2.9.2 - Least Squares Refinement - JdlLeastSquares

John W. Campbell

2.9.2.1 Introduction

Class, fields and methods:

2.9.2.2 Class Details

Package:

Jdl.JdlLib;

Class name:

JdlLeastSquares

Class definition:

public class JdlLeastSquares

Extends:

Object

Implements:

none

Actions:

none

2.9.2.3 Accessible Fields

2.9.2.4 Refinement Methods

2.9.2.4.1 Introduction

Methods:

2.9.2.4.2 Least squares refinement - leastSquaresMQL

The function minimised is:

               m-1
       F(x) = Sigma [f(j)(x[0],x[1],...x[n-1])]**2         (m>=n)
               j=0

The refinement code has been directly derived from SLATEC public domain Fortran routines written by K.L. Hiebert (SNLA).

Method Definition:

public static int leastSquaresMQL (int m, int n, double[] x, double acc, double h, JdlLeastSquaresRefineable lfn, JdlDouble fmin, double[] fvecc)

Parameters List:

m

The number of functions f(j) occuring in the sum of squares.

n

The number of variables x[0] ... x[n-1] to be refined. Estimated values must be given on input and refined values are returned.

x

An array containing the 'n' refineable parameters x[0] ... x[n-1]. Estimated values must be given on input and refined values are returned.

acc

Tolerance in the Function value such that failure to decrease by more than this amount in an iteration signals completness. (Or, more precisely, used as the values for FTOL and XTOL in call to the SLATEC library routine DNLS1).

h

The step size to be used in calculating the partial derivatives (EPSFCN parameter in call to the SLATEC library routine DNLS1).

lfn

The object with the method to calculate the values of the functions f(j), j=0,1,..,m-1.

fmin

Returns the minimum value sum of squares in fmin.value.

fvecc

An array, dimensioned to at least m, returning the m values of f(j), j=0,1,...m-1 calculated using the refined parameter values.

Method Return:

Output flag:
= 0, OK
= -1, error in parameters
= 1, algorithm estimates that the relative error in the sum of squares is at most 'acc'
= 2, algorithm estimates that the relative error between 'x' and the solution is at most 'acc'
= 3, conditions for flag = 1 and flag = 2 both hold
= 4, 'fvecc' is orthogonal to the columns of the Jacobian to machine precision
= 5, number of calls to function method has reached 200*(n+1). = 6, 'acc' is too small. No further reduction in the sum of squares is possible
= 7, 'acc' is too small. No further improvement in the approximate solution 'x' is possible

2.9.2.4.3 Calculate sigma values - leastSquaresSigmas

Method Definition:

public static int leastSquaresSigmas(int m, int n, double[] x, JdlLeastSquaresRefineable lfn, double[] sigma, double[] fvecc)

Parameters List:

m

The number of functions f(j) occuring in the sum of squares.

n

The number of variables x[0] ... x[n-1] refined.

x

An array containing the 'n' refined parameters x[0] ... x[n-1].

lfn

The object with the method to calculate the values of the functions f(j), j=0,1,..,m-1.

sigma

This array returns the 'n' sigma values for the 'n' refined parameters.

fvecc

An array, dimensioned to at least m, returning the m values of f(j), j=0,1,...m-1 calculated using the refined parameter values.

Method Return:

Output flag:
= 0, OK
= 1, Error in the parameters
= 2, Singular matrix

2.9.2.5 Curve Fitting

2.9.2.5.1 Introduction

Methods:

2.9.2.5.2 Fit a curve - fitCurve

Method Definition:

public static JdlChebyshevPolynomial fitCurve(int n, JdlXY[] xy, int iord, JdlDouble fmin, JdlError err)

Parameters List:

n

The number of points to be used in the curve fitting.

JdlXY

The array of points to which the curve is to be fitted. 'x' is taken as the independent variable and 'y' as the dependent variable.

iord

The order of the polynomial to be used in the curve fitting (<= n-1).

fmin

Returns the least squares residual (sum of squares).

err

An error object in which to set error return codes. If an error is found 'err.err' will be set to true, an error message will be returned in 'err.msg1' and the flag 'err.flag' will be set as follows:
= 0, OK
= -1, Invalid parameters
= -2, Order too large for number of points
> 0, Flag as from the leastSquaresMQL(..) method

Method Return:

A JdlChebyshevPolynomial object containing the fitted curve parameters - null if an error was found (see 'err') (Note that the stored 'outlim' parameter will be set to 0.0 by this method).

2.9.2.5.3 Fit a curve - fitCurve

Method Definition:

public static JdlChebyshevPolynomial fitCurve(int n, JdlXY[] xy, double[] wt, int iord, JdlDouble fmin, JdlError err)

Parameters List:

n

The number of points to be used in the curve fitting.

JdlXY

The array of points to which the curve is to be fitted. 'x' is taken as the independent variable and 'y' as the dependent variable.

wt

Weights for the 'n' points to be fitted.

iord

The order of the polynomial to be used in the curve fitting (<= n-1).

fmin

Returns the least squares residual (sum of squares).

err

An error object in which to set error return codes. If an error is found 'err.err' will be set to true, an error message will be returned in 'err.msg1' and the flag 'err.flag' will be set as follows:
= 0, OK
= -1, Invalid parameters
= -2, Order too large for number of points
> 0, Flag as from the leastSquaresMQL(..) method
Values of -1, -2, and 4 are treated as fatal errors.

Method Return:

A JdlChebyshevPolynomial object containing the fitted curve parameters - null if a fatal error was found (see 'err') (Note that the stored 'outlim' parameter will be set to 0.0 by this method).