DOI: 10.17587/prin.8.422-432

Algorithms for Symbolic Solving of Algebraic Equations

I. S. Astapov, velais@imec.msu.ru, Institute of Mechanics Lomonosov Moscow State University, Moscow, 119192, Russian Federation, N. S. Astapov, nika@hydro.nsc.ru, Lavrentyev Institute of Hudrodynamics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, 630090, Russian Federation

Corresponding author: Astapov Nikolay St., Senior Researcher, Lavrentyev Institute of Hudrodynamics SB RAS, Novosibirsk, 630090, Russian Federation, E-mail: nika@hydro.nsc.ru

In physical and technical simulations there is often a need to express symbolically (not numerically) the roots of algebraic equations as functions of the symbolic coefficients of these equations for a subsequent study of the phenomena of interest. In many technical problems we have a cubic equation with three real roots. In this case the well-known solutions (Ferro-Tartaglias and Ferrari-Cardanos formulas) are difficult to implement accurately, since there is no efficient algorithm for finding the cube roots of a complex number. The general polynomial equations of degree five or higher have no solutions in radicals, although they can be expressed in terms of generalized hyper-geometric functions of the coefficients of these equations. However, some special types of equations, for example, xn - a = 0, can be solved in radicals. Therefore, it is important to find simple algorithms to solve algebraic equations, in particular, of degree three to eight. The bulk of this work is devoted to factorizations of special polynomials of degree five and six. A modular equation is considered. A polynomial of degree six with a single coefficient parametrically dependent of the other arbitrary coefficients is factorized. Factorizations are found for some polynomials of higher degree. A study of symbolic solutions to a reciprocal equation has resulted in some new algorithms for solving equations of degree three and four. These algorithms are based on solutions to equations with discriminants and resolvents different from those of the original equations. New methods of reducing the equations of degree three and four to reciprocal equations are proposed. Particular attention is given to equations solved by square radicals. The performance of these methods is shown by a comparison with solutions generated in the software system Mathematica. These results can be used in the design of simple computer programs to be used as supplementary to the programs of exact symbolic solving of algebraic equations available in the conventional software.

Keywords: reciprocal equations, solution in radicals, Cardano’s formula, resolvent, De Moivre’s polynomial, modular equation, computer algebra, software

pp. 422–432

For citation:
Astapov I. S., Astapov N. S. Algorithms for Symbolic Solving of Algebraic Equations, Programmnaya Ingeneria, 2017, vol. 8, no. 9, pp. 422—432.