Current Inverse Iteration Software can Fail | Center for Big Data Analytics

Abstract: Inverse Iteration is widely used to compute the eigenvectors of a matrix once accurate eigenvalues are known. We discuss various issues involved in any implementation of inverse iteration for real, symmetric matrices. Current implementations resort to reorthogonalization when eigenvalues agree to more than three digits relative to the norm. Such reorthogonalization can have unexpected consequences. Indeed, as we show in this paper, the implementations in EISPACK [18] and LAPACK [1] may fail. We illustrate with both theoretical and empirical failures.

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Current Inverse Iteration Software can Fail (pdf, software)
I. Dhillon.
BIT Numerical Mathematics 38(4), pp. 685-704, December 1998.

Bibtex:
@article{dhillon1998currentin, author = "Inderjit S. Dhillon", title = "Current Inverse Iteration Software can Fail", journal = "BIT Numerical Mathematics", page = "685–704", volume = "38", issue = "4", year = "1998", month = "dec", abstract = "Inverse Iteration is widely used to compute the eigenvectors of a matrix once accurate eigenvalues are known. We discuss various issues involved in any implementation of inverse iteration for real, symmetric matrices. Current implementations resort to reorthogonalization when eigenvalues agree to more than three digits relative to the norm. Such reorthogonalization can have unexpected consequences. Indeed, as we show in this paper, the implementations in EISPACK [18] and LAPACK [1] may fail. We illustrate with both theoretical and empirical failures." }