Solves a nonlinear least-squares problem using a modified Levenberg-Marquardt algorithm.

The type double function is imsl_d_nonlin_least_squares.

A pointer to the solution x of the nonlinear least-squares problem. To release this space, use imsl_free. If no solution can be computed, then NULL is returned.

The function imsl_f_nonlin_least_squares is based on the MINPACK routine LMDER by Moré et al. (1980). It uses a modified Levenberg-Marquardt method to solve nonlinear least-squares problems. The problem is stated as follows:

where m ≥ n, F : ℜn → ℜm, and fi(x) is the i-th component function of F(x). From a current point, the algorithm uses the trust region approach,

to get a new point xn, which is computed as

where μc = 0 if δc ≥ ∥(J(xc)T J(xc))-1 J(xc)T F(xc)∥2 and μc > 0, otherwise. The value μc is defined by the function. The vector and matrix F(xc) and J(xc) are the function values and the Jacobian evaluated at the current point xc, respectively. This function is repeated until the stopping criteria are satisfied.

The first stopping criterion for imsl_f_nonlin_least_squares occurs when the norm of the function is less than the absolute function tolerance afcn_tol. The second stopping criterion occurs when the norm of the scaled gradient is less than the given gradient tolerance grad_tol. The third stopping criterion for imsl_f_nonlin_least_squares occurs when the scaled distance between the last two steps is less than the step tolerance step_tol. For more details, see Levenberg (1944), Marquardt (1963), or Dennis and Schnabel (1983, Chapter 10).

On some platforms, imsl_f_nonlin_least_squares can evaluate the user-supplied functions fcn and jacobian in parallel. This is done only if the function imsl_omp_options is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables

In this example, the nonlinear data-fitting problem found in Dennis and Schnabel (1983, p. 225),

where

is solved with the data t =(1, 2, 3) and y =(2, 4, 3).

In this example, imsl_f_nonlin_least_squares is first invoked to fit the following nonlinear regression model discussed by Neter et al. (1983, pp. 475-478):

where the ɛi’s are independently distributed each normal with mean zero and variance σ2. The estimate of σ2 is then computed as

Finally, the diagonal elements of this matrix are used together with imsl_f_t_inverse_cdf (see Chapter 9, Special Functions) to compute 95% confidence intervals on θ1 and θ2.