The calculation of best linear one-sided $L_{p}$ approximations
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- by G. A. Watson PDF
- Math. Comp. 27 (1973), 607-620 Request permission
Abstract:
The calculation of linear one-sided approximations is considered, using the discrete ${L_p}$ norm. For $p = 1$ and $p = \infty$, this gives rise to a linear programming problem, and for $1 < p < \infty$, to a convex programming problem. Numerical results are presented, including some applications to the approximate numerical solution of ordinary differential equations, with error bounds.References
- Ian Barrodale and Andrew Young, Algorithms for best $L_{1}$ and $L_{\infty }$ linear approximations on a discrete set, Numer. Math. 8 (1966), 295β306. MR 196912, DOI 10.1007/BF02162565
- I. Barrodale and F. D. K. Roberts, Applications of mathematical programming to $l_{p}$ approximation, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp.Β 447β464. MR 0271594
- R. BojaniΔ and R. DeVore, On polynomials of best one sided approximation, Enseign. Math. (2) 12 (1966), 139β164. MR 213790
- Lothar Collatz, Approximation in partial differential equations, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Publication of the Mathematics Research Center, U.S. Army, the University of Wisconsin, no. 1, University of Wisconsin Press, Madison, Wis., 1959, pp.Β 413β422. Edited by R. E. Langer. MR 0103593
- L. Collatz, Monotonicity and related methods in non-linear differential equations problems, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, N.Y., 1966, pp.Β 65β87. MR 0210330
- Ronald DeVore, One-sided approximation of functions, J. Approximation Theory 1 (1968), no.Β 1, 11β25. MR 230018, DOI 10.1016/0021-9045(68)90054-3
- Anthony V. Fiacco and Garth P. McCormick, Nonlinear programming: Sequential unconstrained minimization techniques, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0243831
- R. Fletcher, J. A. Grant, and M. D. Hebden, The calculation of linear best $L_{p}$ approximations, Comput. J. 14 (1971), 276β279. MR 303948, DOI 10.1093/comjnl/14.3.276
- R. Fletcher, Minimizing general functions subject to linear constraints, Numerical methods for non-linear optimization (Conf., Dundee, 1971), Academic Press, London, 1972, pp.Β 279β296. MR 0408244
- G. Hadley, Linear programming, Addison-Wesley Series in Industrial Management, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962. MR 0135622
- James T. Lewis, Computation of best one-sided $L_{1}$ approximation, Math. Comp. 24 (1970), 529β536. MR 273780, DOI 10.1090/S0025-5718-1970-0273780-5
- G. Marsaglia, One-sided approximations by linear combinations of functions, Approximation Theory (Proc. Sympos., Lancaster, 1969) Academic Press, London, 1970, pp.Β 233β242. MR 0266401
- B. A. Murtagh and R. W. H. Sargent, A constrained minimization method with quadratic convergence, Optimization (Sympos., Univ. Keele, Keele, 1968) Academic Press, London, 1969, pp.Β 215β245. MR 0284213
- M. R. Osborne and G. A. Watson, On the best linear Chebyshev approximation, Comput. J. 10 (1967), 172β177. MR 218808, DOI 10.1093/comjnl/10.2.172
- David L. Phillips, A note on best one-sided approximations, Comm. ACM 14 (1971), 598β600. MR 0297100, DOI 10.1145/362663.362743
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- J. B. Rosen, The gradient projection method for nonlinear programming. I. Linear constraints, J. Soc. Indust. Appl. Math. 8 (1960), 181β217. MR 112750, DOI 10.1137/0108011
- J. B. Rosen, Approximate computational solution of non-linear parabolic partial differential equations by linear programming, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, N.Y., 1966, pp.Β 265β296. MR 0207232
- J. B. Rosen, Approximate solution and error bounds for quasi-linear elliptic boundary value problems, SIAM J. Numer. Anal. 7 (1970), 80β103. MR 264861, DOI 10.1137/0707004
- G. A. Watson, On the best linear one-sided Chebyshev approximation, J. Approximation Theory 7 (1973), 48β58. MR 342929, DOI 10.1016/0021-9045(73)90051-8
- Philip Wolfe, Methods of nonlinear programming, Recent advances in mathematical programming, McGraw-Hill, New York, 1963, pp.Β 67β86. MR 0155683
- P. Wolfe, Methods of nonlinear programming, Nonlinear Programming (NATO Summer School, Menton, 1964) North-Holland, Amsterdam, 1967, pp.Β 97β131. MR 0216868
- 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), 401β406. MR 0240957, DOI 10.1145/363347.363364 I. Barrodale & F. D. K. Roberts, An Improved Algorithm for Discrete ${l_1}$ Linear Approximation, MRC Technical Summary Report #1172, January 1972.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 607-620
- MSC: Primary 65D15
- DOI: https://doi.org/10.1090/S0025-5718-1973-0343537-8
- MathSciNet review: 0343537