A computational comparison of the first nine members of a determinantal family of root-finding methods

Bahman Kalantari, Seungyoung Park

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

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, Bm (k) defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287-318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96-109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials Bm (k-1) is more efficient than Bm (k), but as the degree increases, Bm (k) becomes more efficient than Bm (k-1). The most efficient of the nine methods is B4 (4), having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.

Original languageEnglish (US)
Pages (from-to)197-204
Number of pages8
JournalJournal of Computational and Applied Mathematics
Volume130
Issue number1-2
DOIs
StatePublished - May 1 2001

Fingerprint

Root-finding
Polynomials
Order of Convergence
Newton-Raphson method
Natural number
Newton Methods
Iteration Function
Halley's Method
Derivatives
Secant Method
Polynomial
Less than or equal to
Computational Results
Determinant
Derivative
Family

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

@article{03389206b60e4ef6a61b2184bcf3baef,
title = "A computational comparison of the first nine members of a determinantal family of root-finding methods",
abstract = "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, Bm (k) defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287-318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96-109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials Bm (k-1) is more efficient than Bm (k), but as the degree increases, Bm (k) becomes more efficient than Bm (k-1). The most efficient of the nine methods is B4 (4), having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.",
author = "Bahman Kalantari and Seungyoung Park",
year = "2001",
month = "5",
day = "1",
doi = "10.1016/S0377-0427(99)00383-0",
language = "English (US)",
volume = "130",
pages = "197--204",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",
number = "1-2",

}

A computational comparison of the first nine members of a determinantal family of root-finding methods. / Kalantari, Bahman; Park, Seungyoung.

In: Journal of Computational and Applied Mathematics, Vol. 130, No. 1-2, 01.05.2001, p. 197-204.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A computational comparison of the first nine members of a determinantal family of root-finding methods

AU - Kalantari, Bahman

AU - Park, Seungyoung

PY - 2001/5/1

Y1 - 2001/5/1

N2 - 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, Bm (k) defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287-318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96-109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials Bm (k-1) is more efficient than Bm (k), but as the degree increases, Bm (k) becomes more efficient than Bm (k-1). The most efficient of the nine methods is B4 (4), having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.

AB - 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, Bm (k) defined as the ratio of two determinants that depend on the first m - k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287-318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96-109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials Bm (k-1) is more efficient than Bm (k), but as the degree increases, Bm (k) becomes more efficient than Bm (k-1). The most efficient of the nine methods is B4 (4), having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.

UR - http://www.scopus.com/inward/record.url?scp=0035336353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035336353&partnerID=8YFLogxK

U2 - 10.1016/S0377-0427(99)00383-0

DO - 10.1016/S0377-0427(99)00383-0

M3 - Article

VL - 130

SP - 197

EP - 204

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -