 
Summary: "
SIAM J. NUMER. ANAL.
Vol. 31, No. 1, pp. 252260, 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 NewtonKantorovich 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, NewtonKantorovich theorem, validation of solutions
AMS subject classifications. 65HI0,65GlO
1. Introduction. Newton's method is the bestknown algorithm for solving non
linear systems of equations. The most famous theoretical result on the convergence of
