Homotopy continuation method for solving systems of nonlinear and polynomial equations.

*(English)*Zbl 1337.65044The well-known homotopy continuation method has been developed during the last decades and it has been proved to be a reliable and efficient numerical algorithm for solving systems of nonlinear equations numerically. It is an important tool for this problem and most important, it is global in the sense that solutions of a smooth map whose zeros can easily be obtained may not need to be anywhere close to the solution of the system which we are interested.

In this paper, the authors choose a special category of nonlinear systems, the systems of polynomial equations, and they use the homotopy continuation method. The need to solve those type of systems arises very frequently in various field of science and engineering, and thus, the problem of solving polynomial systems is one of the most important subjects in pure and applied mathematics.

The biggest advantage of the homotopy continuation method in solving polynomial systems is its natural parallelism in the sense that each isolated zero is computed independently of the others.

Since considering polynomial systems in real spaces is beneficial numerically speaking, in this paper, the authors pay a special attention in solving real polynomial systems by real homotopies.

In this paper, the authors also describe the adaptation of homotopy continuation algorithms to a variety of parallel computation environments. Only the most current parallel computing technologies for solving very large polynomial systems are presented.

In this paper, the authors choose a special category of nonlinear systems, the systems of polynomial equations, and they use the homotopy continuation method. The need to solve those type of systems arises very frequently in various field of science and engineering, and thus, the problem of solving polynomial systems is one of the most important subjects in pure and applied mathematics.

The biggest advantage of the homotopy continuation method in solving polynomial systems is its natural parallelism in the sense that each isolated zero is computed independently of the others.

Since considering polynomial systems in real spaces is beneficial numerically speaking, in this paper, the authors pay a special attention in solving real polynomial systems by real homotopies.

In this paper, the authors also describe the adaptation of homotopy continuation algorithms to a variety of parallel computation environments. Only the most current parallel computing technologies for solving very large polynomial systems are presented.

Reviewer: Sonia Pérez Díaz (Madrid)

##### MSC:

65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |

65Y05 | Parallel numerical computation |

65H04 | Numerical computation of roots of polynomial equations |

12Y05 | Computational aspects of field theory and polynomials (MSC2010) |