| Optimization Toolbox | |
Nonlinear Equations Implementation
Both the Gauss-Newton and trust-region dogleg methods are implemented in the Optimization Toolbox. Details of their implementations are discussed below.
Gauss-Newton Implementation
The Gauss-Newton implementation is the same as that for least-squares optimization. It is described in Gauss-Newton Implementation.
Trust-Region Dogleg Implementation
The key feature of this algorithm is the use of the Powell dogleg procedure for computing the step 
, which minimizes Eq. 3-25. For a detailed description, see Powell [36].
The step 
is constructed from a convex combination of a Cauchy step (a step along the steepest descent direction) and a Gauss-Newton step for 
. The Cauchy step is calculated as
where 
is chosen to minimize Eq. 3-24.
The Gauss-Newton step is calculated by solving
using the MATLAB \ (matrix left division) operator.
where 
is the largest value in the interval [0,1] such that 
. If 
is (nearly) singular, 
is just the Cauchy direction.
The dogleg algorithm is efficient since it requires only one linear solve per iteration (for the computation of the Gauss-Newton step). Additionally, it can be more robust than using the Gauss-Newton method with a line search.
| | Nonlinear Systems of Equations | Constrained Optimization | |
is chosen so that