What Is Optimization Toolbox?

4 Large-Scale AlgorithmsNonlinear Least-SquaresAn important special case for f(x) is the nonlinear least-squares problem(4-12)where is a vector-valued function with component i of equal toThebasicmethodusedtosolvethisproblemisthesameasinthegeneralcase described in “Trust-Region Methods for Nonlinear Minimization” onpage 4-3. However, the structure of the nonlinear least-squares problem isexploited to enhance efficiency. In particular, an approximate Gauss-Newtondirection, i.e., a solution s to(4-13)(where J is the Jacobian of ) is used to help define the two-dimensionalsubspace . Second derivatives of the component function are not used.In each iteration the method of preconditioned conjugate gradients is usedto approximately solve the normal equations, i.e.,although the normal equations are not explicitly formed.4-12