scholarly journals Accelerated Solution of Multivariate Polynomial Systems of Equations

2003 ◽  
Vol 32 (2) ◽  
pp. 435-454 ◽  
Author(s):  
B. Mourrain ◽  
V. Y. Pan ◽  
O. Ruatta
Keyword(s):  
Polynomial Systems ◽  
Systems Of Equations ◽  
Multivariate Polynomial

Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators

1995 ◽  
Vol 117 (B) ◽  
pp. 71-79 ◽  
Author(s):  
M. Raghavan ◽  
B. Roth
Keyword(s):  
Kinematic Analysis ◽  
State Of The Art ◽  
Polynomial Systems ◽  
The State ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Synthesis Of Mechanisms ◽  
Analysis And Synthesis ◽  
Systems Of Polynomial Equations ◽  
And Robotics

Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

Numerical solution of multivariate polynomial systems by homotopy continuation methods

Acta Numerica ◽  
1997 ◽  
Vol 6 ◽  
pp. 399-436 ◽  
Author(s):  
T. Y. Li
Keyword(s):  
Power Flow ◽  
Polynomial Systems ◽  
Continuation Methods ◽  
Homotopy Continuation ◽  
Multivariate Polynomial ◽  
Science And Engineering ◽  
Flow Problems ◽  
Homotopy Continuation Methods ◽  
Buchberger Algorithm ◽  
Image Position

Let P(x) = 0 be a system of n polynomial equations in n unknowns. Denoting P = (p1,…, pn), we want to find all isolated solutions offor x = (x1,…,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Gröbner bases, are the classical approach to solving (1.1), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.

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