TY - GEN
T1 - New families of higher order iterative methods for solving equations
AU - Hasan, Mohammed A.
PY - 2006
Y1 - 2006
N2 - In this paper, several one-parameter families of root-finding algorithms that have higher order convergence to simple and/or multiple roots have been derived. Specifically, the rth root iterations for simple and multiple zeros are analyzed. The rth root iteration family is an infinite family of rth order methods for every positive integer r, and uses only the first r - 1 derivatives. This family includes Newton's method and the square root iteration as the first and second member, respectively. In addition, this work provides analyses and generalizations of Halley's and Laguerre's iterations, and develops a procedure of deriving higher order methods of any desired order. Many important properties of the rth root iteration family and its variants are established. Some of these variants maintain a high order of convergence for multiple roots whether the multiplicity is known or not. Based on individual methods, disks containing at least one zero are derived.
AB - In this paper, several one-parameter families of root-finding algorithms that have higher order convergence to simple and/or multiple roots have been derived. Specifically, the rth root iterations for simple and multiple zeros are analyzed. The rth root iteration family is an infinite family of rth order methods for every positive integer r, and uses only the first r - 1 derivatives. This family includes Newton's method and the square root iteration as the first and second member, respectively. In addition, this work provides analyses and generalizations of Halley's and Laguerre's iterations, and develops a procedure of deriving higher order methods of any desired order. Many important properties of the rth root iteration family and its variants are established. Some of these variants maintain a high order of convergence for multiple roots whether the multiplicity is known or not. Based on individual methods, disks containing at least one zero are derived.
KW - Halley's method
KW - Hansen-patrick's family
KW - Konig's family
KW - Laguerre's method
KW - Newton's method
KW - Order of convergence
KW - Root-finding
KW - Schwartz derivative
KW - Square root iteration
KW - Zeros of analytic functions
KW - Zeros of polynomials
KW - rth root iterations
UR - http://www.scopus.com/inward/record.url?scp=39649100824&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=39649100824&partnerID=8YFLogxK
U2 - 10.1109/cdc.2006.377537
DO - 10.1109/cdc.2006.377537
M3 - Conference contribution
AN - SCOPUS:39649100824
SN - 1424401712
SN - 9781424401710
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 6379
EP - 6384
BT - Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 45th IEEE Conference on Decision and Control 2006, CDC
Y2 - 13 December 2006 through 15 December 2006
ER -