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The DDR Method of Polynomial Multiple Root Determination. The links below are to the Sections of the paper on a new method, the Differential Division Remainder, (DDR), method, for the precise determination of the roots of high order polynomials, when those roots occur in multiple pairs. The method utilises the fact that the remainder ratios of the result of dividing the polynomial by its first derivative, are polynomic functions of the primary root. |
Introduction - The first section contains a title page, a brief abstract, and the main introduction. The Differential Division Remainder Method - The second section provides details of those polynomials to which this method is applicable. A decription of the method itself is also provided, in the form of the analysis of a representative 7th order polynomial, containing a pair of roots in the combination 4 times primary root plus 3 times secondary root. Algorithms for all applicable root combinations for all polynomials of orders 3 to 10, is then presented in tabular form. Generalisation - The third Section provides a series of algorithms for extending the method to any order polynomial. These include algorithms for determining the applicable differential division remainder ratios, and algorithms for calculation of the primary and secondary roots, including the special case of the root combination of (n-2) multiple roots, and two singular roots in an nth order polynomial. Implementation of the Method - The fourth Section provides a brief description of manual implementation of the method, which is augmented in the first Appendix with two detailed examples. The computer implementation of the method, via a macro driven EXCEL spreadsheet, is then described in detail. The main emphesis in this description, is the avoidance of problems associated with EXCEL's rounding errors, due to its restriction of utilising only 15 digits of significant places in the representation of all numbers. This avoidance is achieved by utilising all the unused differential remainder term ratios, together with a check on the polynomial constant term, as the root verification process. Conclusions - Concluding Remarks. Appendix A - The first Appendix provides two examples of the manual implementation of the method, via the analysis of two representative 7th order polynomials, containing two forms of multiple root pairs. It also provides a comparison of the improved accuracy the DDR Method against that of the Bairstow Root Finding Method Appendix B - The second Appendix provides a complete listing of all the differential division remainder term ratios for all applicable polynomials of orders 3 to 10. |
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