A COMBINED MODIFICATION OF THE NEWTON'S METHOD FOR SYSTEMS OF NONLINEAR EQUATIONS



M.T. Monteiro and Edite M.G.P. Fernandes

Universidade do Minho, Braga, Portugal


ABSTRACT


For relatively small problems Newton's method seems to be a good choice for
solving systems of nonlinear equations, F(x)=0. Quadratic convergence can be
proved from good starting approximations. When the jacobian matrix, J, is 
singular or badly conditioned, the Newton equations do not give a reliable
step and need not converge. Secant methods use secant approximations to the
jacobian and although singularity is not present in general, two fundamental
problems can occur in implementing the iterative process. It is not easy to
get a good initial approximation to the jacobian and less satisfactory points
can be generated if the step happens to be an uphill direction for f(x), a
function defined by 1/2 ||F(x)||^2. The objective function, f(x), of the
related minimum problem takes on its absolute minimum 0 at a solution of
F(x)=0. To improve the performance of Newton's method on general problems,
the Newton step is modified and safeguards are included so that the
corresponding step, d, is pointing towards the root of F(x)=0, in the sense
that (J^T F)^T d < 0, where J^T F is the gradient of f(x). 

We motivate our approach by using the same simple analysis that we used in 
analysing nonlinear optimization problems, Fernandes (1995). The direction d 
is then taken as a suitable linear combination of Newton and steepest descent
directions,

d = - [ w (J^T J)^(-1) + (1-w) I ] J^T F.

In this paper we describe how the coefficients of the linear combination can
be generated to make effective use of the two component steps. Finally we
show some numerical results that demonstrate the usefulness of the
combined modification presented in the paper.


Edite M.G.P. Fernandes, Newton based exact penalty techniques for nonlinear
optimization with constraints, Operations Research Proceedings 1994, 
(39 - 44), Springer-Verlag, 1995.



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The speaker will be: Edite M.G.P. Fernandes