Some efficient algorithms for solving systems of nonlinear equations
14. R. P. Brent,
Some efficient algorithms for solving systems of nonlinear equations,
SIAM J. Numerical Analysis 10 (1973), 327-344
Forsythe memorial issue).
We compare the Ostrowski efficiency of some methods for solving systems
of nonlinear equations without explicitly using derivatives. The methods
considered include the discrete Newton method, Shamanskii's method,
the two-point secant method, and Brown's methods. We introduce a class
of secant methods and a class of methods related to Brown's methods,
but using orthogonal rather than stabilized elementary transformations.
The idea of these methods is to avoid finding a new approximation to the
Jacobian matrix of the system at each step, and thus increase the efficiency.
Local convergence theorems are proved, and the efficiencies of the methods
are calculated. Numerical results are given, and some possible extensions
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