# On the Order of Convergence of a Determinantal Family of Root-Finding Methods

Research output: Contribution to journalArticle

18 Citations (Scopus)

### Abstract

Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m-k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.

Original language English (US) 96-109 14 BIT Numerical Mathematics 39 1 https://doi.org/10.1023/A:1022321325108 Published - Jan 1 1999

### Fingerprint

Root-finding
Order of Convergence
Natural number
Determinant
Generalized Fibonacci numbers
Iteration Function
Halley's Method
Hessenberg Matrix
Toeplitz matrix
Characteristic polynomial
Less than or equal to
Newton Methods
Newton-Raphson method
Roots
Derivative
Decrease
Polynomials
Derivatives
Family

### All Science Journal Classification (ASJC) codes

• Software
• Computer Networks and Communications
• Computational Mathematics
• Applied Mathematics

### Keywords

• Generalized Fibonacci numbers
• Halley's method
• Interpolation
• Iteration functions
• Newton's method
• Roots
• Taylor's theorem

### Cite this

title = "On the Order of Convergence of a Determinantal Family of Root-Finding Methods",
abstract = "Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m-k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.",
keywords = "Generalized Fibonacci numbers, Halley's method, Interpolation, Iteration functions, Newton's method, Roots, Taylor's theorem",
author = "Bahman Kalantari",
year = "1999",
month = "1",
day = "1",
doi = "10.1023/A:1022321325108",
language = "English (US)",
volume = "39",
pages = "96--109",
journal = "BIT",
issn = "0006-3835",
publisher = "Springer Netherlands",
number = "1",

}

In: BIT Numerical Mathematics, Vol. 39, No. 1, 01.01.1999, p. 96-109.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the Order of Convergence of a Determinantal Family of Root-Finding Methods

AU - Kalantari, Bahman

PY - 1999/1/1

Y1 - 1999/1/1

N2 - Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m-k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.

AB - Recently, we have shown that for each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, defined as the ratio of two determinants that depend on the first m-k derivatives of the given function. For each k the corresponding matrices are upper Hessenberg matrices. Additionally, for k = 1 these matrices are Toeplitz matrices. The goal of this paper is to analyze the order of convergence of this fundamental family. Newton's method, Halley's method, and their multi-point versions are members of this family. In this paper we also derive these special cases. We prove that for fixed m, as k increases, the order of convergence decreases from m to the positive root of the characteristic polynomial of generalized Fibonacci numbers of order m. For fixed k, the order of convergence increases in m. The asymptotic error constant is also derived in terms of special determinants.

KW - Generalized Fibonacci numbers

KW - Halley's method

KW - Interpolation

KW - Iteration functions

KW - Newton's method

KW - Roots

KW - Taylor's theorem

UR - http://www.scopus.com/inward/record.url?scp=0004967376&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0004967376&partnerID=8YFLogxK

U2 - 10.1023/A:1022321325108

DO - 10.1023/A:1022321325108

M3 - Article

AN - SCOPUS:0004967376

VL - 39

SP - 96

EP - 109

JO - BIT

JF - BIT

SN - 0006-3835

IS - 1

ER -