Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

Hung Yuan Fan; Eric King wah Chu

doi:10.1016/j.cam.2016.10.018

Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

Hung Yuan Fan, Eric King wah Chu^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton's method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P₂P₂^⊤A−γI_n or P_l2P_l2^⊤(A−γB), for some orthonormal P₂ or P_l2 from R^n×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.

Original language	English
Pages (from-to)	70-86
Number of pages	17
Journal	Journal of Computational and Applied Mathematics
Volume	315
DOIs	https://doi.org/10.1016/j.cam.2016.10.018
Publication status	Published - 2017 May 1

Keywords

Deflating subspace
Invariant subspace
Large-scale problem
Nonsymmetric algebraic Riccati equation
Sparse matrix
Sylvester equation

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2016.10.018

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title = "Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces",

abstract = "We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton's method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P2P2⊤A−γIn or Pl2Pl2⊤(A−γB), for some orthonormal P2 or Pl2 from Rn×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.",

keywords = "Deflating subspace, Invariant subspace, Large-scale problem, Nonsymmetric algebraic Riccati equation, Sparse matrix, Sylvester equation",

author = "Fan, {Hung Yuan} and Chu, {Eric King wah}",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier B.V.",

year = "2017",

month = may,

day = "1",

doi = "10.1016/j.cam.2016.10.018",

language = "English",

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journal = "Journal of Computational and Applied Mathematics",

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T1 - Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

AU - Fan, Hung Yuan

AU - Chu, Eric King wah

PY - 2017/5/1

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N2 - We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton's method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P2P2⊤A−γIn or Pl2Pl2⊤(A−γB), for some orthonormal P2 or Pl2 from Rn×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.

AB - We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton's method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P2P2⊤A−γIn or Pl2Pl2⊤(A−γB), for some orthonormal P2 or Pl2 from Rn×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.

KW - Deflating subspace

KW - Invariant subspace

KW - Large-scale problem

KW - Nonsymmetric algebraic Riccati equation

KW - Sparse matrix

KW - Sylvester equation

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JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

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Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

Abstract

Keywords

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