Abstract
We consider the problem of determination of decay decrements and amplitudes of exponential functions form observations with errors of the sum of these functions on a nonuniform grid. The integro-differential methods proposed by other authors lead to numerically unstable ill-posed problems, and make it impossible to estimate the errors in the results. New approaches are proposed that overcome these difficulties. The effectiveness of the proposed solutions is demonstrated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. N. Zaikin and E. V. Khoroshilova “Stable iterative method for resolution of multiplets” in: Mathematical Problems of Experimental Data Processing [in Russian], Izd. MGU, Moscow (1984), pp. 19–43.
A. N. Tikhonov and V. Ya. Arsenin, Methods of Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
L. Lantzos, Practical Methods of Applied Analysis [Russian translation], IL, Moscow (1962).
P. N. Zaikina and V. N. Moiseev, “Stable method of data interpretation for isotopic analysis,” Zh. Vychisl. Matem. i Mat. Fiziki,18, No. 2, 487–492 (1978).
R. Prony, “Essai experimental e. analytique...,” J. L'Ecole Polytechnique, Paris,1, No. 2, 24–76 (1795).
D. S. Ross, “Orthogonal exponential,” IEEE Trans. Comm. Electron.,CE-1, 173–176 (1964).
L. Aleksandrov, “Regularized computational process for the analysis of exponential dependences,” Zh. Vychisl. Matem. i Mat. Fiziki.10, No. 5, 1285–1287 (1970).
S. W. Provencher, “A function expansion method for the analysis of exponential decay curves,” J. Chem. Phys.,64, 2772–2778 (1976).
R. Piessens, “A new numerical method for the inversion of Laplace transform,” J. Inst. Math. Appl.,10, 185–197 (1972).
D. G. Gardner, “Resolution of multicomponent exponential decay curves using Fourier transform,” Ann. NY Acad. Sci.,108, 105–107 (1968).
S. D. Foss, “A method of exponential curve fitting by numerical integration,” Biometrics,26, No. 4, 815–821 (1970).
E. V. Khoroshilova, Development of Stable Computer-Aided Algorithms for Solving Spectrometric Decomposition Problems [in Russian], abstract of thesis, Dubna (1986).
I. A. Koltun, Decomposition Methods for the Analysis of Exponential Dependences and Their Application [in Russian], abstract of thesis, Dubna (1988).
V. G. Kritskii, “Resolution of two components in an exponential representation of a function,” in: Processing and Interpretation of Physical Experiments [in Russian], No. 8, Izd. MGU, Moscow (1980), pp. 104–115.
I. G. Petrovskii, Lectures in the Theory of Ordinary Differential Equations [in Russian], GTTI, Moscow (1952).
J. H. Golub and Ch. F. Van Loan, Matrix Computations, Johns Hopkins Univ. Press (1983).
H. J. Wilkinson and C. Reinsch, Linear Algebra, Handbook for Automatic Computation Series, Springer, Berlin (1971).
B. Efron, Nontraditional Methods of Multivariate Statistical Analysis [in Russian], Finansy i Statistika, Moscow (1988).
Additional information
Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University. Moscow, 1996, pp. 129–136.
Rights and permissions
About this article
Cite this article
Zaikin, P.N., Ufimtsev, M.V. Development of integro-differential methods of parameter determination for a sum of exponential functions. Comput Math Model 8, 165–171 (1997). https://doi.org/10.1007/BF02405168
Issue Date:
DOI: https://doi.org/10.1007/BF02405168