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

Hung Yuan Fan, Eric King wah Chu

Research output: Contribution to journalArticle

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 P2P2A−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.

Original languageEnglish
Pages (from-to)70-86
Number of pages17
JournalJournal of Computational and Applied Mathematics
Volume315
DOIs
Publication statusPublished - 2017 May 1

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Nonsymmetric Algebraic Riccati Equation
Riccati equations
Refining
Subspace
Sylvester Equation
Invariant
Orthonormal
Newton-Raphson method
Error Analysis
Newton Methods
Estimate
Error analysis
Inversion
Refinement
Engine
Numerical Solution
Engines
Numerical Examples

Keywords

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

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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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.",
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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.

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