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

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class NonlinLeastSquares
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class NonlinLeastSquares
Visual C++
[SerializableAttribute]
public ref class NonlinLeastSquares

Remarks

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

\mathop {\min }\limits_{x \in R^n } \frac{1}{2}F\left( x \right)^T F\left( x \right) = \frac{1}{2}\sum \limits_{i = 1}^m {f_i } \left( x \right)^2

where m \ge n, F:\,\,R^n \to R^m , and f_i(x) is the i-th component function of F(x). From a current point, the algorithm uses the trust region approach:

\mathop {\min }\limits_{x_n \in R^n } \left \| {F\left( {x_c } \right) + J\left( {x_c } \right)\left( {x_n - x_c } \right)} \right\|_2

subject to

\left\| {x_n - x_c } \right\|_2 \le \delta _c

to get a new point x_n, which is computed as

x_n = x_c - \left( {J\left( {x_c } \right)^T J\left( {x_c } \right) + \mu _c I} \right)^{ - 1} J\left( {x_c } \right) ^T F\left( {x_c } \right)

where \mu _c = 0 if \ \delta _c \ge \left \|{\left( {J\left( {x_c } \right)^T J\left( {x_c } \right)} \right)^{-1} \,\,J\left( {x_c } \right)^T F\left( {x_c } \right)} \right\|_2 and \mu _c > 0 otherwise. F(x_c) and J(x_c) are the function values and the Jacobian evaluated at the current point x_c . This procedure is repeated until the stopping criteria are satisfied. The first stopping criteria occurs when the norm of the function is less than the property AbsoluteTolerance. The second stopping criteria occurs when the norm of the scaled gradient is less than the proprty GradientTolerance. The third stopping criteria occurs when the scaled distance between the last two steps is less than the property StepTolerance. For more details, see Levenberg (1944), Marquardt (1963), or Dennis and Schnabel (1983, Chapter 10).

A finite-difference method is used to estimate the Jacobian when the user supplied function, f, defines the least-squares problem. Whenever the exact Jacobian can be easily provided, f should implement NonlinLeastSquares.Jacobian.

Inheritance Hierarchy

System..::.Object
Imsl.Math..::.NonlinLeastSquares

See Also