There are two objectives for this paper. Firstly, we shall introduce the Thukral-determinantal formula, and secondly, we shall demonstrate the similarities between the well-established algorithms, namely the Aitkin Δ^{2} algorithm and the Durbin sequence transformation. In fact, we have found that the solution of the Thukral-determinantal formula is equivalent to the Thukral-sequence transformation formula.

In almost all areas of numerical mathematics, there are convergence problems. In such cases, either the convergence is slow or even divergence is observed and accordingly many techniques for accelerating convergence have been devised. For example; Aitkin

The structure of this paper is as follows: In Section 2, we state the essential definitions relevant to the present work. In Section 3, we shall define the Thukral-determinantal formula, and in process we will demonstrate the similarities between the Aitkin

In order to construct an iterative scheme, the following assumptions are essential and are given in [

Let us assume that

The basic assumption of all sequence transformation is that a sequence element

The conventional approach of evaluating an infinite series consists of adding up so many terms that the error

Furthermore, let us assume the two sequences

In this section, we shall define the Thukral-determinantal formula for accelerating the convergence of sequences. In the process, we shall demonstrate that the Thukral-determinantal formula is in fact equivalent to the Thukral-sequence transformation formula and show a connection with the Brezinski-Durbin-Redivo-Zaglia [

In order to obtain the identical results between the Thukral-determinantal formula and the Thukral-sequence transformation we will modify the Brezinski-Durbin-Redivo-Zaglia formula, and define a new version, namely Thukral-sequence transformation as

We shall demonstrate the similarities between the Thukral-determinantal formula and Thukral-sequence transformation for several values of

For case

By cancelling common factors and simplifying

Using the Thukral-sequence transformation

Expanding and simplifying

The Aitkin

We progress to

Expanding the numerator and denominator of

Again, we cancel the common factors and simplify

The second scheme of the Thukral-sequence transformation

The

Here also we find that the expressions

For the purpose and motivation of this paper, we shall demonstrate similarities between the Thukral-determinantal formula and the Durbin sequence transformation for the next case.

When

Evaluating the above equation, we get

As before, we cancel the common factors of

We expand

By cancelling similar terms and simplifying

Which is identical to the Durbin sequence transformation [

Expanding and simplifying

Consequently, for

The Thukral-determinantal formula for

Expanding

Eliminating the common factors and simplifying

The Thukral-sequence transformation

It is apparent that

The Thukral-determinantal formula for

Similarly, we expand

By cancelling these factors and simplifying

Expanding the Thukral-sequence transformation

We observe that the equations of the Thukral-determinantal formula and the Thukral-sequence transformation formula given by

In general, we define the Thukral-determinantal formula as

Mathematically we have demonstrated that the Thukral-determinantal

In this study, we have introduced the Thukral-determinantal formula and the Thukral-sequence transformation formula for accelerating the convergence of a sequence. It is well established that the Durbin sequence transformation has not been presented in a ratio of two determinants [