# A numerical method of fitting a multiparameter nonlinear function to experimental data in the ${L}_{1}$ norm

Aplikace matematiky (1988)

• Volume: 33, Issue: 3, page 161-170
• ISSN: 0862-7940

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## Abstract

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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 ${\sum }_{i}{\left({x}_{i}^{2}+{a}^{2}\right)}^{1/2}$, 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.

## How to cite

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Jakeš, Jaromír. "A numerical method of fitting a multiparameter nonlinear function to experimental data in the $L_1$ norm." Aplikace matematiky 33.3 (1988): 161-170. <http://eudml.org/doc/15534>.

@article{Jakeš1988,
abstract = {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.},
author = {Jakeš, Jaromír},
journal = {Aplikace matematiky},
keywords = {nonlinear function; adjustment of parameters by $L_1$ norm; photon correlation spectroscopy; analysis of experimental data; nonlinear function; adjustment of parameters by norm; photon correlation spectroscopy; analysis of experimental data},
language = {eng},
number = {3},
pages = {161-170},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A numerical method of fitting a multiparameter nonlinear function to experimental data in the $L_1$ norm},
url = {http://eudml.org/doc/15534},
volume = {33},
year = {1988},
}

TY - JOUR
AU - Jakeš, Jaromír
TI - A numerical method of fitting a multiparameter nonlinear function to experimental data in the $L_1$ norm
JO - Aplikace matematiky
PY - 1988
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 33
IS - 3
SP - 161
EP - 170
AB - 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.
LA - eng
KW - nonlinear function; adjustment of parameters by $L_1$ norm; photon correlation spectroscopy; analysis of experimental data; nonlinear function; adjustment of parameters by norm; photon correlation spectroscopy; analysis of experimental data
UR - http://eudml.org/doc/15534
ER -

## References

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1. K. Zimmermann M. Delaye P. Licinio, 10.1063/1.448316, J. Chem. Phys. 82 (1985) 2228. (1985) DOI10.1063/1.448316
2. G. B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey 1963. (1963) Zbl0108.33103MR0201189
3. K. Zimmermann J. Jakeš, [unknown], Unpublished results.
4. J. Jakeš P. Štěpánek, [unknown], To be published.
5. A. J. Pope, Two approaches to nonlinear least squares adjustments, Can. Surveyor 28 (1974) 663. (1974)
6. M. J. Box, 10.1093/comjnl/9.1.67, Computer J. 9 (1966) 67. (1966) Zbl0146.13304MR0192645DOI10.1093/comjnl/9.1.67
7. B. A. Murtagh R. W. H. Sargent, 10.1093/comjnl/13.2.185, Computer J. 13 (1970) 185. (1970) MR0266403DOI10.1093/comjnl/13.2.185
8. K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Mathematics 2 (1944) 164. (1944) Zbl0063.03501MR0010666
9. D. W. Marquardt, 10.1137/0111030, J. Soc. Indust. Appl. Math. 11 (1963) 431. (1963) Zbl0112.10505MR0153071DOI10.1137/0111030
10. P. Bloomfield W. L. Steiger, Least Absolute Deviations. Theory, Applications, and Algorithms, Birkhäuser, Boston-Basel-Stuttgart 1983. (1983) MR0748483
11. T. S. Arthanari Y. Dodge, Mathematical Programming in Statistics, Wiley, New York 1981. (1981) MR0607328

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