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An efficient method for the discrete linear $L_{1}$ approximation problem


Author: Nabih N. Abdelmalek
Journal: Math. Comp. 29 (1975), 844-850
MSC: Primary 65D15; Secondary 90C10
DOI: https://doi.org/10.1090/S0025-5718-1975-0378354-8
MathSciNet review: 0378354
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Abstract: An improved dual simplex algorithm for the solution of the discrete linear ${L_1}$ approximation problem is described. In this algorithm certain intermediate iterations are skipped. This method is comparable with an improved simplex method due to Barrodale and Roberts, in both speed and number of iterations. It also has the advantage that in case of ill-conditioned problems, the basis matrix can lend itself to triangular factorization and can thus ensure a stable solution. Numerical results are given.


References [Enhancements On Off] (What's this?)

  • Nabih N. Abdelmalek, On the discrete linear $L_{1}$ approximation and $L_{1}$ solutions of overdetermined linear equations, J. Approximation Theory 11 (1974), 38–53. MR 388750, DOI https://doi.org/10.1016/0021-9045%2874%2990037-9
  • N. N. ABDELMALEK, "${L_1}$ solution of overdetermined system of linear equations by a dual simplex method," Comm. ACM (Submitted.) N. N. ABDELMALEK, "${L_1}$ solution of overdetermined system of linear equations by a dual simplex method and LU decomposition," Comm. ACM (Submitted.)
  • Ian Barrodale and Andrew Young, Algorithms for best $L_{1}$ and $L_{\infty }$ linear approximations on a discrete set, Numer. Math. 8 (1966), 295–306. MR 196912, DOI https://doi.org/10.1007/BF02162565
  • I. BARRODALE & F. D. K. ROBERTS, An Improved Algorithm For Discrete ${l_1}$. Approximation, The University of Wisconsin, MRC Tech. Report no. 1172, 1972.
  • 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 overdetermined system of equations in the ${l_1}$ norm," Comm. ACM, v. 17, 1974, pp. 319-920. R. H. BARTELS & G. H. GOLUB, "The simplex method of linear programming using LU decomposition," Comm. ACM, v. 12, 1969, pp. 266-268. MR 39 #2302.
  • Karl H. Usow, On $L_{1}$ approximation. II. Computation for discrete functions and discretization effects, SIAM J. Numer. Anal. 4 (1967), 233–244. MR 217499, DOI https://doi.org/10.1137/0704022
  • Harvey M. Wagner, Linear programming techniques for regression analysis, J. Amer. Statist. Assoc. 54 (1959), 206–212. MR 130753

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Additional Information

Keywords: Discrete linear <IMG WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img11.gif" ALT="${L_1}$"> approximation, overdetermined system of linear equations, linear programming, dual simplex algorithm
Article copyright: © Copyright 1975 American Mathematical Society