GenMinRes Example Using the Second Householder Implementation

Example 5: The Second Householder Implementation

The coefficient matrix in this example corresponds to the five-point discretization of the 2-d Poisson equation with the Dirichlet boundary condition. Assuming the natural ordering of the unknowns, and moving all boundary terms to the right hand side, we obtain a block tridiagonal matrix. (Consider the tridiagonal matrix T which has the value 4.0 down the main diagonal and -1.0 along the upper and lower co-diagonals. Then the coefficient matrix is the block tridiagonal matrix consisting of T's down the main diagonal and -I along the upper and lower codiagonals where I is the identity matrix.) Discretizing on a 20 x 20 grid implies that the coefficient matrix is 400 x 400. In the solution, the second Householder implementation is selected and we choose to update the residual vector by direct evaluation.

using System;
using Imsl.Math;
public class GenMinResEx5 : GenMinRes.IFunction
{
	//Creates a new instance of GenMinResEx5
	public GenMinResEx5()
	{
	}
	public void  Amultp(double[] p, double[] z)
	{
		int n = z.Length;
		int k = (int) Math.Sqrt(n);
		// Multiply by diagonal blocks
		for (int i = 0; i < n; i++)
		{
			z[i] = 4.0 * p[i];
		}
		for (int i = 0; i < n - 2; i++)
		{
			z[i] = (- 1.0) * p[i + 1] + z[i];
		}
		for (int i = 0; i < n - 2; i++)
		{
			z[i + 1] = (- 1.0) * p[i] + z[i + 1];
		}
		// Correct for terms not properly in block diagonal
		for (int i = k - 1; i < n - k; i = i + k)
		{
			z[i] += p[i + 1];
			z[i + 1] += p[i];
		}
		// Do the super and subdiagonal blocks, the -I's
		for (int i = 0; i < n - k; i++)
		{
			z[i] = (- 1.0) * p[i + k] + z[i];
		}
		for (int i = 0; i < n - k; i++)
		{
			z[i + k] = (- 1.0) * p[i] + z[i + k];
		}
	}
	
	public static void  Main(String[] args)
	{
		int n = 400;
		double[] b = new double[n];
		double[] xguess = new double[n];
		
		GenMinResEx5 atp = new GenMinResEx5();
		
		// Construct a GenMinRes object
		GenMinRes gnmnrs = new GenMinRes(n, atp);
		// Set right hand side and initial guess to ones    */
		for (int i = 0; i < n; i++)
		{
			b[i] = 1.0;
			xguess[i] = 1.0;
		}
		gnmnrs.SetGuess(xguess);
		gnmnrs.Method =
         Imsl.Math.GenMinRes.ImplementationMethod.SecondHouseholder;
		gnmnrs.ResidualUpdating =
         Imsl.Math.GenMinRes.ResidualMethod.DirectAtRestartOnly;
		// Solve Ax = b
		gnmnrs.Solve(b);
		int iterations = gnmnrs.Iterations;
		Console.Out.WriteLine("The number of iterations used = " + iterations);
		double resnorm = gnmnrs.ResidualNorm;
		Console.Out.WriteLine("The final residual norm is " + resnorm);
	}
}

Output

The number of iterations used = 92
The final residual norm is 2.52648529541037E-07

Link to C# source.