NonlinLeastSquares Class

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

Inheritance Hierarchy

SystemObject
Imsl.MathNonlinLeastSquares

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class NonlinLeastSquares

<SerializableAttribute>
Public Class NonlinLeastSquares

[SerializableAttribute]
public ref class NonlinLeastSquares

[<SerializableAttribute>]
type NonlinLeastSquares =  class end

The NonlinLeastSquares type exposes the following members.

Constructors

	Name	Description
	NonlinLeastSquares	Creates an object to solve a nonlinear least squares problem.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetFscale	Sets the diagonal scaling matrix for the functions.
	SetGuess	Sets the initial guess of the minimum point of the input function.
	SetXscale	Set the diagonal scaling matrix for the variables.
	Solve	Solve a nonlinear least-squares problem using a modified Levenberg-Marquardt algorithm and a Jacobian.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	AbsoluteTolerance	The absolute function tolerance.
	Digits	The number of good digits in the function.
	ErrorStatus	Get information about the performance of NonlinLeastSquares.
	FalseConvergenceTolerance	The false convergence tolerance.
	GradientTolerance	The scaled gradient tolerance.
	InitialTrustRegion	The initial trust region radius.
	MaximumIterations	The maximum number of iterations allowed.
	MaximumStepsize	The maximum allowable stepsize to use.
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	Parallel	Enable or disable performing NonlinLeastSquares.IFunction.F in parallel.
	RelativeTolerance	The relative function tolerance.
	StepTolerance	The scaled step tolerance.

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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 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 , 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. and are the function values and the Jacobian evaluated at the current point . 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 property 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.

Reference

Imsl.Math Namespace

Other Resources

Finite Differences Example

Jacobian Example