Directional Newton methods in 𝑛 variables

Levin, Yuri; Ben-Israel, Adi

doi:10.1090/S0025-5718-01-01332-1

Directional Newton methods in $n$ variables
HTML articles powered by AMS MathViewer

by Yuri Levin and Adi Ben-Israel PDF

Math. Comp. 71 (2002), 251-262 Request permission

Abstract:

Directional Newton methods for functions $f$ of $n$ variables are shown to converge, under standard assumptions, to a solution of $f(\mathbf {x})=0$. The rate of convergence is quadratic, for near-gradient directions, and directions along components of the gradient of $f$ with maximal modulus. These methods are applied to solving systems of equations without inversion of the Jacobian matrix.

References

Adi Ben-Israel, A Newton-Raphson method for the solution of systems of equations, J. Math. Anal. Appl. 15 (1966), 243–252. MR 205445, DOI 10.1016/0022-247X(66)90115-6
Adi Ben-Israel, Newton’s method with modified functions, Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995) Contemp. Math., vol. 204, Amer. Math. Soc., Providence, RI, 1997, pp. 39–50. MR 1442993, DOI 10.1090/conm/204/02621
I. S. Berezin and N. P. Shidkov, Computing methods. Vols. I, II, Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Translated by O. M. Blunn; translation edited by A. D. Booth. MR 0174165
Wendell Fleming, Functions of several variables, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1977. MR 0422527
Carl-Erik Fröberg, Numerical mathematics, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1985. Theory and computer applications. MR 790312
Y. Levin and A. Ben-Israel, Maple programs for directional Newton methods are available at: ftp://rutcor.rutgers.edu/pub/bisrael/Newton-Dir.mws.
Gábor Lukács, The generalized inverse matrix and the surface-surface intersection problem, Theory and practice of geometric modeling (Blaubeuren, 1988) Springer, Berlin, 1989, pp. 167–185. MR 1042329
A. M. Ostrowski, Solution of equations in Euclidean and Banach spaces, Pure and Applied Mathematics, Vol. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Third edition of Solution of equations and systems of equations. MR 0359306
Anthony Ralston and Philip Rabinowitz, A first course in numerical analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Bogotá, 1978. MR 0494814
J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543