1.0  Introduction.

There are many methods of determining the roots of linear and non-linear equations of a single variable as delineated in [1]. Perhaps the most widely known is the Newton-Raphson Method, within a population that ranges from the very simple Bi-Section Method, up to the sophisticated Jenkins-Traub Method, which can deal directly with equations with complex coefficients.

All of these methods use a process of iteration starting from one or more ïnitial guesses", to home in on one or two roots simultaneously, and then reducing the original equation accordingly to repeat the process until all the roots have been found. They all enjoy varying degrees of success, depending upon their sophistication and complexity. However, they are also subject to a number of failure mechanisms, and restrictions in application, and there does not appear to be one single process that will reliably deal with all types of equations.

The method to be studied here is known as Bairstow's Method for polynomials, and was developed by Sir Leonard Bairstow and first published in an Appendix in his book, Äpplied Aerodynamics" in 1920. This method is particularly appealing because its convergence is quadratic, and its mathematical technique can easily be implemented in a computer spreadsheet, so eliminating a considerable amount of computer code development. However, Bairstow's Method also suffers from a number of restrictions and failure mechanisms, which have resulted in it being of limited use.

It is the purpose of this paper to analyse these problems and eliminate them in two experimental spreadsheets that can be downloaded from the website as a ZIP file. Of the two spreadsheets provided, the first, "Bairstow.xls", is the main one which will determine the roots of any polynomial with real coefficients up to order 10, from the input of its coefficients. It is similar to, but more extensive than, the spreadsheet application of Bairstow's Method presented in [2]. The second spreadsheet, "Polynomial Construction.xls", serves three purposes, (i) construction of any polynomial up to order 10 from the input of its roots, both real and complex, (ii) iteration of the roots of a polynomial to make minor adjustments to its generated coefficients, and (iii) the multiplication together of two polynomials each of order up to 5, again with both real and complex coefficients. The in depth details and usage of both of these spreadsheets is described more fully in the later text.

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