Example 2: Bounded Least Squares

The nonlinear least-squares problem

\min \frac{1}{2}\sum\limits_{i = 0}^1 {f_i \left( x \right)^2 }

- 2 \le x_0 \le 0.5

-1 \le x_1 \le 2

where

f_0 (x) = 10(x_1 - x_0^2 ) \,\, {\rm{and}} \,\, f_1 (x) = (1 - x_0 )

is solved. An initial guess (-1.2, 1.0) is supplied, as well as the analytic Jacobian. The residual at the approximate solution is returned.

using System;
using Imsl.Math;

public class BoundedLeastSquaresEx2 : BoundedLeastSquares.IJacobian
{
	public void F(double[] x, double[] f)
	{
		f[0] = 10.0 * (x[1] - x[0] * x[0]);
		f[1] = 1.0 - x[0];
	}

	public void Jacobian(double[] x, double[] fjac)
	{
		fjac[0] = - 20.0 * x[0];
		fjac[1] = 10.0;
		fjac[2] = - 1.0;
		fjac[3] = 0.0;
	}


	public static void  Main(String[] args)
	{
		int m = 2;
		int n = 2;
		int ibtype = 0;
		double[] xlb = new double[]{- 2.0, - 1.0};
		double[] xub = new double[]{0.5, 2.0};
		
		BoundedLeastSquares.IJacobian rosbck = 
			new BoundedLeastSquaresEx2();
		BoundedLeastSquares zf = 
			new BoundedLeastSquares(rosbck, m, n, ibtype, xlb, xub);
		zf.SetGuess(new double[]{- 1.2, 1.0});
		zf.solve();
		new PrintMatrix("Solution").Print(zf.GetSolution());
		new PrintMatrix("Residuals").Print(zf.GetResiduals());
	}
}

Output

Solution
    0    
0  0.5   
1  0.25  

Residuals
    0   
0  0    
1  0.5
Link to C# source.