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SIAM J. NUMER. ANAL. Vol. 31, No. 1, pp. 252-260, February 1994
 

Summary: "
SIAM J. NUMER. ANAL.
Vol. 31, No. 1, pp. 252-260, February 1994
@ 1994 Society for lndustrial and Applied Mathematics
013
EFFICIENT NUMERICAL VALIDATION OF SOLUTIONS OF
NONLINEAR SYSTEMS.
G. ALEFELDt, A. GIENGERt, AND F. POTRAt
Abstract. A new stopping criterion for Newton's method is derived by combining the prop-
erties oi the Krawczyk operator and a corollary of the Newton-Kantorovich theorem. When this
criterion is satisfied the authors use the last three Newton iterates to compute an interval vector
that is very likely to contain a solution of the given nonlinear system. The existence oisuch a
solution is tested using Krawczyk's operator. Furthermore, each element from this interval vector
considered as an approximation to the solution has a relative error that is of the order oi the machine
precision. Extensive numerical testing has shown that the proposed method has very good practical
performance.
Keywords. nonlinear systems, Newton-Kantorovich theorem, validation of solutions
AMS subject classifications. 65HI0,65GlO
1. Introduction. Newton's method is the best-known algorithm for solving non-
linear systems of equations. The most famous theoretical result on the convergence of

  

Source: Alefeld, Götz - Institut für Angewandte und Numerische Mathematik & Fakultät für Mathematik, Universität Karlsruhe

 

Collections: Mathematics