Journal "Software Engineering"
a journal on theoretical and applied science and technology
ISSN 2220-3397
Journal "Software Engineering"
a journal on theoretical and applied science and technology
ISSN 2220-3397
Issue N10 2014 year
Various methods of expressing the roots of third- and fourth-order algebraic equations as exact functions of the coefficients are analyzed. A special case of a third-order equation or a fourth-order equation resolvent function with three real roots is investigated. In this case Cardano's formulas are difficult to implement accurately, since there is no efficient algorithm for finding the cube roots of a complex number. The trigonometrical (Viete's) formula for the roots is too cumbersome for use in analytical manipulations.
In some particular cases of additional constraints imposed on the coefficients the roots may sometimes be obtained as from the coefficients with formulas that are much simpler than those in, e. g., software system Mathematica 8.0. Such formulas are proposed in the present paper. These are based on Theorems 2 and 3. Theorem 2 finds the solutions to an infinite set of cubic equation from the solution to a single equation. In Theorem 3 it is proved that a fourth-order equation with complex coefficients can be reduced to a generalized reciprocal equation by linear substitution. This equation can be easily solved in symbolic form. For fourth-order algebraic equation a resolvent function is found in the theorem. The function differs from those found by the conventional methods of solving these equations. Hence, some particular fourth-order equations can be effectively solved. The distribution and multiplicity of real roots of fourth-order equations with real coefficients are completely studied in the paper.
These results can be used for the creation of computer programs in additions to the programs of solving third- and fourth-order algebraic equations available in conventional software.