From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

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Introduction to Numerical Continuation Methods

This paper presents a unique approach to the kinematic analysis of the most general six-degree-of-freedom, six-revolute-joint manipulators (Previously, the problem of computing all possible configurations of a manipulator corresponding to a given hand position was approached by reducing the problem to that of solving a high degree polynomial equation in one variable. -In-this-paper Jt is shown that the problem can be reduced to that of solving a system of eight second-degree equations in eight unknowns, it-is-fttrther-demrmstrate^-thafmis second-degree system can be routinely solved using a continuation algorithm. To complete the general analysis, a second numerical method—a continuation heuristic —is shown to generate partial solution sets quickly. Finally, in some special cases, closed form solutions wereobtained for some commonly used industrial manipulators. The results can be applied to the~analysis~of--both-six-~and-~fme'degreeuf-freedmn manipulators composed of mixed revolute and prismatic joints. The numerical stability of continuation on small second-degree systems opens the way for routine use in offline robot programming applications. \\

Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom Manipulators by Continuation Methods

A homotopy for solving general polynomial systems that respects m-homogeneous structures

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 Grobner bases, are the classical approach to solving (11), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems

http://dsec.pku.edu.cn/~tieli/notes/numer_anal/ActaNumer_TYLI_Homotopy.pdf

Numerical solution of multivariate polynomial systems by homotopy continuation methods

Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

In this paper 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. The method employed is that of "embedding", i.e. the systemT is embedded in a set of systems which are successively solved, starting with one having solutions easily to compute and proceding toT in a finite series of steps. An estimation of the number of steps necessary is given. The practicability of the method is proved for all systemsT. Numerical examples and results are contained.

Franz-Josef Drexler

Papers

Eine Methode zur berechnung sämtlicher Lösungen von Polynomgleichungssystemen