Journal ArticleDOI
On the Number of Solutions to Polynomial Systems of Equations
C. B. Garcia,Tien-Yien Li +1 more
TLDR
In this paper, it was shown that for almost all polynomial systems of n complex variables, the number of solutions is equal to √ √ n q_i, where q is the degree of equation i. The proof of this result was done in such a way that all q solutions can be explicitly calculated.Abstract:
It is shown that for almost every system of n polynomial equations in n complex variables, the number of solutions is equal to $q \equiv \Pi _{i = 1}^n q_i $, where $q_i $ is the degree of equation i. The proof of this result is done in such a way that all q solutions can be explicitly calculated for almost all such systems.It is further shown that if the polynomial system obtained by retaining only the terms of degree $q_i $ in each equation i has only the trivial solution, then the number of solutions is equal to q.read more
Citations
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Book
Introduction to Numerical Continuation Methods
Eugene L. Allgower,Kurt Georg +1 more
TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
Journal ArticleDOI
A homotopy for solving general polynomial systems that respects m-homogeneous structures
TL;DR: A new method for defining a homotophy to find all solutions to a polynomial system, F(z) = 0, of n equations in n unknowns using a generic homotopy, which represents a significant advance over previous homotopies.
Book ChapterDOI
Numerical Solution of Polynomial Systems by Homotopy Continuation Methods
TL;DR: In this article, the numerical solution of polynomial systems by homotopy continuation methods is presented, given the complexity of the problem, e standard machine arithmetic to obtain efficient programs is used.
Journal ArticleDOI
Tensor Methods for Nonlinear Equations.
Robert B. Schnabel,Paul D. Frank +1 more
TL;DR: In extensive computational tests, a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model, both on standard nonsingular test problems and on problems where the Jacobian at the solution is singular.
Journal ArticleDOI
A methodology for solving chemical equilibrium systems
TL;DR: The problem can be solved by a unified approach, quickly and reliably, when many systems with the same structure must be solved quickly, as in finite-difference models of fluid flow and combustion.
References
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Journal ArticleDOI
Inverse eigenvalue problems
TL;DR: In this article, the classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied using general results on the solvability of polynomial systems and it is shown that in the complex case these problems are always solvable by a finite number of solutions.
Journal ArticleDOI
Determining All Solutions to Certain Systems of Nonlinear Equations
C. B. Garcia,Willard I. Zangwill +1 more
TL;DR: The technique utilized was the so-called “complementarity” or “fixed point” approach to the Brouwer fixed point theorem, which was extended to calculate for certain systems not just one but all possible solutions.
Journal ArticleDOI
Eine Methode zur berechnung sämtlicher Lösungen von Polynomgleichungssystemen
TL;DR: In this article, a numerical method is given to compute all solutions of systemsT ofn polynomial equations inn unknowns on the only premises that the sets of solutions of these systems are finite.
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