com.imsl.math
Class NonlinLeastSquares

java.lang.Object
  com.imsl.math.NonlinLeastSquares

All Implemented Interfaces:

Serializable, Cloneable

public class NonlinLeastSquares
extends Object
implements Serializable, Cloneable

Nonlinear least squares.

NonlinLeastSquares is based on the MINPACK routine LMDIF 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 , , and is the i-th component function of F(x). From a current point, the algorithm uses the trust region approach:

subject to

to get a new point , which is computed as

where if and 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 value set in method setAbsoluteTolerance. The second stopping criteria occurs when the norm of the scaled gradient is less than the value set in method setGradientTolerance. The third stopping criteria occurs when the scaled distance between the last two steps is less than the value set by method setStepTolerance. 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.

See Also:

Finite Differences Example, Jacobian Example, Serialized Form

Nested Class Summary

static interface	NonlinLeastSquares.Function Public interface for the user supplied function to the NonlinLeastSquares object.
static interface	NonlinLeastSquares.Jacobian Public interface for the user supplied function to the NonlinLeastSquares object.
static class	NonlinLeastSquares.NoProgressException The algorithm is not making any progress, try new initial guess.
static class	NonlinLeastSquares.TooManyIterationsException Too many iterations.

Constructor Summary

NonlinLeastSquares(int m, int n)
Creates an object to solve a nonlinear least squares problem.

Method Summary

int	getErrorStatus() Get information about the performance of NonlinLeastSquares.
void	setAbsoluteTolerance(double absoluteTolerance) Set the absolute function tolerance.
void	setDigits(int ngood) Set the number of good digits in the function.
void	setFalseConvergenceTolerance(double falseConvergenceTolerance) Set the false convergence tolerance.
void	setFscale(double[] fscale) Set the diagonal scaling matrix for the functions.
void	setGradientTolerance(double gradientTolerance) Set the scaled gradient tolerance stopping critierion.
void	setGuess(double[] xguess) Set the initial guess of the minimum point of the input function.
void	setInitialTrustRegion(double initialTrustRegion) Set the initial trust region radius.
void	setMaximumStepsize(double maximumStepsize) Set the maximum allowable stepsize to use.
void	setMaxIterations(int maxIterations) Set the maximum number of iterations allowed.
void	setRelativeTolerance(double relativeTolerance) Set the relative function tolerance.
void	setStepTolerance(double stepTolerance) Set the scaled step tolerance.
void	setXscale(double[] xscale) Set the diagonal scaling matrix for the variables.
double[]	solve(NonlinLeastSquares.Function F) Solve a nonlinear least-squares problem using a modified Levenberg-Marquardt algorithm and a Jacobian.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

NonlinLeastSquares

public NonlinLeastSquares(int m,
                          int n)

Creates an object to solve a nonlinear least squares problem.

Parameters:

m - is the number of functions

n - is the number of variables. n must be less than or equal to m.

Method Detail

getErrorStatus

public int getErrorStatus()

Get information about the performance of NonlinLeastSquares.

Returns:

an int specifying information about convergence.

VALUE	DESCRIPTION
0	All convergence tests were met.
1	Scaled step tolerance was satisfied. The current point may be an approximate local solution, or the algorithm is making very slow progress and is not near a solution, or StepTolerance is too big.
2	Scaled actual and predicted reductions in the function are less than or equal to the relative function convergence tolerance RelativeTolerance.
3	Iterates appear to be converging to a noncritical point. Incorrect gradient information, a discontinuous function, or stopping tolerances being too tight may be the cause.
4	Five consecutive steps with the maximum stepsize have been taken. Either the function is unbounded below, or has a finite asymptote in some direction, or the maximum stepsize is too small.

See Also:

setRelativeTolerance(double), setStepTolerance(double)

setAbsoluteTolerance

public void setAbsoluteTolerance(double absoluteTolerance)

Set the absolute function tolerance.

Parameters:

absoluteTolerance - a double scalar value specifying the absolute function tolerance. When the norm of the function is less than absoluteTolerance the procedure stops.

Default: absoluteTolerance = 1.0e-32

Throws:

IllegalArgumentException - is thrown if absoluteTolerance is less than or equal to 0

setDigits

public void setDigits(int ngood)

Set the number of good digits in the function. If this member function is not called, the number of good digits is set to 7.

Parameters:

ngood - an int specifying the number of good digits in the user supplied function which defines the least-squares problem

Throws:

IllegalArgumentException - is thrown if ngood is less than or equal to 0

setFalseConvergenceTolerance

public void setFalseConvergenceTolerance(double falseConvergenceTolerance)

Set the false convergence tolerance. If this member function is not called, 100.0e-16 is used as the false convergence tolerance.

Parameters:

falseConvergenceTolerance - a double scalar value specifying the false convergence tolerance

Throws:

IllegalArgumentException - is thrown if falseConvergenceTolerance is less than or equal to 0

setFscale

public void setFscale(double[] fscale)

Set the diagonal scaling matrix for the functions.

Parameters:

fscale - a double array specifying the diagonal scaling matrix for the functions. The i-th component of fscale is a positive scalar specifying the reciprocal magnitude of the i-th component function of the problem. By default, the identity is used.

Throws:

IllegalArgumentException - is thrown if any of the elements of fscale is less than or equal to 0

setGradientTolerance

public void setGradientTolerance(double gradientTolerance)

Set the scaled gradient tolerance stopping critierion.

Parameters:

gradientTolerance - a double specifying the scaled gradient tolerance used in stopping criteria. The i-th component of the scaled gradient at x is calculated as

where

J(x) is the Jacobian, s = xscale, = fscale, and

When the norm of the scaled gradient is less than gradientTolerance the procedure stops.

By default, gradientTolerance = where is machine precision.

Throws:

IllegalArgumentException - is thrown if gradientTolerance is less than or equal to 0

setGuess

public void setGuess(double[] xguess)

Set the initial guess of the minimum point of the input function. If this member function is not called, an initial guess of 0.0 is used.

Parameters:

xguess - a double array specifying the initial guess of the minimum point of the input function

setInitialTrustRegion

public void setInitialTrustRegion(double initialTrustRegion)

Set the initial trust region radius. If this member function is not called, a default is set based on the initial scaled Cauchy step.

Parameters:

initialTrustRegion - a double scalar value specifying the initial trust region radius

Throws:

IllegalArgumentException - is thrown if initialTrustRegion is less than or equal to 0

setMaximumStepsize

public void setMaximumStepsize(double maximumStepsize)

Set the maximum allowable stepsize to use. If this member function is not called, maximum stepsize is set to a default value based on a scaled xguess.

Parameters:

maximumStepsize - a nonnegative double value specifying the maximum allowable stepsize.

Throws:

IllegalArgumentException - is thrown if maximumStepsize is less than or equal to 0

setMaxIterations

public void setMaxIterations(int maxIterations)

Set the maximum number of iterations allowed. If this member function is not called, the maximum number of iterations is set to 100.

Parameters:

maxIterations - an int specifying the maximum number of iterations allowed

Throws:

IllegalArgumentException - is thrown if maxIterations is less than or equal to 0

setRelativeTolerance

public void setRelativeTolerance(double relativeTolerance)

Set the relative function tolerance.

Parameters:

relativeTolerance - a double scalar value specifying the relative function tolerance

Default: relativeTolerance = 1.0e-20

Throws:

IllegalArgumentException - is thrown if relativeTolerance is less than or equal to 0

setStepTolerance

public void setStepTolerance(double stepTolerance)

Set the scaled step tolerance.

Parameters:

stepTolerance - a double scalar value specifying the scaled step tolerance used in stopping criteria. The i-th component of the scaled step between two points x and y is computed as

where s = xscale. When the scaled distance between the last two steps is less than stepTolerance the procedure stops.

By default, stepTolerance = where is machine precision.

Throws:

IllegalArgumentException - is thrown if stepTolerance is less than or equal to 0

setXscale

public void setXscale(double[] xscale)

Set the diagonal scaling matrix for the variables.

Parameters:

xscale - a double array specifying the diagonal scaling matrix for the variables. xscale is used in scaling the gradient and the distance between two points. See methods setGradientTolerance and setStepTolerance for more detail. By default, the identity is used.

Throws:

IllegalArgumentException - is thrown if any of the elements of xscale is less than or equal to 0

solve

public double[] solve(NonlinLeastSquares.Function F)
               throws NonlinLeastSquares.TooManyIterationsException,
                      NonlinLeastSquares.NoProgressException

Solve a nonlinear least-squares problem using a modified Levenberg-Marquardt algorithm and a Jacobian.

Parameters:

F - User supplied function that defines the least-squares problem. If F implements Jacobian then its Jacobian is used. Otherwise, a finite difference Jacobian is used.

Returns:

a double array of length n containing the approximate solution

Throws:

NonlinLeastSquares.TooManyIterationsException - is thrown if the number of iterations exceeds MaxIterations. MaxIterations is set to 100 by default.

NonlinLeastSquares.NoProgressException - is thrown when the algorithm is not making any progress. Try a new initial guess.