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## Accurate Computations of Matrix Eigenvalues with Applications to Differential Operators

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.Numerical examples are presented to demonstrate the high accuracy achieved by the new algorithm.