On the Transformation T+m Due Gray and Clark

We determine the values of the integer m for which the parametric transformation T+m due Gray and Clark is well conditionned. This process of acceleration being quasilinear transformation, we use an adequate definition of the condition numbers that apply to the real sequences of linear convergence. The results obtained for this set are enough meaning.


INTRODUCTION
Nonlinear sequences transformations are generally used for solve the extrapolation of the limit or to accelerate slowly convergent sequences. One can also use them to sum the divergent series [1,11] . But most of the time, the implementation of these transformations provides us with sequences whose terms are sullied with errors due to the limits imposed by the capacities of the computer on the one hand and by the truncation errors on the other hand.
To these errors, we can add the conditioning of the transformation which is used. It is thus significant to make a study of conditioning. We are interested in this work in the conditioning of the transformation T +m of Gray and Clark, which generalizes the famous ∆²-Aitken's process.
For that we use the definition [4] of the number of conditioning when the transformation considered is quasi-linear. Thereafter, we will discuss the values of the integer m for which we obtain the best conditioned transformation.
The first generalization of Aitken's ∆² process is T +m transformation due to Gray [8] and Clark ]. Let m be a strictly positive integer and consider the sequence For 1 m = , Aitken's ∆² process is recovered. Acceleration results were given by the previous authors and by Streit [14] . Let be a convergent sequence of a limit * x and let Φ be a mapping defined on ℝ p+1 , where 2 p ≥ . In the paper, we shall consider Φ as a function of ( ) is associated with the m T + transformation.

Proposition 1:
The transformation m T + of Gray and Clark is a quasilinear transformation.
Proof: For one justification , one can refer to the proof given in [3] . In this work we are interested only in stationary processes defined in the sense of Ortega [9] and Rheinboldt . Let Φ be the transformation defined before by evaluating the partial derivatives at the point i X rather than i ξ , we can give the following definition.

Definition 2:
The punctual condition number th i step of Φ for a sequence x is given by This number is in fact the factor of amplification of the errors made on the term i x . In the case where the sequence of the punctual condition numbers converges, then we can define the asymptotic conditioning of the transformation Φ , applied to the sequence x, by

Calculus of partial derivatives
It is easy to verify that This is a general result. The quasilinear transformations verify this property.

Definition 3:
We call LIN the set of linear sequences, that is the set of convergent sequences  is said to be regular if the transformed sequence is converging and to the same limit * x .

Proposition 3:
On the set LIN, the quasilinear m T + of Gray and Clark is a regular transformation.
Proof: For one justification , one can refer to the proof given in [10] .
Results in particular cases: For 1 m = , the asymptotic condition numbers ( ) This last formula was established in [10] ]and allows us to verify the work done previously.
We observe that the process of Aitken is well conditioned when applied to elements of − LIN ( ) . However, it is ill conditioned if the asymptotic ratio of the sequence (which we have to accelerate) is close to one.
For m=2, we take the following results established in [3] First case: x is an alternating sequence ( ) Concluding Results: According to the sign of ρ and the parity of m, we obtain the results:

Numerical applications
In the first table we give the initial sequence's terms computed with Digits:=30, Digits :=9 and Digits :=6. If we consider the calculus with Digits 30 := as accurate, then that done with Digits 9 := or 6 := will be done with roundoff error which will be a source of a small error. This error will be propagated in the terms of the transformed sequence. The values of transformed sequence are given in the second table. In some cases, we have chosen to study the transformation m T + for the values  The terms of the sequence ( ) n S S = approach to the limit S. Relativement à Digits := 30 , the terms n S are calculed for Digits :=9, with eight exact digits and with five exact digits for digits:= 6. We remark the roundoff errors generated to ninth digit of initial sequence terms n S are reproduced also to ninth digit of transformed

CONCLUSION
In this study and within sight of the results obtained, we can say that the transformation of Gray and Clark is well conditioned for m odd on the set of sequences to alternating convergence whereas for the other cases the asymptotic conditioning is expressed according to the integer m. We can plot the graph of the asymptotic conditioning.