upperbounds 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. Whenthe mass matrix or stiffness matrix is positive definite symmetric matrix, therefore the generalizedeigenvalue problem can be reduced to standard eigenvalue problem by using Choleskydecomposition. The fundamental natural frequency or the highest natural frequency can beobtained from decomposition of mass matrix or stiffness matrix respectively. To verify thetheory, some examples are presented.