Solution formulas for cubic equations without or with constraints

  title={Solution formulas for cubic equations without or with constraints},
  author={Ting Zhao and Dongming Wang and Hoon Hong},
  journal={J. Symb. Comput.},

Figures and Tables from this paper

Analytical formula for the roots of the general complex cubic polynomial
We present a new method to calculate analytically the roots of the general complex polynomial of degree three. This method is based on the approach of clever change of variable involving an arbitrary
Real solution formulas of cubic and quartic equations applied to generate dynamic diagrams with inequality constraints
It is shown that for generating dynamic diagrams automatically the performance of this approach can be enhanced, in terms of stability of numeric computation and quality of generated diagrams, when the used solution formulas of cubic and quartic equations are replaced by newly introduced real solution formulas with inequality constraints.
A rapidly convergent method for solving third-order polynomials
We present a rapidly convergent method for solving cubic polynomial equations with real coefficients. The method is based on a power series expansion of a simplified form of Cardano’s formula using
The second discriminant of a univariate polynomial
We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$
A portable library to support the SBML Layout Extension
The SBML layout extension enables SBML models to encode layout information which describes the graphical depiction of model elements. In this application note, we describe libSBNW, a portable library


Partial Cylindrical Algebraic Decomposition for Quantifier Elimination
A complete discrimination system for polynomials
A generic algorithm is proposed to produce a complete discrimination system for a polynomial with any degrees, which has extensive applications in various fields, and its efficiency was demonstrated by computer implementations.
Explicit Criterion to Determine the Number of Positive Roots of a Polynomial 1)
In a recent article, a complete discrimination system was given for a poly- nomial with symbolic/literal coecients. Such a system is a set of explicit expressions in terms of the coecients, which is
The Complexity of Linear Problems in Fields
A complete discrimination system for polynomials with complex coefficients and its automatic generation
By establishing a complete discrimination system for polynomials and using the automatic generation of root classification, a method to determine the positive definiteness of a polynomial in one or two indeterminates is automatically presented.
Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond
  • V. Weispfenning
  • Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 1997
A new, “elementary” quantifier elimination method for various special cases of the general quantifiers elimination problem for the first-order theory of real numbers, which yields a quantifier Elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurrences of the quantified variables.
Simplification of Quantifier-Free Formulae over Ordered Fields
An overview is given over various methods combining elements of field theory, order theory, and logic that do not require a Boolean normal form computation for simplifying intermediate and final results of automatic quantifier elimination by elimination sets.
GEOTHER 1.1: Handling and Proving Geometric Theorems Automatically
GEOTHER provides an environment for handling and proving theorems in geometry automatically, and several functions are implemented that allow one to translate the specification of a geometric theorem into an English or Chinese statement, into algebraic expressions, and into a logic formula automatically.
Quantifier elimination for real algebra—the cubic case
A special purpose quantifier elimination method that eliminates a quantifier in formulas that has similar upper complexity bounds and offers similar advantages (relatively large preprocessing part, explicit parametric solutions).