NonlinLeastSquares (JMSL Numerical Library)

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:

$mathop {min }limits_{x in R^n } frac{1}{2}Fleft( x right)^T Fleft( x right) = frac{1}{2}sumlimits_{i = 1}^m {f_i } left( x right)^2$

where , , and is the i-th component function of F(x). From a current point, the algorithm uses the trust region approach:

$mathop {min }limits_{x_n in R^n } left| {Fleft( {x_c } right) + Jleft( {x_c } right)left( {x_n - x_c } right)} right|_2$

subject to

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

to get a new point , which is computed as

$x_n = x_c - left( {Jleft( {x_c } right)^T Jleft( {x_c } right) + mu _c I} right)^{ - 1} Jleft( {x_c } right)^T Fleft( {x_c } right)$

where if $delta _c ge left| {left( {Jleft( {x_c } right)^T Jleft( {x_c } right)} right)^{-1} ,,Jleft( {x_c } right)^T Fleft( {x_c } right)} right|_2$ 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

Nested Classes
Modifier and Type	Class and Description
`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.TooManyIterationsException` Too many iterations.

Constructor Summary

Constructors
Constructor and Description

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

Constructors
Constructor and Description
`NonlinLeastSquares(int m, int n)` Creates an object to solve a nonlinear least squares problem.

Method Summary

Methods
Modifier and Type	Method and Description
`int`	`getErrorStatus()` Get information about the performance of NonlinLeastSquares.
`int`	`getNumberOfThreads()` Returns the number of `java.lang.Thread` instances used for parallel processing.
`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`	`setNumberOfThreads(int numberOfThreads)` Sets the number of `java.lang.Thread` instances to be used for parallel processing.
`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.

getNumberOfThreads
```
public int getNumberOfThreads()
```
Returns the number of java.lang.Thread instances used for parallel processing.

Returns:
an int containing the number of java.lang.Thread instances used for parallel processing.

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
$frac{|{g_i}| * max(|x_i|,, frac{1}{s_i} )}{frac{1}{2} ||F(x)||_2^2 }$
where
$g_i = left ( J(x)^TF(x) right )_i * (f_s)^2_i ,,mbox{,}$
J(x) is the Jacobian, s = xscale, = fscale, and
$||F(x)||_2^2 = sum_{i=1}^m f_i(x)^2 ,mbox{.}$
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

setNumberOfThreads
```
public void setNumberOfThreads(int numberOfThreads)
```
Sets the number of java.lang.Thread instances to be used for parallel processing.

Parameters:
numberOfThreads - an int specifying the number of java.lang.Thread instances to be used for parallel processing. If numberOfThreads is greater than 1, then interface Function.f is evaluated in parallel and Function.f must be thread-safe. Otherwise, unexpected behavior can occur.
Default: numberOfThreads = 1.

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
$frac{| x_i - y_i |}{max(|x_i|, frac{1}{s_i})}$
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
```
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.

Class NonlinLeastSquares

Nested Class Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

NonlinLeastSquares

Method Detail

getErrorStatus

getNumberOfThreads

setAbsoluteTolerance

setDigits

setFalseConvergenceTolerance

setFscale

setGradientTolerance

setGuess

setInitialTrustRegion

setMaximumStepsize

setMaxIterations

setNumberOfThreads

setRelativeTolerance

setStepTolerance

setXscale

solve