For F : &RR;ⁿ → &RR;^m, m ≥ n, the problem of minimizing ∥ F ( x ) ∥ ₂ encompasses the nonlinear root-finding problem (when m = n) and the least squares problem (when m > n). If F is approximated in a neighborhood of a current estimate x_c by the linear model M ( x_c + d ) = F ( x_c ) + J ( x_c ) d, and the next estimate x₊ = x_c + d_c is obtained by minimizing ∥ M ( x_c + d ) ∥ ₂ , we obtain the Newton ( m = n ) and Gauss-Newton ( m > n ) methods, provided the Jacobian J ( x_c ) ∈ &RR; ^{m × n} has maximal rank. (The authors choose not to distinguish here between the situations where J ( x_c ) is given analytically as F′ ( x_c ) and those where it is given as a difference approximation thereto.) Tensor analogues of these methods can be obtained by using a quadratic model. Such local methods may be globalized by incorporating a line search or trust region.

This paper describes a code package that implements a family of such methods. Details and underlying theory are, for the most part, cited rather than given here, but the broad outlines are adequately sketched. In particular, a two-dimensional trust region approach for tensor methods is developed. The package permits the user to select various configurations, such as analytical or numerical Jacobians, standard or tensor methods, and line search or trust region approaches. The relative performance of different configurations on a battery of test problems from the literature is evaluated. Since tensor methods are of special interest when the Jacobian is rank deficient, or nearly so, the original test problems were modified to obtain further examples of this sort. Performance comparisons were also made with the well-known nonlinear least squares code NL2SOL.

Tensor methods appear to be both more robust and more efficient than their standard counterparts (especially in rank-deficient cases); the trust region approach is more efficient than the line search method (though the line search approach is superior for some problems); and this code package compares favorably with NL2SOL. The conclusions are much more nuanced than a brief review might suggest, and the reasons for these results remain to be understood. This code package is best suited to moderate-sized problems with no special structure.