Abstract
An efficient refinement algorithm is proposed for symmetric eigenvalue problems. The structure of the algorithm is straightforward, primarily comprising matrix multiplications. We show that the proposed algorithm converges quadratically if a modestly accurate initial guess is given, including the case of multiple eigenvalues. Our convergence analysis can be extended to Hermitian matrices. Numerical results demonstrate excellent performance of the proposed algorithm in terms of convergence rate and overall computational cost, and show that the proposed algorithm is considerably faster than a standard approach using multiple-precision arithmetic.
Highlights
Let A be a real symmetric n ×n matrix
We show that the proposed algorithm converges quadratically if a modestly accurate initial guess is given, including the case of multiple eigenvalues
Numerical results demonstrate excellent performance of the proposed algorithm in terms of convergence rate and overall computational cost, and show that the proposed algorithm is considerably faster than a standard approach using multiple-precision arithmetic
Summary
Let A be a real symmetric n ×n matrix. We are concerned with the standard symmetric eigenvalue problem Ax = λx, where λ ∈ R is an eigenvalue of A and x ∈ Rn is an eigenvector of A associated with λ. Simultaneous iteration or Grassmann–Rayleigh quotient iteration [1] can potentially be used to refine eigenvalue decompositions Such methods require higher-precision arithmetic for the orthogonalization of approximate eigenvectors. Wilkinson [30, Chapter 9, pp.637–647] explained the refinement of eigenvalue decompositions for general square matrices with reference to Jahn’s method [6,19] Such methods rely on a similarity transformation C := X −1 AX with high accuracy for a computed result X for X , which requires an accurate solution of the linear system XC = AX for C, and slightly breaks the symmetry of A due to nonorthogonality of X.
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