Nonlinear curve-fitting in the $L_{1}$ and $L_{\infty }$ norms
Authors:
Richard L. Shrager and Edward Hill
Journal:
Math. Comp. 34 (1980), 529-541
MSC:
Primary 41A45; Secondary 41A50, 65D10
DOI:
https://doi.org/10.1090/S0025-5718-1980-0559201-X
MathSciNet review:
559201
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In extending the Levenberg-Marquardt ${L_2}$ method for nonlinear curve-fitting to the ${L_1}$ and ${L_\infty }$ norms, the following problems arise, but are dealt with successfully: (1) Trial parameters are generated by linear programming, which can be time-consuming. (2) Trial parameters are not uniquely specified in some cases. (3) There are intervals of the search parameter for which the trial parameters remain constant. (4) In ${L_1}$, the trial parameters are discontinuous with respect to the search parameter. It is shown that linear constraints on the parameters are easily included in the computations. Finally, some numerical results are presented.
- D. H. Anderson and M. R. Osborne, Discrete, nonlinear approximation problems in polyhedral norms, Numer. Math. 28 (1977), no. 2, 143–156. MR 448807, DOI https://doi.org/10.1007/BF01394449
- D. H. Anderson and M. R. Osborne, Discrete, nonlinear approximation problems in polyhedral norms. A Levenberg-like algorithm, Numer. Math. 28 (1977), no. 2, 157–170. MR 445788, DOI https://doi.org/10.1007/BF01394450
- I. Barrodale and F. D. K. Roberts, An improved algorithm for discrete $l_{1}$ linear approximation, SIAM J. Numer. Anal. 10 (1973), 839–848. MR 339449, DOI https://doi.org/10.1137/0710069 I. BARRODALE & F. D. K. ROBERTS, “Solution of an over-determined system of equations in the ${L_1}$ norm,” Comm. ACM, v. 17, 1974, pp. 319-320.
- I. Barrodale and C. Phillips, An improved algorithm for discrete Chebyshev linear approximation, Proceedings of the Fourth Manitoba Conference on Numerical Mathematics (Winnipeg, Man., 1974) Utilitas Math., Winnipeg, Man., 1975, pp. 177–190. Congr. Numer., No. XII. MR 0373585 I. BARRODALE & F. D. K. ROBERTS, An Efficient Algorithm for Discrete ${L_1}$ Linear Approximation with Linear Constraints, Tech. Dept. DM-103-IR, Dept. of Math., Univ. of Victoria Victoria, B. C., Canada, July 1977. I. BARRODALE & F. D. K. ROBERTS, Solution of the Constrained ${L_1}$ Linear Approximation Problem, Tech. Dept. DM-104-IR, Dept. of Math., Univ. of Victoria, Victoria, B. C., Canada, July 1977.
- Richard H. Bartels and Gene H. Golub, Stable numerical methods for obtaining the Chebyshev solution to an overdetermined system of equations, Comm. ACM 11 (1968), 401–406. MR 0240957, DOI https://doi.org/10.1145/363347.363364
- Richard H. Bartels, Andrew R. Conn, and James W. Sinclair, Minimization techniques for piecewise differentiable functions: the $l_{1}$ solution to an overdetermined linear system, SIAM J. Numer. Anal. 15 (1978), no. 2, 224–241. MR 501831, DOI https://doi.org/10.1137/0715015
- Richard H. Bartels, Andrew R. Conn, and Christakis Charalambous, On Cline’s direct method for solving overdetermined linear systems in the $l_{\infty }$ sense, SIAM J. Numer. Anal. 15 (1978), no. 2, 255–270. MR 501832, DOI https://doi.org/10.1137/0715017
- E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
- Saul I. Gass, Linear programming, 4th ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1975. Methods and applications. MR 0373586
- P. LaFata and J. B. Rosen, An interactive display for approximation by linear programming, Comm. ACM 13 (1970), 651–659. MR 0267810, DOI https://doi.org/10.1145/362790.362793
- Kenneth Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math. 2 (1944), 164–168. MR 10666, DOI https://doi.org/10.1090/S0033-569X-1944-10666-0
- Donald W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math. 11 (1963), 431–441. MR 153071
- G. F. McCormick and V. A. Sposito, Using the $L_{2}$-estimator in $L_{1}$-estimation, SIAM J. Numer. Anal. 13 (1976), no. 3, 337–343. MR 448808, DOI https://doi.org/10.1137/0713030
- Duane A. Meeter, On a theorem used in nonlinear least squares, SIAM J. Appl. Math. 14 (1966), 1176–1179. MR 207094, DOI https://doi.org/10.1137/0114094 D. D. MORRISON, Methods for Non-Linear Least Squares Problems and Convergence Proofs. Tracking Programs and Orbit Determination, Proc. Jet Propulsion Lab. Seminar, 1970, pp. 1-9.
- M. R. Osborne and G. A. Watson, An algorithm for minimax approximation in the nonlinear case, Comput. J. 12 (1969/70), 63–68. MR 245314, DOI https://doi.org/10.1093/comjnl/12.1.63
- M. R. Osborne and G. A. Watson, On an algorithm for discrete nonlinear $L_{1}$ approximation, Comput. J. 14 (1971), 184–188. MR 278491, DOI https://doi.org/10.1093/comjnl/14.2.184
- M. R. Osborne and G. A. Watson, Nonlinear approximation problems in vector norms, Numerical analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), Lecture Notes in Pure and Appl. Math., vol. 36, Dekker, New York, 1978, pp. 117–132. MR 492763
- John R. Rice, The approximation of functions. Vol. I: Linear theory, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964. MR 0166520
- S. R. Searle, Linear models, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0293792
- Richard I. Shrager, Nonlinear regression with linear constraints: An extension of the magnified diagonal method, J. Assoc. Comput. Mach. 17 (1970), 446–452. MR 278742, DOI https://doi.org/10.1145/321592.321597 R. I. SHRAGER, “Quadratic programming for nonlinear regression,” Comm. ACM, v. 15, 1972, pp. 41-45. R. I. SHRAGER & E. HILL, “Some properties of the Levenberg method in the ${L_1}$ and ${L_\infty }$ norms,” 1979. Available from the authors.
- G. A. Watson, A method for calculating best non-linear Chebyshev approximations, J. Inst. Math. Appl. 18 (1976), no. 3, 351–360. MR 454480
Retrieve articles in Mathematics of Computation with MSC: 41A45, 41A50, 65D10
Retrieve articles in all journals with MSC: 41A45, 41A50, 65D10
Additional Information
Article copyright:
© Copyright 1980
American Mathematical Society