Some computational methods for systems of nonlinear equations and systems of polynomial equations

@article{Frster1992SomeCM,
  title={Some computational methods for systems of nonlinear equations and systems of polynomial equations},
  author={W. Főrster},
  journal={Journal of Global Optimization},
  year={1992},
  volume={2},
  pages={317-356}
}
  • W. Főrster
  • Published 1 December 1992
  • Mathematics, Computer Science
  • Journal of Global Optimization
This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed… 
Parallel schemes of computation for Bernstein coefficients and their application
  • Z. Garczarczyk
  • Mathematics
    Proceedings. International Conference on Parallel Computing in Electrical Engineering
  • 2002
TLDR
An approach to the range evaluation of a function over an interval is established and coefficients of Bernstein polynomials are effectively calculated in some parallel process for obtaining all solutions of nonlinear equations.
A Geometrical Root Finding Method for Polynomials, with Complexity Analysis
The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to
Interval Methods for Analog Circuits
TLDR
In this chapter a version of the predictor-corrector method for computing points of continua‐ tion path of a nonlinear equation is presented and Krawczyk operator is used in ndimensional box-searching of all solutions.
Chapter 4 Interval Methods for Analog Circuits
TLDR
In this chapter a version of the predictor-corrector method for computing points of continua‐ tion path of a nonlinear equation is presented and Krawczyk operator is used in ndimensional box-searching of all solutions.
Solving Polynomial Systems by Penetrating Gradient Algorithm Applying Deepest Descent Strategy
TLDR
The most prominent feature of penetrating gradient algorithm is its ability to see and penetrate through the obstacles in error space along the line of search direction and to jump to the global minimizer in a single step.
Quantifier Elimination and Cylindrical Algebraic Decomposition
1 Introduction to the Method.- 2 Importance of QE and CAD Algorithms.- 3 Alternative Approaches.- 4 Practical Issues.- Acknowledgments.- Quantifier Elimination by Cylindrical Algebraic Decomposition

References

SHOWING 1-10 OF 116 REFERENCES
Computation of all solutions to a system of polynomial equations
TLDR
The method is based on piecewise linear approximation and complementarity theory and utilizes a skilful artificial map and two copies of the triangulationJ3 with continuous refinement of grid size to increase the computational efficiency and to avoid the necessity of determining the grid size a priori.
Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations
This paper presents a digest of recently developed simplicial and continuation methods for approximating fixed-points or zero-points of nonlinear finite-dimensional mappings. Underlying the methods
Numerical linear algebra aspects of globally convergent homotopy methods
Probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one. These methods are theoretically powerful, and
Global Continuation Methods for Finding all Solutions to Polynomial Systems of Equations in N Variables
TLDR
A new approach is presented based upon the continuation method and differential equations for finding all solutions to a system of n nonlinear equations in n unknowns which creates new theoretical insights especially relative to the underlying homotopy and to globality.
A Simplicial Approximation Algorithm for Solving Systems of Nonlinear Equations.
Abstract : A simplicial approximation algorithm with a variable initial point and a restart procedure is presented for solving systems of nonlinear equations. The algorithm can employ any labeling
Finding all solutions to polynomial systems and other systems of equations
TLDR
This paper utilizes a different approach that eliminates the requirement of the previous paper for a leading dominating term in each equation, and adds a dominating term artificially and then fades it to permit the calculation of all solutions to arbitrary polynomials and to various other systems ofn equations inn complex variables.
Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems
TLDR
This introduction to polynomial continuation remains a useful starting point for the reader interested in learning how to solve practical problems without advanced mathematics.
Finding zeroes of maps: homotopy methods that are constructive with probability one
We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive
The octahedral algorithm, a new simplicial fixed point algorithm
TLDR
A new variable dimension simplicial algorithm using the restrart technique of Merrill to improve the accuracy of the solution is presented and is shown to converge quadratically under certain conditions.
Iterative solution of nonlinear equations in several variables
TLDR
Convergence of Minimization Methods An Annotated List of Basic Reference Books Bibliography Author Index Subject Index.
...
...