### Abstract

It is well known that Halley's method can be obtained by applying Newton's method to the function f/f′. Gerlach (SIAM Rev. 36 (1994) 272-276) gives a generalization of this approach, and for each m≥2, recursively defines an iteration function G_{m}(x) having order m. Kalantari et al. (J. Comput. Appl. Math. 80 (1997) 209-226) and Kalantari (Technical Report DCS-TR 328, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1997) derive and characterize a determinantal family of iteration functions, called the Basic Family, B_{m}(x), m≥2. In this paper we prove, G_{m}(x)=B_{m}(x). On the one hand, this implies that G_{m}(x) enjoys the previously derived properties of B_{m}(x), i.e., the closed formula, its efficient computation, an expansion formula which gives its precise asymptotic constant, as well as its multipoint versions. On the other, this gives a new insight on the Basic Family and Newton's method.

Original language | English (US) |
---|---|

Pages (from-to) | 195-200 |

Number of pages | 6 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 116 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1 2000 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Keywords

- 65H05
- Newton's method
- Polynomial roots
- Rootfinding

### Cite this

}

*Journal of Computational and Applied Mathematics*, vol. 116, no. 1, pp. 195-200. https://doi.org/10.1016/S0377-0427(99)00361-1

**Newton's method and generation of a determinantal family of iteration functions.** / Kalantari, Bahman; Gerlach, Jürgen.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Newton's method and generation of a determinantal family of iteration functions

AU - Kalantari, Bahman

AU - Gerlach, Jürgen

PY - 2000/4/1

Y1 - 2000/4/1

N2 - It is well known that Halley's method can be obtained by applying Newton's method to the function f/f′. Gerlach (SIAM Rev. 36 (1994) 272-276) gives a generalization of this approach, and for each m≥2, recursively defines an iteration function Gm(x) having order m. Kalantari et al. (J. Comput. Appl. Math. 80 (1997) 209-226) and Kalantari (Technical Report DCS-TR 328, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1997) derive and characterize a determinantal family of iteration functions, called the Basic Family, Bm(x), m≥2. In this paper we prove, Gm(x)=Bm(x). On the one hand, this implies that Gm(x) enjoys the previously derived properties of Bm(x), i.e., the closed formula, its efficient computation, an expansion formula which gives its precise asymptotic constant, as well as its multipoint versions. On the other, this gives a new insight on the Basic Family and Newton's method.

AB - It is well known that Halley's method can be obtained by applying Newton's method to the function f/f′. Gerlach (SIAM Rev. 36 (1994) 272-276) gives a generalization of this approach, and for each m≥2, recursively defines an iteration function Gm(x) having order m. Kalantari et al. (J. Comput. Appl. Math. 80 (1997) 209-226) and Kalantari (Technical Report DCS-TR 328, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1997) derive and characterize a determinantal family of iteration functions, called the Basic Family, Bm(x), m≥2. In this paper we prove, Gm(x)=Bm(x). On the one hand, this implies that Gm(x) enjoys the previously derived properties of Bm(x), i.e., the closed formula, its efficient computation, an expansion formula which gives its precise asymptotic constant, as well as its multipoint versions. On the other, this gives a new insight on the Basic Family and Newton's method.

KW - 65H05

KW - Newton's method

KW - Polynomial roots

KW - Rootfinding

UR - http://www.scopus.com/inward/record.url?scp=0033880273&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033880273&partnerID=8YFLogxK

U2 - 10.1016/S0377-0427(99)00361-1

DO - 10.1016/S0377-0427(99)00361-1

M3 - Article

AN - SCOPUS:0033880273

VL - 116

SP - 195

EP - 200

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -