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**Recursive algorithms for solving systems of nonlinear equations.**
*(English)*
Zbl 0674.65026

A way of generalizing one-dimensional root-finding algorithms for nonlinear equation to the multidimensional case by means of recursion is described.

The paper deals with the exploitation of the abilities of the PASCAL language to express the true recursion and to define special nonnumerical types of data in such a way that the basic structure of the algorithms is simply described and easily understood. It is also shown how a failure of the root finding algorithm can be prevented. In the second part, the algorithm is modified so as to exploit sparsity of large systems of equations for reducing the recursion depth and consequently decreasing the computational requirements of the method.

The above algorithm in its robust version is proved to converge and to find a root under very general (and sometimes unfavourable) conditions. In comparison with conventional methods, (Newton-Raphson iteration, iteration by components) it has some advantages: no need for calculating and inverting the Jacobi matrix, and for special means against divergence.

The paper deals with the exploitation of the abilities of the PASCAL language to express the true recursion and to define special nonnumerical types of data in such a way that the basic structure of the algorithms is simply described and easily understood. It is also shown how a failure of the root finding algorithm can be prevented. In the second part, the algorithm is modified so as to exploit sparsity of large systems of equations for reducing the recursion depth and consequently decreasing the computational requirements of the method.

The above algorithm in its robust version is proved to converge and to find a root under very general (and sometimes unfavourable) conditions. In comparison with conventional methods, (Newton-Raphson iteration, iteration by components) it has some advantages: no need for calculating and inverting the Jacobi matrix, and for special means against divergence.

Reviewer: J.Hřebíček

### MSC:

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

### Keywords:

recursive algorithms; root-finding algorithms; comparison; Newton-Raphson iteration; iteration by components
Full Text:
EuDML

### References:

[1] | J. Jan: Recursive method of numerical analysis of inertialess nonlinear circuits. (in Czech). Library of research and scientific writings, Technical University Brno, B-57, 1975. |

[2] | J. Jan J. Holčík J. Kozumplík: Recursive method and general purpose program RANG to analyze nonlinear circuits. (in Czech). Research report, project no. III-3-1/1, Technical University Brno, 1975. |

[3] | J. Jan O. Gotfrýd J. Holčík J. Kozumplík: Analysis of nonlinear circuits by means of the generalized recursive method. Proc. of the II-nd Int. Conference on Electronic Circuits, Prague 1976. |

[4] | P. Hladký: Use of the recursive method in analysis of transients in nonlinear circuits. (in Czech). Thesis, Dept. of Computers, Technical University of Brno, 1976. |

[5] | J. Jan O. Gotfrýd J. Holčík J. Kozumplík: Recursive analysis of nonlinear circuits. (in Czech). Slaboproudý obzor 39, 1978, no. 1. |

[6] | J. Jan: Recursive algorithms to solve systems of nonlinear equations. Proc. of the 7-th European Conference on Circuit Theory and Design, Prague 1985. · Zbl 0585.65041 |

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