L1 Norm Based Computational Algorithms

Keywords: L1 norm, Regression, Algorithm, Computer

Abstract

This paper gives a rather general review of the L1 norm algorithms. The chronology and historical development of the L1 norm estimation theory for the period of 1632-1928 will be surveyed and the algorithms belonging to the after 1928 period will be categorized into three main classes of direct descent, simplex type, and other algorithms.

References

N.N. Abdelmalek (1971) Linear approximation for a discrete point set and L1 solutions of overdetermined linear equations. J. ACM, 18, 41-47.

N.N. Abdelmalek (1974) On the discrete linear L1 approximation and L1 solutions of overdetermined linear equations. J. of Approx. Theory, 11, 38-53.

N.N. Abdelmalek (1975a) An efficient method for the discrete L1 approximation problem. Math. Comput., 29, 844-850.

N.N. Abdelmalek (1980a) L1 solution of overdetermined systems of linear equations. ACM Trans. Math. Soft., 6, 220-227.

N.N. Abdelmalek (1980b) A Fortran subroutine for the L1 solution of overdetermined systems of linear equations. ACM Trans. Math. Soft., 6, 228-30.

D.H. Anderson, M.R. Osborne (1976) Discrete linear approximation problems in polyhedral norms. Numer. Math. 26, 179-189.

R.D. Armstrong, E.L. Frome, D.S. Kung (1979) A revised simplex algorithm for the absolute deviation curve fitting problem. Commun. Stat. B8, 175-190.

R.D. Armstrong, J. Godfrey (1979) Two linear programming algorithms for the discrete L1 problem. Math. Comput., 33, 289-300.

R.D. Armstrong, D.S. Kung (1978) AS132: Least absolute value estimates for a simple linear regression problem. Appl. Stat., 27, 363-366.

R.D. Armstrong, D.S. Kung (1982b) A dual algorithm to solve linear least absolute value problems. J. Oper. Res. Soc., 33, 931-936.

T.S. Arthanari, Y. Dodge (1981) Mathematical programming in statistics. John Wiley, Interscience division, New York.

S. Baboolal, G.A. Watson (1981) Computational experience with an algorithm for discrete L1 approximation. Computing, 27, 245-252.

S.C. Banks, H.L. Taylor (1980) A modification to the discrete L1 linear approximation algorithm of Barrodale and Roberts. SIAM J. on Scientific and Stat. Comput. 1, 187-190.

Barrodale (1970) On computing best L1 approximations. In A. Talbot, Approximation theory Academic Press, New York, 205-215.

Barrodale, F.D.K. Roberts (1973) An improved algorithm for discrete L1 linear approximation. SIAM J. Numer. Anal., 10, 839-848.

Barrodale, F.D.K. Roberts (1974) Algorithm 478: Solution of an overdetermined system of equations in the L1 norm. Commun. ACM, 17, 319-320.

Barrodale, A. Young (1966) Algorithms for best L1 and L∞ linear approximations on a discrete set. Numer. Math., 8, 295-306.

R.H. Bartels, A.R. Conn (1977) LAV regression: A special case of piecewise linear minimization. Commun. Stat., B6, 329-340.

R.H. Bartels, A.R. Conn, J. Sinclair (1976) The L1 solution to an overdetermined linear system. Proc. 9th Ann. Symp. Interface Statist. In C.D.C. Hoaglin (ed.) Boston, Prindle, Weber and Schmidt Inc., 120-7.

R.H. Bartels, A.R. Conn, J. Sinclair (1978) Minimization technique for piecewise differentiable functions: The L1 solution to an overdetermined linear system. SIAM J. Numer. Anal. 15, 224-241.

R.H. Bartels, G.H. Golub (1969) The simplex method of linear programming using LU decomposition. Commun. ACM, 12, 266-268.

J. Bejar (1956) Regression en mediana y la programación lineal, Trabajos de Estadistica 7, 141-58.

J. Bejar (1957) Calculo practico de la regression en mediana Trabajos de Estadistica, 8, 157-173.

Bijan Bidabad (1987a) Least absolute error estimation. The First International Conference on Statistical Data Analysis Based on the L1‎‎ norm and Related Methods, Neuchatel, Switzerland. http://www.bidabad.com/doc/lae-I.pdf

Bijan Bidabad (1987b) Least absolute error estimation, part II. Submitted to the First International Conference on Statistical Data Analysis Based on the L1‎‎ norm and Related Methods, Neuchatel, Switzerland. http://www.bidabad.com/doc/lae-II.pdf

Bijan Bidabad (1988a) A proposed algorithm for least absolute error estimation. Proc. of the Third Seminar of Mathematical Analysis. Shiraz Univ., 24-34, Shiraz, Iran.

Bijan Bidabad (1988b) A proposed algorithm for least absolute error estimation, part II. Proc. of the Third Seminar of Mathematical Analysis, Shiraz Univ., 35-50, Shiraz, Iran.

Bijan Bidabad (1989a) Discrete and continuous L1‎‎ norm regressions, proposition of discrete approximation algorithms and continuous smoothing of concentration surface, Ph.D. thesis, Islamic Azad Univ., Tehran, Iran. http://www.bidabad.com/doc/L1-norm-thesis-en.pdf

Bijan Bidabad (1989b) Discrete and continuous L1‎‎ norm regressions, proposition of discrete approximation algorithms and continuous smoothing of concentration surface, Ph.D. thesis, Islamic Azad Univ., Tehran, Iran. Farsi translation. http://www.bidabad.com/doc/L1-norm-thesis-fa.pdf

Bijan Bidabad (2005). L1 norm based computational algorithms. http://www.bidabad.com/doc/l1-article6.pdf

Bijan Bidabad (2005). L1 norm solution of overdetermined system of linear equations. http://www.bidabad.com/doc/l1-article5.pdf

Bijan Bidabad (2005). L1 norm based data analysis and related methods. http://www.bidabad.com/doc/l1-articl1.pdf

Bijan Bidabad (2005). New algorithms for the L1 norm regression. http://www.bidabad.com/doc/l1-article2.pdf

Bijan Bidabad (2005). Comparative study of the L1 norm regression algorithms. http://www.bidabad.com/doc/l1-articl3.pdf

Bijan Bidabad (2005). Continuous L1 norm estimation of Lorenz curve. http://www.bidabad.com/doc/l1-articl4.pdf

Bijan Bidabad (1993). Estimating Lorenz curve for Iran by using continuous L1 norm estimation, Economics and Management Journal, Islamic Azad University, No. 19, winter 1993, pp. 83-101. http://www.bidabad.com/doc/iraninc-l1.pdf

Bijan Bidabad (2005). Continuous L1 norm estimation of Lorenz curve when probability density function is known.

Bijan Bidabad (2005). USA Income distribution counter-business-cyclical trend (Estimating Lorenz curve using continuous L1 norm estimation). First meeting of the Society for the Study of Economic Inequality (ECINEQ), Palma de Mallorca, Spain, July 20-22, 2005.

http://www.uib.es/congres/ecopub/ecineq/general.html

http://www.uib.es/congres/ecopub/ecineq/papers/039Bidabab.pdf

http://www.bidabad.com/doc/estimating-lorenz-us.pdf

Bijan Bidabad, Hamid Shahrestani. (2008) An implied inequality index using L1 norm estimation of Lorenz curve. Global Conference on Business and Finance Proceedings. Mercedes Jalbert, managing editor, ISSN 1931-0285 CD, ISSN 1941-9589 Online, Volume 3, Number 2, 2008, The Institute for Business and Finance Research, Ramada Plaza Herradura, San Jose, Costa Rica, May 28-31, 2008, pp. 148-163. Global Journal of Business Research, Vol. 4, No. 1, 2010, pp.29-45.

http://www.bidabad.com/doc/L1-Implied-inequality-index-4.pdf

http://www.theibfr.com/archive/ISSN-1941-9589-V3-N2-2008.pdf

http://www.bidabad.com/doc/SSRN-id1631861.pdf

P. Bloomfield, W. Steiger (1980) Least absolute deviations curve fitting. SIAM J. Sci. Statist. Comput. 1, 290-301.

P. Bloomfield, W. Steiger (1983) Least absolute deviations: theory, applications and algorithms. Birkhauser, Boston.

R.J. Boscovich (1757) De litteraria expeditione per pontificiam ditionem, et synopsis amplioris operis..., 'Bononiensi' scientiarum et artum instituto atque academia commetarii, vol.4, 353-396. Reprinted with a Serbo-Croatian translation by N. Cubranic, Institute of higher geodesy, University of Zagreb 1961.

R.J. Boscovich (1760) De recentissimis graduum dimensionibus et figura, ac magnitudine terrae inde derivanda. Philosophiae recentioris, a benedicto stay in Romano archigynasis publico eloquentare professore, vesibus traditae, libri X, cum adnotianibus et supplementas P. Rugerii Joseph Boscovich, S.J., 2, 406-426.

A.L. Bowley (1902) Methods of representing the statistics of wages and other groups not fulfilling the normal law of error, II: applications to wage statistics and other groups. J. of the Roy. Stat. Soc., 65, 331-54.

A.L. Bowley (1928) F.Y. Edgeworth's contributions to mathematical statistics. London, Roy. Stat. Soc..

D. Bradu (1987a) L1 fit, median polish and conjugate gradients. CSIR Tech. Rep. TWISK 509, National Res. Inst. For Math Sci. CSIR, Pretoria.

D. Bradu (1987b) An ε-median polish algorithm. CSDA, 5, 327-336.

J.J. Brennan, L.M. Seiford (1987) Linear programming and L1 approximation using the method of vanishing Jacobians. CSDA, 5, 263-276.

B.M. Brown (1980) Median estimates in a simple linear regression. Australian J. of Stat., 22, 154-165.

C. Bruen (1938) Methods for the combination of observations modal points or most lesser-deviations, mean loci or least squares, and mid point of least range or least greatest-deviation. Metron 13, 61-140.

J. Chamber (1971) Algorithm 410: partial sorting. Comm. ACM, 14, 357-358.

J.M. Chambers (1977) Computational methods for data analysis Wiley, New York.

Charnes, W.W. Cooper, R.O. Ferguson (1955) Optimal estimation of executive compensation by linear programming. Manag. Sci. 1, 138-151.

E.W. Cheney (1966) Introduction to approximation theory, McGraw-Hill, New York.

F.H. Clarke (1983) Optimization and nonsmooth analysis. Wiley, New York

A.R. Conn (1976) linear programming via a nondifferentiable penalty function. SIAM J. Numer. Anal., 13, 145-154.

D.C. Crocker (1969) Linear programming technique in regression analysis, the hidden danger. A.I.E.E. Trans., 1, 112-126.

M. Davies (1976) Linear approximation using the criterion of least total deviations. J. Roy. Stat. Soc. B29, 101-109.

T.E. Dielman (1984) Least absolute value estimation in regression models: An annotated bibliography. Comm. Stat. 13, 513-41.

T. Dielman, R. Pfaffenberger (1982) LAV (Least Absolute Value) estimation in linear regression: A review, TIMS studies in the Manag. Sci.,19, 31-52.

T. Dielman, R. Pfaffenberger (1984) Computational algorithms for calculating least absolute value and Chebyshev estimates for multiple regression. Amer. J. Math. Manag. Sci., 4, 169-197.

P.J. Dhrymes (1978) Mathematics for econometrics. Springer-Verlag, New York.

Y. Dodge (1987) An introduction to statistical data analysis L1-norm based. In Y. Dodge (ed.) Statistical data analysis based on the L1 norm and related methods. North-Holland. Reprinted in CSDA, 5, 239-254.

A.F. Dufton (1928) Correlation. Nature, 121, 866.

F.Y. Edgeworth (1883) The method of least squares. Philosophical Magazine, 16, 360-375.

F.Y. Edgeworth (1887a) On observations relating to several quantities. Hermathena, 6, 279-285.

F.Y. Edgeworth (1887b) A new method of reducing observations relating to several quantities. Philosophical Magazine, 24, 222-223.

F.Y. Edgeworth (1888) On a new method of reducing observation relating to several quantities. Philosophical Magazine, 25, 184-191.

F.Y. Edgeworth (1902) Method of representing statistics of wage and other groups not fulfilling the normal law of error, I: mathematical considerations. J. Roy. Stat. Soc., 65, 325-331.

F.Y. Edgeworth (1923) On the use of medians for reducing observations relating to several quantities. Philosophical Magazine, 6th series, 46, 1074-1088.

Eisenhart (1961) Boscovich and the combination of observations. Ch. 9 of Whyte (1961, 200-212) reprinted in Kendall and Plackett (1977) studies in the history of statistics and probability, vol.II, Charles Griffin and Co. Ltd., High Wycombe 88-100.

J.E. Estienne (1926-28) Introduction a une theorie rationnelle des erreurs d'observation. Revue d'artillerie 97(1926), 421-441; 98(1928), 542-562; 100(1927), 471-487.

R.C. Fair (1974) On the robust estimation of econometric models. Ann. Econ. Soc. Measurement, 3, 667-77.

R.W. Farebrother (1987b) The historical development of the L1 and L∞ estimation procedures. In Y. Dodge (ed.) Statistical data analysis based on the L1 norm and related methods. North-Holland. 37-64.

R.W. Farebrother (1987c) A simple recursive procedure for the L1 norm fitting of a straight line. Work. Pap., Dept. of Econometrics and Social Stat. University of Manchester, Manchester, M13 9PL, UK.

W.D. Fisher (1961) A note on curve fitting with minimum deviations by linear programming. JASA, 11, 359-362.

R. Fourer (1985a) A simplex algorithm for piecewise-linear programming I: derivation and proof. Math. Prog., 33, 204-233.

R. Fourer (1985b) A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy. Tech. Rep., 85-03 (revised), Dept. of Ind. Engin. and Manag. Sci.,The Tech. Inst., Northwestern University, Evanston, Illinois.

R. Fourer (1986) A simplex algorithm for piecewise-linear programming III: Computational analysis and applications. Tech. Rep., 86-03, Dept. of Ind. Engin. and Manag. Sci., The Tech. Inst., Northwestern University, Evanston, Illinois.

J.B.I. Fourier (1824) Solution d'une question particuliere au calcul des inegalites, second extrait. Histoire de l'academie des sciences pour 1824, 47-55. Reprinted in oeuvres de Fourier, 2. Paris, 1980, Gauthier-Villars, 325-328.

G. Galilei (1632) Dialogo dei massimi sistemi.

C.B. Garcia, F.G. Gould (1983) An application of homotopy to solving linear programs. Math. Prog. 27, 263-282.

C.F. Gauss (1809) Theoria motus corporum coelestium. In F. Perthes, I.H. Besser, Sectionbus conicis solem ambientium, Hamburg. Reprinted in his werke, vol. 7, F. Pethes, Gotha 1871. English translation by C.H. Davis, Little, Brown and Co., Boston, 1857. Reprinted by Dover Pub. New York, 1963.

T.E. Harris (1950) Regression using minimum absolute deviations. Am. Statist., 4, 14-15.

H.L. Harter (1974a) The method of least squares and some alternative, I. Int. Stat. Rev., 42, 147-174.

H.L. Harter (1974b) The method of least squares and some alternative, II. Int. Stat. Rev., 42, 235-264.

H.L. Harter (1975a) The method of least squares and some alternative, III. Int. Stat. Rev., 43, 1-44.

H.L. Harter (1975b) The method of least squares and some alternative, IV Int. Stat. Rev., 43, 125-190, 273-278.

H.L. Harter (1975c) The method of least squares and some alternative, V. Int. Stat. Rev., 43, 269-272.

H.L. Harter (1976) The method of least squares and some alternative, VI. Int. Stat. Rev., 44, 113-159.

P.W. Holland, R.E. Welsch (1977) Robust regression using iteratively reweighted least-squares. Comm. Stat., A6, 813-827.

L. Horvath (1987) Asymptotic normality of Lp-norms of density estimators. Tech. Rep. series of Lab Res. Stat. Prob., no.3, Carleton University, Ottawa, Canada.

H. Imai, K. Kato, P. Yamamoto (1987) A linear-time algorithm for linear L1 approximation of points. Tech. Rep. CSCE-87-C30. Dept. of Comp. Sci. and Commun. Engin., Kyushu University 36, Fukuoka 812, Japan.

L.A. Josvanger, V.A. Sposito (1983) L1-norm estimates for the simple regression problem. Comm. Stat. B12, 215-21.

O.J. Karst (1958) Linear curve fitting using least deviations. JASA, 53, 118-132.

Y. Kawara (1979) Straight line fitting by minimizing the sum of absolute deviations. J. of the Japan Stat. Soc., 9, 47-64.

J.H.B. Kemperman (1984) Least absolute value and median polish. In Y.L. Tong (ed.), Inequalities in statistics and probability (IMS Lecture notes monograph series, vol.5), Inst. of Math. Stat., Hayward, CA, 84-113.

W.J. Kennedy, J.E. Gentle (1980) Statistical computing. New York, Marcel Dekker.

P.S. Laplace (1793) Sur quelques points du system du monde. Memoires de l'Academie Royale des Science de Paris. Annee 1789, 1-87. Reprinted in Oeuvres completes de Laplace II. Paris, Gauthier-Villars, 1985, 477-558.

P.S. Laplace (1799) Traite des mecanique celeste, 2. Paris; J.B.M. Depart. Reprinted as oeuvres completes de Laplace, 2. Paris; Gauthier-Villars 1878, 116-165.

P.S. Laplace (1812) Theorie analytique des probabilites, Mme courcier Paris 1820 Reprinted in his oeuvres, vol.7, Imprimerie Royale, Paris, 1847, and Gauthier-Villars et fils, Paris 1886.

P.S. Laplace (1818) Duexieme supplement to Laplace (1812).

C.L. Mathieu (1816) Sur les experiences du pendule, faites par les navigateurs espagnol, en differens points du globe. Connaissance des tems, 314-332.

C.R. McConnell (1987) On computing a best discrete L1 approximation using the method of vanishing Jacobians. CSDA, 5, 277-288.

G.F. McCormick, V.A. Sposito (1975) A note on L1 estimation based on the median positive quotient. Appl. Stat., 24, 347-350.

M.S. Meketon (1986) Least absolute value regression. Work. Pap., AT&T Bell Laboratories, Holmdel, N.J.

R.M. Moroney (1961) The Haar problem in L1. Proc. Amer. Math. Soc., 12, 793-795.

S.C. Narula, J.F. Wellington (1985) Interior analysis for the minimum sum of absolute errors regression. Technometrics, 27, 181-188.

S.C. Narula, J.F. Wellington (1987) An efficient algorithm for the MSAE and MMAE regression problems. Work. Pap., Virginia CommonWealth University, Richmond, VA 23284.

M.R. Osborne (1987) The reduced gradient algorithm. In Y. Dodge (ed.) Statistical data analysis based on the L1 norm and related methods. North-Holland. 95-108.

M.R. Osborne, S.A. Pruess, R.S. Womersley (1986) Concise representation of generalized gradients. J. of Austra. Math. Soc., Ser. B, 28, 57-74.

M.R. Osborne, G.A. Watson (1985) An analysis of the total approximation problem in separable norms, and an algorithm for the total L1 problem. SIAM J. Sci. Stat. Comp., 6, 410-424.

M.J. Panik (1976) Classical optimization: foundation and extensions. North-Holland, Amsterdam.

U. Peters, C. Willms (1983) Up- and down-dating procedures for linear L1 regression. OR Spektrum 5, 229-239.

P. Pilibossian (1987) A direct solving algorithm for a linear regression according to L1-norm criteria. Work. Pap., L.S.T.A. Universite, Paris VI

P. Rabinowitz (1968) Application of linear programming to numerical analysis. SIAM Rev., 10, 121-159.

P. Rabinowitz (1970) Mathematical programming and approximation. In A. Talbot (ed.) Approximation Theory. Academic Press, 217-231.

M.R. Rao, V. Srinivasan (1972) A note on Sharpe's algorithm for minimum sum of absolute deviations in a simple regression problem. Manag. Sci., 19, 222-225.

E.C. Rhodes (1930) Reducing observations by the method of minimum deviations. Philo. Mag., 7th series, 9, 974-92.

J.R. Rice (1964c) The approximation of functions, vol. I, linear theory. Reading Mass:, Addison-Wesley.

P.D. Robers, A. Ben-Israel (1969) An interval programming algorithm for discrete linear L1 approximation problem. J. Approx. Theory, 2, 323-336.

P.D. Robers, S.S. Robers (1973) Algorithm 458: discrete linear L1 approximation by interval linear programming. Comm. ACM, 16, 629-633.

E. Ronchetti (1987) Bounded influence in regression: a review. In Y. Dodge (ed.) Statistical data analysis based on the L1 norm and related methods. North-Holland, 65-80.

A.N. Sadovski (1974) AS74: L1-norm fit of a straight line. Appl. Stat. 23, 244-248.

J.P. Schellhorn (1987) Fitting data through homotopy methods In Y. Dodge (ed.) Statistical data analysis based on the L1 norm and related methods. North-Holland. 131-138.

E.J. Schlossmacher (1973) An iterative technique for absolute deviations curve fitting. JASA 68, 857-865.

E. Seneta (1983) The weighted median and multiple regression. Austral. J. Stat., 25(2), 370-377.

E. Seneta, W.L. Steiger (1984) A new LAD curve-fitting algorithm: slightly overdetermined equation system in L1. Discrete Applied Math., 7, 79-91.

Shanno, R.L. Weil (1970) Linear programming with absolute value functionals. Oper. Res., 19, 120-124.

W.F. Sharpe (1971) Mean-absolute deviation characteristic lines for securities and portfolios. Manag. Sci., 18, B1-B13.

H.D. Sherali, B.O. Skarpness, B. Kim (1987) An assumption-free convergence analysis for a perturbation of the scaling algorithm for linear programs, with application to the L1 estimation problem. Dept. of Ind. Engin. and OR, Virginia Polytechnic Inst. and State University, Blacksburg, Virginia.

O.B. Sheynin (1973) R.J. Boscovich's work on probability. Archive for history of exact sciences, vol. 9, 306-324, and vol. 28, 173.

R.R. Singleton (1940) A method for minimizing the sum of absolute values of deviations. Annals of math. Stat., 11, 301-310.

S.A. Soliman, G.S. Christensen, A. Rouhi (1988) A new technique for curve fitting based on minimum absolute deviations. CSDA, 6(4), 341-352.

V.A. Sposito (1976) A remark on algorithm AS74, L1 norm fit of a straight line. Appl. Stat., 25, 96-97.

V.A. Sposito (1987a) On median polish and L1 estimators.CSDA, 5, 155-162.

V.A. Sposito, W.J. Kennedy, J.E. Gentle (1977) AS110: Lp norm fit of a straight line. Appl. Stat., 26, 114-118.

V.A. Sposito, G.F. McCormick, W.J. Kennedy (1975) L1 estimation strategies based on the simplex algorithm. In Proc. of the eighth symposium on the interface, J.W. France (ed.) Health science computing facility. UCLA, Los Angeles.

V.A. Sposito, W.C. Smith (1976) On a sufficient and necessary condition for L1 estimation. Appl. Stat., 25, 154-157.

K. Spyropoulos, E. Kiountouzis, A. Young (1973) Discrete approximation in the L1 norm. Comp. J., 16, 180-186.

W.L. Steiger (1980) Linear programming via L1 curve fitting beats simplex. Abstracts, AMS, 80T-C26, 385-386.

S.M. Stigler (1981) Gauss and invention of least squares. Annals of Stat., 9, 465-474.

S.M. Stigler (1984) Studies in the history of probability and statistics XL, Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation. Biometrica, 71, 3, 615-620.

J. Svanberg (1805) Exposition des operations faites en lappnie pour la determination d'un arc du meridien en 1801, 1802 et 1803,... Stockholm.

J.W. Tukey (1977) Exploratory data analysis. Reading, Mass. Addison-Wesley.

H.H. Turner (1887) On Mr. Edgeworth's method of reducing observations relating to several quantities. Phil. Mag. (5th series), 24, 466-470.

K.H. Usow (1967a) On L1 approximation: computation for continuous functions and continuous dependence. SIAM J. of Numer. Anal., 4, 70-88.

K.H. Usow (1967b) On L1 approximation: computation for discrete functions and discretization effect. SIAM J. Numer. Anal., 4, 233-244.

J.F. Van Beeck-Calkoen (1816) Ver de theoric der Gemiddelde Waardij. Verhandlingen der K. Nederlandandsch Instituut Can Wetenschappen, 2, 1-19.

B.A. Von Lindenau (1806) Uber den Gebrauch der Gradmessungen zur bestimmung der gestalt der erde. Monatliche correspondenz zur befar derung der Erd-und Himmels-kunde, 14, 113-158.

H.M. Wagner (1959) Linear programming technique for regression analysis. JASA, 54, 202-212.

G.A. Watson (1981) An algorithm for linear L1 approximation of continuous functions. IMA J. Num. Anal., 1, 157-167.

G.O Wesolowsky (1981) A new descent algorithm for least absolute value regression problem. Comm. Stat., B10, 479-491.

Published
2019-06-13
How to Cite
Bidabad, B. (2019). L1 Norm Based Computational Algorithms. Bangladesh Journal of Multidisciplinary Scientific Research, 1(1), 50-68. https://doi.org/10.46281/bjmsr.v1i1.315
Section
Research Paper/Theoretical Paper/Review Paper/Short Communication Paper