3.0  Conclusions.

The enhancements of the basic Bairstow Method of polynomial root extraction introduced here has, it is believed, improved both its functionality and its applicability. In its spreadsheet application, elimination of the divergence and hunting problems have ensured that it will always converge to a solution. Accuracy has in some cases been improved by the introduction of the algorithms to identify both n, and (n - 1) identical roots, and the part multiplicity root algorithm. However, this still leaves the inaccuracies for all other configurations of multiple roots, albeit they can be corrected via the Polynomial Construction spreadsheet facility. This however, in the case of complex roots, and/or high precision, can be a difficult and time consuming task, and ideally, if these roots could be detected accurately, could be dispensed with.

Further improvement can be achieved in essentially three ways, as follows,

(i) Improving accuracy for all other configurations of multiple roots. Clearly this is the most important improvement target, and further algorithms for this purpose are being developed.

(ii) Extending the order of polynomials that can be handled. This is merely an extension of spreadsheet and macro coding. However, as the order is increased, the amount of extra coding in the Polynomial Construction spreadsheet becomes, for one individual, prohibitively high.

(iii) Extending the precision of the roots found. This can initially be achieved by simply increasing the number of decimal points displayed in the output. However, if very high precision were required, it would need modification of Bairstow's original concept to include the second order terms in the Taylor series expansion of r and s. This has been suggested elsewhere.

On to the Next Section:- Appendix A

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