Numerical Continuation Methods for Solving Polynomial Systems Arising in Kinematics

1990 ◽  
Vol 112 (1) ◽  
pp. 59-68 ◽  
Author(s):  
C. W. Wampler ◽  
A. P. Morgan ◽  
A. J. Sommese
Keyword(s):  
Polynomial Systems ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Basic Material ◽  
Polynomial Equations ◽  
All Solutions ◽  
Recent Developments ◽  
Position Problem ◽  
Main Ideas ◽  
Systems Of Polynomial Equations

Many problems in mechanism design and theoretical kinematics can be formulated as systems of polynomial equations. Recent developments in numerical continuation have led to algorithms that compute all solutions to polynomial systems of moderate size. Despite the immediate relevance of these methods, they are unfamiliar to most kinematicians. This paper attempts to bridge that gap by presenting a tutorial on the main ideas of polynomial continuation along with a section surveying advanced techniques. A seven position Burmester problem serves to illustrate the basic material and the inverse position problem for general six-axis manipulators shows the usefulness of the advanced techniques.

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.

scholarly journals Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Daniel Lichtblau
Keyword(s):  
Relative Error ◽  
Gröbner Bases ◽  
Approximate Solutions ◽  
Polynomial Systems ◽  
Grobner Bases ◽  
Polynomial Equations ◽  
Secondary Feature ◽  
Approximate Polynomial ◽  
Polynomial Gcd ◽  
Systems Of Polynomial Equations

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.

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.

A combined method for enclosing all solutions of nonlinear systems of polynomial equations

1995 ◽  
Vol 1 (1) ◽  
pp. 41-64 ◽  
Author(s):  
Christine Jäger ◽  
Dietmar Ratz ◽  
К. йегер ◽  
Л. Рац
Keyword(s):  
Nonlinear Systems ◽  
Polynomial Equations ◽  
Combined Method ◽  
All Solutions ◽  
Systems Of Polynomial Equations

scholarly journals Partition regularity of polynomial systems near zero

2021 ◽  
Author(s):  
Lorenzo Luperi Baglini
Keyword(s):  
Polynomial Systems ◽  
Polynomial Equations ◽  
Systems Of Polynomial Equations

AbstractRecently, S. Kanti Patra and Md. Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups $$(S,+)$$ ( S , + ) of $$(\mathbb {R}^{+},+)$$ ( R + , + ) . We extend their results to a much larger class of systems whilst weakening the requests on S, using solely basic results about ultrafilters.

A complexity chasm for solving sparse polynomial equations over p -adic fields

2020 ◽  
Vol 54 (3) ◽  
pp. 86-90
Author(s):  
J. Maurice Rojas ◽  
Yuyu Zhu
Keyword(s):  
Number Theory ◽  
Linear Optimization ◽  
Circuit Complexity ◽  
Polynomial Systems ◽  
Fixed Number ◽  
Local Fields ◽  
Polynomial Equations ◽  
Sparse Polynomial ◽  
Input Size ◽  
Systems Of Polynomial Equations

The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems --- such as those with a fixed number of monomial terms or invariance with respect to a group action --- arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [8]). Clearly, if we are to fully understand the complexity of solving sparse polynomial systems, then we should at least be able to settle the univariate case, e.g., classify when it is possible to separate and approximate roots in deterministic time polynomial in the input size.

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