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**Solving polynomial systems using continuation for engineering and scientific problems.**
*(English)*
Zbl 0733.65031

Englewood Cliffs, NJ: Prentice-Hall, Inc. xiv, 546 p. $ 54.00 (1987).

As its title implies, this book is concerned with constructing continuation codes for finding all the roots of multivariable polynomials. The author has in mind problems arising in a variety of scientific and engineering contexts but the methods described and codes listed are not restricted to specific problems. The basic idea used is to discard all the terms in the polynomial except those of highest and lowest degree. The resulting polynomial is then factorized to give starting points on continuation paths for all the roots of the original polynomial, except in pathological cases. (The pathological cases are treated separately.) Then, a simple continuation construction and algorithm is used to recover the roots of the original system.

The first three chapters describe the approach for one, two and many variables, respectively. Chapters 4 and 5 consider the implementation questions arising in the continuation algorithm, including the scaling of equations and variables necessary for a successful implementation. Chapter 6 contains a description of continuation methods for more general problems and is peripheral to the general theme of the text. Chapter 7 discusses techniques for “reducing” the system by removing solutions, especially “solutions at infinity”. The last three chapters are concerned with applications to geometric intersections, chemical kinetics, and the kinematics of manipulators.

The second half of the book consists of appendices. After fifty pages of mainly elementary material on numerical techniques, the last two hundred pages contain listings of the continuation codes. These consist of complete programs calling subroutines for specific tasks. The subroutines are quite well documented and laid out. All the codes are written in IBM specific Fortran 66, with an occasional Fortran 77 construct occurring. The codes can be obtained for a handling fee from the author of the book.

The first three chapters describe the approach for one, two and many variables, respectively. Chapters 4 and 5 consider the implementation questions arising in the continuation algorithm, including the scaling of equations and variables necessary for a successful implementation. Chapter 6 contains a description of continuation methods for more general problems and is peripheral to the general theme of the text. Chapter 7 discusses techniques for “reducing” the system by removing solutions, especially “solutions at infinity”. The last three chapters are concerned with applications to geometric intersections, chemical kinetics, and the kinematics of manipulators.

The second half of the book consists of appendices. After fifty pages of mainly elementary material on numerical techniques, the last two hundred pages contain listings of the continuation codes. These consist of complete programs calling subroutines for specific tasks. The subroutines are quite well documented and laid out. All the codes are written in IBM specific Fortran 66, with an occasional Fortran 77 construct occurring. The codes can be obtained for a handling fee from the author of the book.

Reviewer: Ian Gladwell

### MSC:

65H10 | Numerical computation of solutions to systems of equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

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

26C10 | Real polynomials: location of zeros |

30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |

00A69 | General applied mathematics |

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