### Abstract

One of the solution methods for algebraic equations is the continuous iterative method. This is an iterative method in which a differential equation is constructed with the solution of a given algebraic equation as the equilibrium solution, and the solution of the difference equation obtained by discretizing that equation is sought until the solution converges. The method is an extension of various basic iterative methods, and coincides with the original iterative method when the discretization step width is 1. This paper shows first that the continuous iterative method converges globally if the function of the given algebraic equation is a product of a positive-definite matrix, and the gradient of a scalar function and the step width is somewhat small. Especially, for the case where the algebraic equation is linear, it is shown that the continuous interative method may converge even if the original iterative method diverges, or that it converges faster than the original iterative method if the parameters are appropriately set. Then a case is considered where the function of the given algebraic equation is the gradient of a scalar function. A method is proposed which adjusts the coefficients of the continuous iterative method adaptively so that the convergence is as fast as possible. The effectiveness of the method is verified by simple examples.

Original language | English |
---|---|

Pages (from-to) | 70-78 |

Number of pages | 9 |

Journal | Electronics and Communications in Japan (Part III: Fundamental Electronic Science) |

Volume | 74 |

Issue number | 2 |

Publication status | Published - Feb 1991 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Electrical and Electronic Engineering

### Cite this

*Electronics and Communications in Japan (Part III: Fundamental Electronic Science)*,

*74*(2), 70-78.

**Convergence properties of continuous iterative methods.** / Urahama, Kiichi.

Research output: Contribution to journal › Article

*Electronics and Communications in Japan (Part III: Fundamental Electronic Science)*, vol. 74, no. 2, pp. 70-78.

}

TY - JOUR

T1 - Convergence properties of continuous iterative methods

AU - Urahama, Kiichi

PY - 1991/2

Y1 - 1991/2

N2 - One of the solution methods for algebraic equations is the continuous iterative method. This is an iterative method in which a differential equation is constructed with the solution of a given algebraic equation as the equilibrium solution, and the solution of the difference equation obtained by discretizing that equation is sought until the solution converges. The method is an extension of various basic iterative methods, and coincides with the original iterative method when the discretization step width is 1. This paper shows first that the continuous iterative method converges globally if the function of the given algebraic equation is a product of a positive-definite matrix, and the gradient of a scalar function and the step width is somewhat small. Especially, for the case where the algebraic equation is linear, it is shown that the continuous interative method may converge even if the original iterative method diverges, or that it converges faster than the original iterative method if the parameters are appropriately set. Then a case is considered where the function of the given algebraic equation is the gradient of a scalar function. A method is proposed which adjusts the coefficients of the continuous iterative method adaptively so that the convergence is as fast as possible. The effectiveness of the method is verified by simple examples.

AB - One of the solution methods for algebraic equations is the continuous iterative method. This is an iterative method in which a differential equation is constructed with the solution of a given algebraic equation as the equilibrium solution, and the solution of the difference equation obtained by discretizing that equation is sought until the solution converges. The method is an extension of various basic iterative methods, and coincides with the original iterative method when the discretization step width is 1. This paper shows first that the continuous iterative method converges globally if the function of the given algebraic equation is a product of a positive-definite matrix, and the gradient of a scalar function and the step width is somewhat small. Especially, for the case where the algebraic equation is linear, it is shown that the continuous interative method may converge even if the original iterative method diverges, or that it converges faster than the original iterative method if the parameters are appropriately set. Then a case is considered where the function of the given algebraic equation is the gradient of a scalar function. A method is proposed which adjusts the coefficients of the continuous iterative method adaptively so that the convergence is as fast as possible. The effectiveness of the method is verified by simple examples.

UR - http://www.scopus.com/inward/record.url?scp=0026104021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026104021&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0026104021

VL - 74

SP - 70

EP - 78

JO - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

JF - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

SN - 1042-0967

IS - 2

ER -