Ion Păvăloiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
nonlinear equations in R; iterative methods; Steffensen method.
I. Păvăloiu, On computational complexity in solving equations by Steffensen-type methods, Rev. Anal. Numér. Théor. Approx., 24 (1995) no. 2, pp. 215-220.
[1] Ostrowski, A.M., Solution of Equations and Systems of Equations, Academic Press. New York and London, 1960.
[2] Turowicz, B.A., Sur le derivee d’ordre superior d’une fonction inverse, Colloq. Math. (1959), pp. 83-87.
[3] Păvăloiu I., Optimal problems Concerning Interpolation Methods of Solution of Equations. Publications de l’Institut Mathematique, Novelle serie, 52(66) (1992), pp. 113-126.
[4] Păvăloiu I., On Computational Complexity in Solving Equations by Interpolation Methods. Revue d’analyse Numerique et Theorie de l’Approximation, 24, 1-2 (1995), pp. 201-213.
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On Computational Complexity in Solving Equations by Steffensen-Type Methods
On Computational Complexity in Solving Equations
by Steffensen-Type Methods
Ion Păvăloiu
(Cluj-Napoca)
1. Introduction
This note is a continuation of the paper [5]. We shall establish
here the optimal methods for the efficiency index of the class of
Steffensen-type methods.
We adopt the efficiency index of an iterative process as being the number
given in [2] by:
where is the convergence order of the iterative method and
represents the number of function evaluations that must be performed at each
step. As it results from [2] and [5], the
efficiency index can be defined as in (1) if we admit that the
number of function evaluations is constant beginning from a certain step.
Let denote an interval of the real axis, and consider equation
where . Suppose that equation (2) possesses a
unique root . Also suppose that admits derivatives up to the
order , the -th derivative of
is bounded on , and for all .
If , then there exists the function and .
It is obvious that for approximating the solution of (2) it is
sufficient to approximate at .
From the derivability hypotheses concerning it follows that also
possesses derivatives up to the order , which are given by [3]:
| (3) |
|
|
|
, where the above sum extends over all the integer
nonnegative solutions of the system:
| (4) |
|
|
|
|
|
|
|
|
We shall consider the following general iterative process for solving the
equation (2):
| (5) |
|
|
|
where is a function whose restriction to the diagonal
of coincides with a function , whose fixed point is
, i.e. for all
and .
In order to establish the optimal efficiency index of the class of Steffensen
methods we shall adopt, as in [5], the following assumptions:
We consider as a function evaluation:
a) the evaluation of the function or of any of its derivatives at a certain point;
b) the evaluation by (3) of any of the derivatives of at a
certain point;
c) the evaluation of from (5) at a certain point.
2. Generalized Steffensen Method
Let
be interpolation nodes from and
the values of at .
Consider natural numbers such that
, and .
Supposing that at each , we know that values of
and of its derivatives up to the order i.e. we know , by (3) we can get the values of
and of its derivatives up to the order .
We can now construct the Hermite inverse interpolation polynomial
corresponding to , nodes (7), i.e. the following polynomial
exists and is unique:
| (8) |
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where
| (9) |
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If denotes the value of at we have
| (10) |
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where
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If are approximations of ,
then a new approximation can be obtained by (8):
| (11) |
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with the error evaluation
| (12) |
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Method (11) is called Hermite-like iterative method.
Consider a function whose fixed point from is
i.e. , and suppose there
exists a real number such that
| (13) |
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Let be the iterations up to the order of
the function .
To increase the convergence order of method (11) we can do as if follows.
Let be a certain approximation of the solution of
equation (2) and . Consider the
values as
interpolation nodes in (8). Then , the next approximation
of , is given by:
| (14) |
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Repeating this process, called Steffensen type iterative method, we obtain a
sequence of approximations of .
Using (13) and (12) it can be easily seen that the
convergence order of (14) is .
3. The Efficiency Index of Steffensen-Type Methods
As it can be seen above, at each iteration step in (14) we have the
following function evaluations:
1) values of to obtain the interpolation nodes ;
2) values of at the nodes ;
3) at each interpolation node we compute the values
of successive derivatives of up to the order , altogether
function evaluations:
4) by (3) we evaluate the successive derivatives of at
up to the order
altogether function evaluations;
5) finally, consider (14) as a single function evaluation.
Summing up, we obtain altogether function evaluations.
Using (1) we obtain the following expression for the efficiency
index of the class of Steffensen-type methods:
| (15) |
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Elementary considerations on the behaviour of the function
lead us to the conclusion that the function attains its maximum at .
Note that the efficiency index (15) does not depend on the number of
interpolation nodes.
From and it
follows that .
We shall successively analyse all the cases that lead us to the optimal
methods form (14).
A. i.e. . Then
(8) becomes the Lagrange’s inverse interpolation polynomial, and
(14) is written:
| (16) |
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where respectively denote
the first, respectively the second order divided differences of .
B. i.e. or and
. When we obtain the following method:
| (17) |
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and when it follows:
| (18) |
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C. . In this case we get from (14) the third order
Chebyshev iterative method, studied in [5].
In conclusion, the following theorem holds:
Theorem 3.1.
Under the assumptions a)–c) from 1., in the class of
Steffensen-type iterative methods any of the methods (16),
(17) or (18) is optimal, i.e. has the greatest
efficiency index.
References
-
[1]
-
[2]
Ostrowski, A.M., Solution of Equations and
Systems of Equations. Academic Press. New York and London, 1960.
-
[3]
Turowicz, B.A., Sur le dérivée d’ordre
supérior d’une fonction inverse. Colloq. Math. (1959), pp. 83–87.
-
[4]
††margin:
clickable
Păvăloiu, I.,
Optimal problems concerning
interpolation methods of solution of equations, Publications de l’Institut
Mathématique, Nouvelle série, 52 (66), (1972), pp. 113–126.
-
[5]
††margin:
clickable
Păvăloiu, I.,
On computational complexity
in solving equations by interpolation methods, Rev. Anal. Numér.
Théor. Approx., 24, 1–2 (1995), pp. 201–213.
