Polynomial root isolation
The calculators isolates real roots of the input univariate polynomial using Sturm and VAS-CF methods.
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According to Abel–Ruffini theorem, a general polynomial equation of degree 5 or higher has no solution in radicals, contrary to lower-degree equations, see Solution of quadratic equation, Cubic equation, Quartic equation solution, which can be evaluated by the algebraic solution.
To solve the 5-degree or higher degree equations, several numeric methods can be used, e.g.: Newton's method, Bisection method, False position method or Secant method.
Every numeric method listed above requires the approximate root location intervals to be known. The calculators listed below can solve this task. Both calculators find root location intervals by different methods.
The first calculator uses a more effective method, developed by Akritas and Strzebonski. The method finds root isolation intervals with the aid of continued fractions based on the Vincent theorem. The algorithm is known by Vincent-Akritas-Strzebonski Continued Fractions name, or shortly VAS-CF1.
An input polynomial must be square-free. To ensure your input polynomial meets this condition, use Squarefree polynomial factorization calculator.
The next calculator produces the Sturm sequence and calculates the number of sign changes in all Sturm polynomials for different points. The difference in Sturm sequence polynomial sign changes between two points gives the real root number of the input polynomial located in interval bounded by these points according to Sturm theorem2.