LevenbergMarquardt Class

A class for finding minimum of a least squared problem on the form

f(x) = ||fv(x)||

where x and fv are multi-dimensional, of dimension dimx and dimf respectively.

It is using a Levenberg-Marquardt method or a Newton-Raphson method, according to UseLm.

Finding zeros of a nonlinear system of equations, i.e.

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fv(x) = [0]

is a special case of a least square minimization problem, where the minimum is in the zero point and fv(x*) = (0).

If the Jacobian is available, it can be provided, otherwise it works by numerically calculating the Jacobian.

Inheritance Hierarchy

SystemObject
DHIMath.NumericsLevenbergMarquardt

Namespace: DHIMath.Numerics
Assembly: DHI.Mike1D.Generic (in DHI.Mike1D.Generic.dll) Version: 16.0.0.0 (11.1.1.1111)

Syntax

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public class LevenbergMarquardt

The LevenbergMarquardt type exposes the following members.

Constructors

	Name	Description
	LevenbergMarquardt(Int32, Int32, VectorFunction)	Constructor for minimizing/finding zeros of the function
	LevenbergMarquardt(Int32, Int32, VectorFunction, VectorFunction)	Constructor for minimizing/finding zeros of the function The jacobian must return the jacobian as a vector in column major ordering containing the coefficients [df1_dx1, df2_dx1, df1_dx2, df2_dx2].

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Properties

	Name	Description
	Fx	Current function value in iteration
	Jacobian	Setter for Jacobian.
	JacobianRecalc	Integer specifying how often the Jacobian is completely recalculated by a forward difference approximation. Every iteration can either performs a rank-1 update of the the Jacobian or completely recalculate the Jacobian. This number specifies how often the Jacobian should be completely recalculated. Set to 1 to recalculate in every iteration. Set to zero to never recalculate, but only use rank-1 updates. Default is Max(dimx,10). When a Jacobian function is provided, this parameter is not used.
	LambdaScaling	Scaling factor in J'J+lambdaI, replacing the diagonal of I by those in LambdaScaling By default they are [1,...,1]. They can not be negative.
	MaxFunctionEval	Maximum number of function evaluations. Default 200 This may not be hit exactly, since one iteration can trigger several function evaluations. In case a Jacobian is not provided, this can be set app. Copy MaxIter + dimx*MaxIter/JacobianRecalc, since the number of evaluations per iteration is 1 per default and then dimx evaluations per 1/JacobianRecalc iterations
	MaxIter	Maximum number of iterations. Default 100
	MaxLineSearchIter	Maximum number of line search iterations. Default 0.
	NumFunctionEval	Hold the number of function evaluations for the last solve.
	NumIter	Hold the actual number of iterations for the last solve.
	NumJacobianEval	Hold the number of Jacobian evaluations for the last solve.
	RelativeDx	Relative dx used when calculating Jacobian numerically by a forward difference approximation. Default value is 1e-6; The forward difference has the form: Copy dx = RelativeDx*Math.Abs(x) df/dx = (f(x+dx)-f(x)) / dx;
	Tau	used in starting value for Marquardt parameter: Copy lambda = Tau * max( diag(J(x0)'*J(x0)) ) Default value is 1e-3
	TolG	Tolerance value for steepest descent direction, used in stopping criteria, stopping when: Copy \|\|g\|\|_inf <= TolG Default value is 1e-6. It is recommended to set a lower value when not using LM (UseLm).
	TolX	Tolerance value for change in parameter vector, used in stopping criteria, stopping when: Copy \|\|dx\|\|_2 <= TolX*(\|\|x\|\|_2 + TolX) It combines a relative and absolute stopping criteria Default value is 1e-9.
	UseLineSearch	Flag specifying whether a line search approach is to be used. Default false.
	UseLm	Flag whether a LM algorithm should be used. Default true. If set to false the Newton-Raphson method will be used. The Newton-Raphson method can only be used for finding zeros, not for minimizing in generel, and only for square systems, dimx = dimf.
	X	Current value in iteration

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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.)
	JacobianTest(Int32, Int32, Double, VectorFunction, VectorFunction)	Calculates the 2-norm difference between the exact and the numeric jacobian, on an element-wise basis (not matrix norm, but element/vector norm)
	JacobianTest(Int32, Int32, Double, VectorFunction, VectorFunction, Double)	Calculates the 2-norm difference between the exact and the numeric jacobian, on an element-wise basis (not matrix norm, but element/vector norm)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	Solve	Minimize/Solves for zeros, using x as starting point and updating the value of x as a result. The user must check whether the solution is acceptable before using it.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Fields

	Name	Description
	Iterations	Stores all intermediate x-array solution values for all iterations, in case StoreIterations is set. If the StoreIterations is not set, this is null. The residual is decreasing with each x-array in the list.
	LambdaCutRel	Relative lambda value deceding when to set lambda to zero, in order to try to get quadratic convergence close to the solution. Default value is zero (disabled). Values must be positive and reasonable values are in the range [0,1], where a value close to zero will change behavior seldomly.
	LambdaDecreaseMinFac	When decreasing lambda, this sets the minimum decrease factor Default value is 1.0/5.0. Values must be positive and in the range ]0,1[.
	LambdaUpdateFactor	When updating lambda, an optimal update factor is calculated. This scales the update factor. Set greater than 1 in order to get a larger lambda, which can increase stability. Default value is 1. Value must be positive and larger than 1.
	NewValueAction	If set, the action method is called every time a new x value has been computed. If the x-array argument is changed, the method must return true, otherwise it should return false. Changing the x-values can be used to constrain the x values to a certain set of values. Changing the x-values may significantly decrease convergence speed, and it may produce a suboptimal solution, even though an optimal solution is nearby.
	StoreIterations	Flag indicating whether x-array solution values for all iterations should be stored. .

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Reference

DHIMath.Numerics Namespace