### 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 P_{2}P_{2}^{⊤}A−γI_{n} or P_{l2}P_{l2}^{⊤}(A−γB), for some orthonormal P_{2} 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 |
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Pages (from-to) | 70-86 |

Number of pages | 17 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 315 |

DOIs | |

Publication status | Published - 2017 May 1 |

### Fingerprint

### 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

**Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces.** / Fan, Hung Yuan; Chu, Eric King wah.

Research output: Contribution to journal › Article

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TY - JOUR

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

Y1 - 2017/5/1

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

UR - http://www.scopus.com/inward/record.url?scp=84996555426&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84996555426&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2016.10.018

DO - 10.1016/j.cam.2016.10.018

M3 - Article

AN - SCOPUS:84996555426

VL - 315

SP - 70

EP - 86

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -