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**Computing all solutions to polynomial systems using homotopy continuation.**
*(English)*
Zbl 0635.65058

The authors establish the principles of implementation and report on the performance of computer programs that use the homotopies described in their previous paper [ibid. 24, 101-113 (1987; reviewed above)]. Numerical results are presented on three typical problems from geometric modeling, chemical equilibrium studies and kinematics of mechanisms. Generally the new approach is faster and more reliable.

Reviewer: C.Schmidt-Lainé

### MSC:

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

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

### Keywords:

zeros of polynomial functions; homotopy; continuation; computer programs; Numerical results; geometric modeling; chemical equilibrium; kinematics### Citations:

Zbl 0635.65057### Software:

HOMPACK### References:

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[2] | Fulton, W., Intersection Theory (1984), Springer: Springer New York · Zbl 0541.14005 |

[3] | Jenkins, M. A.; Traub, J. F., A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration, Numer. Math., 14, 252-263 (1970) · Zbl 0176.13701 |

[4] | Meintjes, K.; Morgan, A. P., A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22, 333-361 (1987) · Zbl 0616.65057 |

[5] | Morgan, A. P., A transformation to avoid solutions at infinity for polynomial systems, Appl. Math. Comput., 18, 77-86 (1986) · Zbl 0597.65045 |

[6] | Morgan, A. P., A homotopy for solving polynomial systems, Appl. Math. Comput., 18, 87-92 (1986) · Zbl 0597.65046 |

[7] | Morgan, A. P., Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems, ((1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J) · Zbl 1170.65316 |

[8] | A. P. Morgan and A. J. Sommese, A homotopy for solving general polynomial systems that respects \(m\)Appl. Math. Comput.,; A. P. Morgan and A. J. Sommese, A homotopy for solving general polynomial systems that respects \(m\)Appl. Math. Comput., · Zbl 0635.65057 |

[9] | Tsai, L.-W.; Morgan, A. P., Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods, ASME J. Mechanisms, Transmissions and Automation in Design, 107, 48-57 (1985) |

[10] | L. T. Watson, C. S. Billups, and A. P. Morgan, hompack; L. T. Watson, C. S. Billups, and A. P. Morgan, hompack · Zbl 0626.65049 |

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