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A system of nonlinear equations is expressed in the form F(x)=0, where
F is a vectorvalued function of the vector variable x:
.
Given an estimate x^{(k)} of a solution x^{*},
Newton's method computes the (hopefully improved) estimate x^{(k+1)} by
setting the local linear approximation to F at x^{(k)} to zero and solving
for x:
In this calculation,
J=J(x^{(k)}) is the Jacobian matrix of F at
x^{(k)}. Therefore x^{(k+1)} is defined by the formula

(1) 
If J happens to be singular, then the Newton step is undefined, and
a robust algorithm must be prepared to deal with such a situation. I will
deal with this issue later. For now, I will simply assume that J(x^{*})is nonsingular, in which case the continuity of J will ensure that
J(x^{(k)}) is nonsingular for any x^{(k)} sufficiently near x^{*}.
Mark S. Gockenbach
20030123