Title: | A numerical method of fitting a multiparameter nonlinear function to experimental data in the $L_1$ norm (English) |

Author: | Jakeš, Jaromír |

Language: | English |

Journal: | Aplikace matematiky |

ISSN: | 0373-6725 |

Volume: | 33 |

Issue: | 3 |

Year: | 1988 |

Pages: | 161-170 |

Summary lang: | English |

Summary lang: | Russian |

Summary lang: | Czech |

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Category: | math |

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Summary: | A numerical method of fitting a multiparameter function, non-linear in the parameters which are to be estimated, to the experimental data in the $L_1$ norm (i.e., by minimizing the sum of absolute values of errors of the experimental data) has been developed. This method starts with the least squares solution for the function and then minimizes the expression $\sum_i (x^2_i + a^2)^{1/2}$, where $x_i$ is the error of the $i$-th experimental datum, starting with an $a$ comparable with the root-mean-square error of the least squares solution and then decreasing it gradually to a negligibly small value, which yields the desired solution. The solution for each fixed $a$ is searched by using the Hessian matrix. If necessary, a suitable damping of corrections is initially used. Examples are given of an application of the method to the analysis of some data from the field of photon correlation spectroscopy. (English) |

Keyword: | nonlinear function |

Keyword: | adjustment of parameters by $L_1$ norm |

Keyword: | photon correlation spectroscopy |

Keyword: | analysis of experimental data |

MSC: | 65D10 |

MSC: | 65K05 |

idZBL: | Zbl 0654.65010 |

idMR: | MR0944780 |

DOI: | 10.21136/AM.1988.104299 |

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Date available: | 2008-05-20T18:34:27Z |

Last updated: | 2020-07-28 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/104299 |

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Reference: | [1] K. Zimmermann M. Delaye P. Licinio: Analysis of multiexponential decay by a linear programming method: Application to light scattering spectroscopy.J. Chem. Phys. 82 (1985) 2228. 10.1063/1.448316 |

Reference: | [2] G. B. Dantzig: Linear Programming and Extensions.Princeton University Press, Princeton, New Jersey 1963. Zbl 0108.33103, MR 0201189 |

Reference: | [3] K. Zimmermann J. Jakeš: .Unpublished results. |

Reference: | [4] J. Jakeš P. Štěpánek: .To be published. |

Reference: | [5] A. J. Pope: Two approaches to nonlinear least squares adjustments.Can. Surveyor 28 (1974) 663. 10.1139/tcs-1974-0111 |

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Reference: | [7] B. A. Murtagh R. W. H. Sargent: Computational experience with quadratically convergent minimisation methods.Computer J. 13 (1970) 185. MR 0266403, 10.1093/comjnl/13.2.185 |

Reference: | [8] K. Levenberg: A method for the solution of certain non-linear problems in least squares.Quart. Appl. Mathematics 2 (1944) 164. Zbl 0063.03501, MR 0010666, 10.1090/qam/10666 |

Reference: | [9] D. W. Marquardt: An algorithm for least-squares estimation of nonlinear parameters.J. Soc. Indust. Appl. Math. 11 (1963) 431. Zbl 0112.10505, MR 0153071, 10.1137/0111030 |

Reference: | [10] P. Bloomfield W. L. Steiger: Least Absolute Deviations. Theory, Applications, and Algorithms.Birkhäuser, Boston-Basel-Stuttgart 1983. MR 0748483 |

Reference: | [11] T. S. Arthanari Y. Dodge: Mathematical Programming in Statistics.Wiley, New York 1981. MR 0607328 |

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