用最優(yōu)化方法求解大型矩陣特征值問題.pdf

1、南京航空航天大學碩士學位論文用最優(yōu)化方法求解大型矩陣特征值問題姓名：錢小燕申請學位級別：碩士專業(yè)：計算數(shù)學指導教師：汪曉虹20070301用最優(yōu)化方法求解大型矩陣特征值問題 iiAbstract Firstly the source and the development of computing eigenvalues is summarized. Then a new truncated Newton method is propo

2、sed to solving the extreme eigenvalue of large- scale sparse symmetric matrix. It is well known that the Hessian matrix of Rayleigh quotient function is nearly singular and bad conditioned as iteration approaches the opt

3、imal. To overcome this, a new strategy which requires that the correction vector is orthogonal to the current approximate eigenvector is proposed. Under the strategy, the Hessian matrix of the Rayleigh quotient is positi

4、ve definite as the iteration sufficiently approaches the minimal extreme eigenvalue, thus the sufficient optimal condition holds. In order to accelerate the convergence rate of the minimal extreme eigenvalue, we augment

5、the subspace by the computed modified Newton direction, thus a subspace accelerated truncated Newton method is given. The convergence of the method is proved; further numerical experiments are done to demonstrate the con

6、vergence analysis. Similar to the single eigenvalue case, block truncated Newton and subspace accelerated block truncated Newton method in several extreme eigenvalues case are proposed. Theoretical analysis and numerica