edu.rit.numeric

## Class NonLinearLeastSquares

- java.lang.Object
- edu.rit.numeric.NonLinearLeastSquares

public class NonLinearLeastSquaresextends Object

Class NonLinearLeastSquares provides a method for minimizing the sum of the squares of a series of nonlinear functions. There are*M*functions, each of which has*N*inputs. These functions are represented by an object that implements interface VectorFunction. The`solve()`method finds a vector**x**such that Σ_{i}[*f*_{i}(**x**)]^{2}is minimized. The inputs to and outputs from the`solve()`method are stored in the fields of an instance of class NonLinearLeastSquares. The Levenberg-Marquardt method is used to find the solution.The Java code is a translation of the Fortran subroutine

`LMDER`from the MINPACK library. MINPACK was developed by Jorge Moré, Burt Garbow, and Ken Hillstrom at Argonne National Laboratory. For further information, see http://www.netlib.org/minpack/.

### Field Summary

Fields Modifier and Type Field and Description `VectorFunction`

**fcn**The nonlinear functions*f*_{i}(**x**) to be minimized.`double[][]`

**fjac**The*M*×*N*-element Jacobian matrix.`double[]`

**fvec**The*M*-element result vector.`int`

**info**Information about the outcome.`int[]`

**ipvt**The*N*-element permutation vector for`fjac`.`int`

**M**The number of functions.`int`

**N**The number of arguments for each function.`int`

**nprint**Debug printout flag.`double`

**tol**Tolerance.`double[]`

**x**The*N*-element**x**vector for the least squares problem.

### Constructor Summary

Constructors Constructor and Description **NonLinearLeastSquares**(VectorFunction theFunction)Construct a new nonlinear least squares problem for the given functions.

### Field Detail

#### fcn

public final VectorFunction fcn

The nonlinear functions*f*_{i}(**x**) to be minimized.

#### M

public final int M

The number of functions.

#### N

public final int N

The number of arguments for each function.

#### x

public final double[] x

The*N*-element**x**vector for the least squares problem. On input to the`solve()`method,`x`contains an initial estimate of the solution vector. On output from the`solve()`method,`x`contains the final solution vector.

#### fvec

public final double[] fvec

The*M*-element result vector. On output from the`solve()`method, for*i*= 0 to*M*−1,`fvec`_{i}=*f*_{i}(**x**), where**x**is the final solution vector.

#### fjac

public final double[][] fjac

The*M*×*N*-element Jacobian matrix. On output from the`solve()`method, the upper*N*×*N*submatrix of`fjac`contains an upper triangular matrix**R**with diagonal elements of nonincreasing magnitude such that**P**^{T}(**J**^{T}**J**)**P**=**R**^{T}**R**where

**P**is a permutation matrix and**J**is the final calculated Jacobian. Column*j*of**P**is column`ipvt[j]`(see below) of the identity matrix. The lower trapezoidal part of`fjac`contains information generated during the computation of**R**.`fjac`is used to calculate the covariance matrix of the solution. This calculation is not yet implemented.

#### ipvt

public final int[] ipvt

The*N*-element permutation vector for`fjac`. On output from the`solve()`method,`ipvt`defines a permutation matrix**P**such that**JP**=**QR**, where**J**is the final calculated Jacobian,**Q**is orthogonal (not stored), and**R**is upper triangular with diagonal elements of nonincreasing magnitude. Column*j*of**P**is column`ipvt[j]`of the identity matrix.

#### tol

public double tol

Tolerance. An input to the`solve()`method. Must be >= 0. Termination occurs when the algorithm estimates either that the relative error in the sum of squares is at most`tol`or that the relative error between`x`and the solution is at most`tol`. The default tolerance is 1×10^{−6}.

#### info

public int info

Information about the outcome. An output of the`solve()`method. The possible values are:- 0 -- Improper input parameters. Also, an IllegalArgumentException is thrown.
- 1 -- Algorithm estimates that the relative error in the sum of squares is at most
`tol`. - 2 -- Algorithm estimates that the relative error between
`x`and the solution is at most`tol`. - 3 -- Conditions for
`info`= 1 and`info`= 2 both hold. - 4 --
`fvec`is orthogonal to the columns of the Jacobian to machine precision. - 5 -- Number of function evaluations has reached 100(
*N*+1). Also, a TooManyIterationsException is thrown. - 6 --
`tol`is too small. No further reduction in the sum of squares is possible. - 7 --
`tol`is too small. No further improvement in the approximate solution`x`is possible.

#### nprint

public int nprint

Debug printout flag. An input to the`solve()`method. If`nprint`> 0, the`subclassDebug()`method is called at the beginning of the first iteration and every`nprint`iterations thereafter and immediately prior to return. If`nprint`<= 0, the`subclassDebug()`method is not called. The default setting is 0.

### Constructor Detail

#### NonLinearLeastSquares

public NonLinearLeastSquares(VectorFunction theFunction)

Construct a new nonlinear least squares problem for the given functions. Field`fcn`is set to`theFunction`. Fields`M`and`N`are set by calling the`resultLength()`and`argumentLength()`methods of`theFunction`. The vector and matrix fields`x`,`fvec`,`fjac`, and`ipvt`are allocated with the proper sizes but are not filled in.- Parameters:
`theFunction`

- Nonlinear functions to be minimized.- Throws:
`NullPointerException`

- (unchecked exception) Thrown if`theFunction`is null.`IllegalArgumentException`

- (unchecked exception) Thrown if*M*<= 0,*N*<= 0, or*M*<*N*.

### Method Detail

#### solve

public void solve()

Solve this nonlinear least squares minimization problem. The`solve()`method finds a vector**x**such that Σ_{i}[*f*_{i}(**x**)]^{2}is minimized. On input, the field`x`must be filled in with an initial estimate of the solution vector, and the field`tol`must be set to the desired tolerance. On output, the other fields are filled in with the solution as explained in the documentation for each field.- Throws:
`IllegalArgumentException`

- (unchecked exception) Thrown if`tol`<= 0.`TooManyIterationsException`

- (unchecked exception) Thrown if too many iterations occurred without finding a minimum (100(*N*+1) iterations).

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