We extend the Aitken-Steffensen method to the Halley transformation. Under some rather simple assumptions we obtain error bounds for each iteration step; moreover, the convergence order of the iterates is 3, i.e. higher than for the Aitken-Steffensen case.
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
nonlinear equations in R; Aitken-Steffensen-Halley method.
[1] Argyros, I.K. On the convergence of some projection methods with perturbations, J. Comp. Appl. Math. 36, (1991), 255–258.
[2] Argyros, I.K. On an application of the Zincenko method to the approximation of implicit functions, Z.A.A. 10, 3, (1991), 391– 396.
[3] Argyros, I.K. and Szidarovszky, F. The Theory and Application of Iteration Methods, C.R.C. Press, Inc., Boca Raton, Florida, 1993.
[4] Catinas, E. On some iterative methods for solving nonlinear equations, Revue d’analyse Numerique et de theorie de l’approximation, 23, 1, (1994), 47–53.
[5] Kantorovich, L.V. The method of successive approximation for functional equations, Acta Math. 71 (1939), 63–97.
[6] Pavaloiu, I. Sur une generalisation de la methode de Steffensen, Revue d’analyse Numerique et de theorie de l’approximation, 21, 1, (1992), 59–65.
[7] Pavaloiu, I. Bilateral approximations for the solutions of scalar equations, Revue d’analyse numerique et de theorie de l’approximation, 23, 1, (1994), 95–100.
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ON SOME AITKEN-STEFFENSEN-HALLEY-TYPE METHODS FOR APPROXIMATING THE ROOTS OF SCALAR EQUATIONS
ON SOME AITKEN-STEFFENSEN-HALLEY-TYPE METHODS
FOR APPROXIMATING THE ROOTS OF SCALAR EQUATIONS
ION PĂVĂLOIU∗
Abstract.
In this note we extend the Aitken-Steffensen method to the Halley
transformation. Under some rather simple assumptions we obtain error bounds
for each iteration step; moreover, the convergence order of the iterates is
, i.e. higher than for the Aitken-Steffensen case.
1991 Mathematics Subject Classification:
65H05
∗”T. Popoviciu” Institute of Numerical Analysis,
P.O. Box 68-1, 3400 Cluj–Napoca, Romania, e-mail:
pavaloiu@ictp-acad.math.ubbcluj.ro.
1. Introduction
Let , ,
and suppose that , and , . Consider the function
given by
As it was shown in [2], the Halley method for solving:
is given by
| (2) |
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|
This sequence is in fact generated by the Newton method for solving
The first and second order derivatives of are given by
| (3) |
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| (4) |
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which yield the following equalities for a solution of
(1):
| (5) |
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| (6) |
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Relation (6) ensures the convergence order for the sequence
.
In the papers [2]–[8] and [12] there are studied
the convergence and the convergence order of some sequences generated by some
interpolatory methods applied to equation .
We shall consider other two equations equivalent to (1) of the form:
| (7) |
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| (8) |
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The Aitken method for solving is given by the
iteration
| (9) |
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In this note we shall study the convergence of these iterates. We shall show
that the functions and may be chosen in order to
obtain bilateral approximations at each iteration step; this fact allows the
control of the errors. On the other hand, the convergence order of given by (9) is at least equal to .
Hypotheses and , imply, taking into account
(5), that there exist such that .
2. ERROR EVALUATION AND LOCAL CONVERGENCE
Consider the interval given above. We shall
make the following assumptions on and :
-
i.
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ii.
equation (1) has the solution
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iii.
the inequality holds for
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iv.
the function verifies the relation for all ,
where denotes the first order divided
difference of on and ;
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v.
the function verifies the relations for all
We can state the following result:
Theorem 2.1.
Assume that i–v hold, and for some sufficiently close to we have
Then the following relations hold:
-
j.
the sequences and converge to ;
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jj.
for any one has
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jjj.
there exists which does not depend on
such that
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Proof.
By it
obviously follows that and Denote by the interval having the extremities
and We notice, taking into account
the mean formula, that
When by iv. and it follows
. Analogously, for
Taking into account v. and we
get for and for It is obvious that in both situations
It can be easily seen that for all we have
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which, for imply and if respectively and
if i.e.,
It is clear now that, analogously, if or if Denoting by the interval
determined by and then
and the element constructed by (9) satisfies
Repeating the above reason we get the interval
being determined by and also that
and It is clear now that jj. holds.
In order to obtain jjj. we shall use the identity
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which, together with (9) and , imply
|
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For the difference , by iv. one
gets
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i.e.,
Analogously, by v. we get
The mean formula for divided differences implies
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For we have
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Since it follows
Denoting the above relations lead
to
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i.e., jjj. for .
Since the initial approximation was supposed sufficiently close to the
solution then
implies, together with jjj., statement j.
∎
3. DETERMINING THE FUNCTIONS AND
Under reasonable hypotheses on , we shall show that there exist two
classes of functions among we can choose the functions and
such that hypotheses iv. and v. to be satisfied.
Besides assumptions i.–iii. on we shall suppose that is strictly
convex, i.e., This
condition implies that is increasing on . If, moreover,
with then we may
consider the functions
| (10) |
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| (11) |
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where may be taken as any real number greater than the left derivative
of at
In the following we shall show that the functions and chosen above obey iv. and v.
The derivatives of these functions are given by
|
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Obviously, Also, the
monotonicity of implies , while implies
Taking into account Remark 2.1, under the above hypotheses it is obvious that
if then condition from Theorem 2.1 is obviously satisfied.
Indeed, this fact follows from
since i.e.,
On the other hand, and so the relations
hold. The hypothesis
must be kept.
References
-
[1]
A. Ben-Israel, Newton’s
method with modified functions, Contemp. Math., 204, pp. 39–50,
1997.
-
[2]
G.H. Brown, Jr., On Halley’s
variation of Newton’s method, Amer. Math. Monthly, 84, pp. 726–728,
1977.
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[3]
V. Candela and A. Marquina,
Recurrence relations for rational cubic methods I: The Halley’s
method, Computing, 44, pp. 169–184, 1990.
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[4]
G. Deslauries and S. Dubuc,
Le calcul de la racine cubique selon Héron, El.
Math., 51, pp. 28–34, 1996.
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[5]
W.F. Ford and J.A. Pennline,
Accelerated convergence in Newton method, SIAM Rev.,
38, pp. 658–659, 1996.
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[6]
J. Gerlach, Accelerated
convergence in Newton’s method, SIAM Rev., 36, pp. 272–276, 1994.
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[7]
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clickable
D. Luca and I. Păvăloiu,
On the Heron’s method for the approximation of the cubic
root of a real number,
Rev. Anal. Numér. Théor. Approx., 28,
pp. 103–108, 1997.
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[8]
A. Melman, Geometry and
convergence of Euler’s and Halley’s methods, SIAM Rev., 39,
pp. 728–735, 1997.
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[9]
A.M. Ostrowski, The Solution
of Equations and Systems of Equations, Academic Press, New York–London,
1960.
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[10]
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I. Păvăloiu,
On the monotonicity of the sequences of approximations obtained by Stef- fensen
method, Mathematica (Cluj), 35 (58), pp. 171–76, 1993.
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[11]
††margin:
clickable
I. Păvăloiu,
Approximation of the roots of equations by Aitken–Steffensen-type
mono- tonic sequences, Calcolo, 32, pp. 69–82, 1995.
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[12]
††margin:
clickable
I. Păvăloiu,
On a Halley-Steffensen method for approximating the solutions of scalar
equations, Rev. Anal. Numér. Théor. Approx., 30, 2001 no. 1, pp. 69–74.
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[13]
T. Popoviciu,
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Sur la
délimitation de l’erreur dans l’approximation des racines d’une
équation par interpolation linéaire ou quadratique, Rev. Roumaine
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[14]
