Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils

Abstract

We consider the refinement of estimates of invariant (or deflating) subspaces for a large and sparse real matrix (or pencil) in \(\mathbb {R}^{n \times n}\), through some (generalized) nonsymmetric algebraic Riccati equations or their associated (generalized) Sylvester equations via Newton’s method. The crux of the method is the inversion of some well-conditioned unstructured matrices via the efficient and stable inversion of the associated structured but near-singular matrices. All computations have complexity proportional to \(n\), under appropriate assumptions, as illustrated by several numerical examples.

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Acknowledgments

The first author has been supported by the NSC, Taiwan Grant Number NSC 102-2115-M-003-009, and the second author by a Monash Graduate Scholarship and a Monash International Postgraduate Research Scholarship. Part of the work was completed when the third author visited the Shanghai Key Laboratory of Contemporary Applied Mathematics at FuDan University and National Taiwan Normal University.

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Correspondence to Peter Chang-Yi Weng.

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Communicated by Miloud Sadkane.

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Fan, HY., Weng, P.CY. & Chu, E.Kw. Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils. Bit Numer Math 54, 147–169 (2014). https://doi.org/10.1007/s10543-014-0469-1

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Keywords

  • Deflating subspace
  • Invariant subspace
  • Large-scale problem
  • Newton’s method
  • Nonsymmetric algebraic Riccati equation
  • Sparse matrix
  • Sylvester equation

Mathematics Subject Classification (2000)

  • 15A18
  • 15A22
  • 15A24
  • 65F15
  • 65F50