[1] Brezinski, C., Comparation de suites convergentes, Revue francaise d’informatique et de recherches operationnelles, Nr. R-2, 95–99, (1971).
[2] Pavaloiu, I., Introducere în teoria aproximarii solutiilor ecuatiilor, Editura Dacia, Cluj-Napoca, (1976).
[3] Pavaloiu, I., Sur les procedes iteratifs a un ordre eleve de convergence, Mathematica, Cluj, 12(35), 2 309–324, (1970).
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On the convergence order of the iterative methods
by
Ion Pavaloiu
In the work [ 1 ] C. Brezinski defined the notion of order of convergence of a sequence and a notion of comparison of the speeds of convergence of two sequences.
In numerical analysis, a set of methods for solving equations leads to the approximation of the solutions of the equations considered by elements of a suitably constructed sequence.
If we can construct a sequence whose terms approximate the solution of the given equation, then it is very important to know the speed of convergence of this sequence towards the solution of the equation considered.
In this note we will look for a relationship between the order of convergence of two sequences that have the same speed of convergence. We will apply the results to a problem of convergence of numerical methods for solving equations.
Eithera metric space anda sequence of elements of space
. We assume that the following
converges to the element.
We designate byany real and positive number.
Definition 1 .
[ 1 ] . We say that the followinghas the order of convergenceif there are two positive constantsAndin such a way that for each
inequalities take place
| (1) |
|
|
|
Concerning the order of convergence of a sequence, in the work [ 1 ] we demonstrate the following theorem.
Theorem 1 .
Yes, please.has the order of convergence, SOis unique.
Be it now Andtwo sequences that converge to the same limit.
Definition 2 .
[ 1 ] . We say that the followingconverges at least as quickly asif the following inequalities are verified:
| (2) |
|
|
|
Oris a positive constant independent of.
Definition 3 .
[ 1 ] We say that the suites Andhave the same speed of convergence if there are two positive constantsAnd
such that, for eachthe inequalities are satisfied:
| (3) |
|
|
|
Now we will prove the following theorem:
Theorem 2 .
If the sequelsAnd
have the same speed of convergence, and if one of the two sequences has an order of convergence, then the other sequence has the same order of convergence.
Demonstration..
To clarify the ideas we designate bythe order of convergence of the sequence. Because the two sequences have the same speed of convergence, the following inequalities result:
| (6) |
|
|
|
and taking into account that the followinghas the order of convergenceresults:
| (7) |
|
|
|
Orare positive constants, independent of.
From ( 6 ) and ( 7 ) we deduce:
| (8) |
|
|
|
Still from ( 6 ) and ( 7 ) we deduce:
| (9) |
|
|
|
From inequalities ( 8 ) and ( 9 ) the double inequality results:
| (10) |
|
|
|
where the constantsAnddo not depend on.
From Theorem 1 it follows that the real and positive numberfor which the inequalities ( 10 ) take place is unique.
Subsequently, we will apply the results presented above to specify the order of convergence of the numerical methods for solving operational equations.
LetAndtwo normed linear spaces and either
an equation, whereis an application and
is the zero element of the space.
We designate bya solution of equation ( 11 ) and we assume that we have constructed a sequenceof elements of space, which converges towards.
In practice we do not always manage to obtain the exact solution.of equation ( 11 ) but we can approximate the elementby a suitable element of the sequence
.
It is also important to know what is the convergence speed of the sequencetowards
and to evaluate what is the number of steps such that
considered as an approximation of the solutioncan replace this solution with a precision given in advance.
But we don't know the elementand this is why it is very difficult to apply the results presented in this note to study the speed of convergence of the sequence.
A simple criterion to verify that,approachis to see what the distance is between the real numberAnd. From this remark we propose to study the speed of convergence of the sequence.
We now assume that the applicationis continuous and then from the fact thatit follows that
.
Using notions similar to those presented in the first part of this work, we can clarify the notion of order of convergence for the followingwhich approaches the solution
of equation ( 11 ). ∎
Definition 4 .
We say that the sequelhas the order of convergencerelative to equation
( 11 ), if there are two positive constants,in such a way that for eachwe have:
| (12) |
|
|
|
It obviously follows from Theorem 1 that if
has the order of convergence, SO
is unique.
We now consider two sequencesAndwhich we assume have the same limitOris the solution to equation ( 11 ).
By analogy with the notions specified in definitions
2 and 3 we present the following definitions:
Definition 5 .
We say that the sequelconverges at least as quickly asrelative to equation ( 11 ) if the inequalities are verified:
| (13) |
|
|
|
Oris a constant independent of.
Definition 6 .
We say that the sequelsAndhave the same speed of convergence relative to equation ( 11 )
if there are two positive constantsAndin such a way that for eachwe have:
|
|
|
A consequence of Theorem 2 is the following.
Theorem 3 .
If the sequelsAnd
have the same speed of convergence relative to equation ( 11 ) and if one of the two sequences has an order of convergence relative to equation
( 11 ) then the other sequence has the same order of convergence relative to equation ( 11 ).
Bibliography
-
[1]
Brezinski, C., Comparison of convergent sequences , French Journal of Computer Science and Operational Research, No. R-2, 95–99, (1971).
-
[2]
Pavaloiu, I.,
††margin:
clickable
Introduction to the theory of approximation of solutions of equations , Dacia Publishing House, Cluj-Napoca, (1976).
-
[3]
Pavaloiu, I.,
††margin:
clickable
On iterative processes with a high order of convergence , Mathematica, Cluj,
12 (35), 2 309–324, (1970).
Institute of Mathematics,
PO Box 68 3400 Cluj-Napoca
