Accurate Computations of Matrix Eigenvalues with Applications to Differential Operators
主 题: Accurate Computations of Matrix Eigenvalues with Applications to Differential Operators
报告人: Qiang Ye（University of Kentucky）
时 间: 2014-01-15 10：00-11:00
地 点: 理科一号楼1479（主持人：蔡云峰）
For matrix eigenvalue problems arising in discretizations of differential operators, it is usually smaller eigenvalues that well approximate the eigenvalues of the differential operators and are of interest. The finite difference discretization leads to a standard eigenvalue problem $Ax=\\lambda x$ and the finite element method results in a generalized eigenvalue problem $Ax = \\lambda Bx$. With the condition number for the discretized problem $A$ (or $B^{-1}A$) typically large, smaller eigenvalues computed are expected to have low relative accuracy.
In this talk, we present our recent works on high relative accuracy algorithms for computing eigenvalues of diagonally dominant matrices. We present an algorithm that computes all eigenvalues of a symmetric diagonally dominant matrix to high relative accuracy. We further consider using the algorithm in an iterative method for a large scale eigenvalue problem and we show how smaller eigenvalues of finite difference discretizations of differential operators can be computed accurately.
