Particular rules for the Θ-algorithm

The $\Theta$-algorithm is an extrapolation algorithm which can be very useful in accelerating some slowly convergent sequences. Like the other acceleration algorithms, the $\Theta$-algorithm is quite sensitive to the propagation of rounding errors due to cancellation in the difference between two almost equal quantities. In order to (partially) avoid this drawback, particular rules are given. They have to be used, instead of the usual rules of the algorithm, when two adjacent quantities in a column are nearly equal. Numerical examples show that these rules can improve the numerical stability of the algorithm in some cases while, in other cases, the improvement is non-existent.