### Abstract

The matrix n-sector function is a generalization of the matrix sign function; it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues. It is known that Newton's method, which can be used for computing the matrix sign function, is not globally convergent for the matrix sector function. The only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix. In this paper, we introduce a new algorithm based on Halley's generalized iteration formula for solving nonlinear equations. It is shown that the iteration has good error propagation properties and high accuracy. Finally, we give two application examples and summarize the results of our numerical experiments comparing Newton's, the continued fraction, and Halley's method.

Original language | English (US) |
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Pages (from-to) | 944-949 |

Number of pages | 6 |

Journal | IEEE Transactions on Automatic Control |

Volume | 40 |

Issue number | 5 |

DOIs | |

State | Published - May 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Automatic Control*,

*40*(5), 944-949. https://doi.org/10.1109/9.384237

**Halley's method for the matrix sector function.** / Kaya Koc, Cetin; Bakkaloglu, Bertan.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 40, no. 5, pp. 944-949. https://doi.org/10.1109/9.384237

}

TY - JOUR

T1 - Halley's method for the matrix sector function

AU - Kaya Koc, Cetin

AU - Bakkaloglu, Bertan

PY - 1995/5

Y1 - 1995/5

N2 - The matrix n-sector function is a generalization of the matrix sign function; it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues. It is known that Newton's method, which can be used for computing the matrix sign function, is not globally convergent for the matrix sector function. The only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix. In this paper, we introduce a new algorithm based on Halley's generalized iteration formula for solving nonlinear equations. It is shown that the iteration has good error propagation properties and high accuracy. Finally, we give two application examples and summarize the results of our numerical experiments comparing Newton's, the continued fraction, and Halley's method.

AB - The matrix n-sector function is a generalization of the matrix sign function; it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues. It is known that Newton's method, which can be used for computing the matrix sign function, is not globally convergent for the matrix sector function. The only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix. In this paper, we introduce a new algorithm based on Halley's generalized iteration formula for solving nonlinear equations. It is shown that the iteration has good error propagation properties and high accuracy. Finally, we give two application examples and summarize the results of our numerical experiments comparing Newton's, the continued fraction, and Halley's method.

UR - http://www.scopus.com/inward/record.url?scp=0029304771&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029304771&partnerID=8YFLogxK

U2 - 10.1109/9.384237

DO - 10.1109/9.384237

M3 - Article

VL - 40

SP - 944

EP - 949

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 5

ER -