International Journal of Advancements in Research & Technology, Volume 2, Issue2, February-2013 1
ISSN 2278-7763
Copyright © 2013 SciResPub.
Optimal Equi-scaled Families of Ostrowski's
Method System of Nonlinear Equations
Sandeep Singh
Mathematics, D. A. V. College, Chandigarh, India.
Email:sbisht402357@gmail.com
ABSTRACT
In this paper, we present many new fourth -order optimal families of Ostrowski's method for computing zeros
of system of nonlinear equations numerically. In this paper, we extending the idea of the proposed families of
Ostrowski's method to system of nonlinear equations .It is proved that the above said families have fourth
order of convergence. Several numerical examples are also given to illustrate the efficiency and the
performance of the presented families.
Keywords: System of Nonlinear equations, Optimal Order of Convergence, Halley's method, Schroder's
method, Ostrowiski's method.
1 INTRODUCTION
ue to the fact that systems of nonlinear
equations arise frequently in science and
engineering they have attracted
researcher's interest. For example, nonlinear
systems of equations, after the necessary
processing step of implicit discretization, are
solved by finding the solutions of systems of
equations. We consider here the problem of
finding a real zero, x*= (x*1, x*2…….; x*n)
T
, of a
system of non linear equations
f1(x1, x2, ...…….., xn) = 0;
f2(x1, x2, ...…….., xn) = 0;
::::::::::::::::::::::::::::::::::::::;
::::::::::::::::::::::::::::::::::::::;
fn(x1, x2, ……….., xn) = 0;
This system can referred in vector form by
F(X) = 0 (1.1)
Where F = (f1, f2, ………, fn)
T
and
X = (x1, x2,.……..,xn)
T
Let the mapping F: D R
n
→R
n
assumed to satisfy
the assumptions (1.1). F(X) is continuously
differentiable in an open neighborhood D of X*.
There exists a solution vector X* of (1.1) in D such
that F(X*) = 0 and F’(X*) ≠ 0. Then the standard
method for finding the solution to equation (1.1) is
the classical Newton's method [2-5] given by
X
k+1
= X
k
-F(X
k
)/F’(X
k
), k = 0, 1, 2,………….. (1.2)
2 CONSTRUCTION OF NOVEL
TECHNIQES WITHOUT MEMORY
Case I: New optimal families of Ostrowski's
method. The well-known Schroder’s method [9] for
multiple zero and Halley's method [6] for simple
zero, are given by
x
୬ାଵ
= x
୬
−
( ୶
) ’( ୶
)
ᇲ
మ
( ୶
) ( ୶
) ᇱᇱ( ୶
)
(2.1)
and
x
୬ାଵ
= x
୬
−
ଶ( ୶
) ’( ୶
)
ଶ
ᇲ
మ
( ୶
) ( ୶
) ᇱᇱ( ୶
)
(2.2)
respectively.
We now intend to develop new optimal families of
Ostrowski's method the having the cubing scaling
factor of function in the correction factor. For this,
we take the Newton’s method
yn = xn - f(xn)/f’(xn) (2.3)
Similarly using the concept, we now expand the
function f(yn) ≌f(xn - f(xn)/f’(xn) ) about the point x =
xn by Taylor's series expansion, we have
f(x
୬
− f(x
୬
)/ f’(x
୬
)) ≌
f
ଶ
(x
୬
) f′′(x
୬
)
2f
ᇱଶ
(x
୬
)
therefore, we obtain
f’′(x
୬
)) ≌
మ
( ୶
)
ᇲᇲ
( ୷
)
ଶ
ᇲమ
( ୶
)
(2.4)
ݔ
ାଵ
= ݔ
−
ଵ
ଶ
[
( ୶
)
ᇲ
( ୶
)
ᇲ
మ
( ୶
) ( ୶
)
ᇲᇲ
( ୶
)
+
ଶ( ୶
)
ᇲ
( ୶
)
ଶ
ᇲ
మ
( ୶
) ( ୶
)
ᇲᇲ
( ୶
)
]
(2.5)
D