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Directional Newton methods in $n$ variables


Authors: Yuri Levin and Adi Ben-Israel
Journal: Math. Comp. 71 (2002), 251-262
MSC (2000): Primary 65H05, 65H10; Secondary 49M15
DOI: https://doi.org/10.1090/S0025-5718-01-01332-1
Published electronically: May 17, 2001
MathSciNet review: 1862998
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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 [Enhancements On Off] (What's this?)

  • 1. A. Ben-Israel, A Newton-Raphon method for the solution of sytems of equations, J. Math. Anal. Appl. 15(1966), 243-252. MR 34:5273
  • 2. A. Ben-Israel, Newton's method with modified functions, Contemp. Math. 204(1997), 39-50. MR 98c:65080
  • 3. I.S. Berezin and N.P. Zhidkov, Computing Methods, Pergamon Press, 1965. MR 30:4372
  • 4. W. Fleming, Functions of Several Variables, 2nd Edition, Springer, 1977. MR 54:10514
  • 5. C.-E. Fröberg, Numerical Mathematics: Theory and Computer Applications, Benjamin, 1985. MR 86h:65001
  • 6. Y. Levin and A. Ben-Israel, MAPLE programs for directional Newton methods are available at: ftp://rutcor.rutgers.edu/pub/bisrael/Newton-Dir.mws.
  • 7. G. Lukács, The generalized inverse matrix and the surface-surface intersection problem, pp. 167-185 in Theory and Practice of Geometric Modeling (W. Strasser and H.-P. Seidel, editors), Springer-Verlag, 1989. MR 91f:65041
  • 8. A.M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, 3rd Edition, Academic Press, 1973. MR 50:11760
  • 9. A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd edition, McGraw-Hill, 1978. MR 58:13599
  • 10. J. Stoer and K. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 1976. MR 83d:65002

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Additional Information

Yuri Levin
Affiliation: RUTCOR–Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Rd, Piscataway, New Jersey 08854-8003
Email: ylevin@rutcor.rutgers.edu

Adi Ben-Israel
Affiliation: RUTCOR–Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Rd, Piscataway, New Jersey 08854-8003
Email: bisrael@rutcor.rutgers.edu

DOI: https://doi.org/10.1090/S0025-5718-01-01332-1
Keywords: Newton method, single equations, systems of equations
Received by editor(s): October 27, 1999
Received by editor(s) in revised form: May 15, 2000
Published electronically: May 17, 2001
Additional Notes: The first author was supported by the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), Rutgers University
Article copyright: © Copyright 2001 American Mathematical Society

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