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Solving nonlinear algebraic systems in MATLAB 83

2

2

2

2 2
1
Figure 2.15 Trajectory of reduced-step Newton method with the guess (2,−1).

The full Newton step at x [k] satisfies the linear system

B [k] (n) =− f x [k] (2.85)
p
Therefore, it is also a global minimum of the quadratic cost function
[k] & [k] & 2
m (p) = 1 & B p + f x [k] & (2.86)
2 2
2 T
& & [k] [k] 1 T [k] [k]
= 1 & f x [k] & + f x · B p + p B B p
2 2 2
[k]
If we were to apply a minimization algorithm to m (p), we would find the full Newton step
(n)
p . We note, however, that p (n) becomes erratic when we are far from the solution. We
therefore expect there to be some trust radius [k] (that we modify at each iteration) such
[k]
that we expect the quadratic model m (p) to be a valid measure of locating the solution
[k]
only if p 2 < . We obtain the update vector by solving the constrained problem
[k]
min m (p) subject to |p| < [k] (2.87)
This technique is the basis of the MATLAB routine fsolve, whose use is demonstrated
below. A further discussion of the trust-region method, and the efficient dogleg method of
implementing it, is provided in Chapter 5.

Solving nonlinear algebraic systems in MATLAB

Although MATLAB contains a built-in routine, fsolve, for solving systems of multiple
nonlinear algebraic equations (using the trust-region Newton method discussed above), it
is part of the Optimization Toolkit and so is not available in every installation of MATLAB.
reduced Newton.m implements the reduced-step Newton algorithm, and can be used if