GenMinRes Example Using the Second Householder Implementation and Preconditioning

Example 6: The Second Householder Implementation With Preconditioning

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. Preconditioning is used with the preconditioning matrix being a diagonal matrix with 4.0 down the main diagonal and -1.0 along the upper and lower co-diagonals. This Preconditioner method solves this tridiagonal matrix.

using System;
using Imsl.Math;
public class GenMinResEx6 : GenMinRes.IPreconditioner
{
    private double[] precondA, precondB, precondC;
    
    // Creates a new instance of GenMinResEx6 
    public GenMinResEx6(int n)
    {
        precondA = new double[n];
        precondB = new double[n];
        precondC = new double[n];
        // Define the preconditioning matrix                    
        for (int i = 0; i < n; i++)
        {
            precondA[i] = 4.0;
            precondB[i] = - 1.0;
            precondC[i] = - 1.0;
        }
    }
    
    public void  Amultp(double[] p, double[] z)
    {
        int m = z.Length;
        int n = (int) Math.Sqrt(m);
        // Multiply by diagonal blocks           
        for (int i = 0; i < m; i++)
        {
            z[i] = 4.0 * p[i];
        }
        for (int i = 0; i < m - 2; i++)
        {
            z[i] -= p[i + 1];
        }
        for (int i = 0; i < m - 2; i++)
        {
            z[i + 1] -= p[i];
        }
        // Correct for terms not properly in block diagonal
        for (int i = n - 1; i < m - n; i = i + n)
        {
            z[i] += p[i + 1];
            z[i + 1] += p[i];
        }
        // Do the super and subdiagonal blocks, the -I's     
        for (int i = 0; i < m - n; i++)
        {
            z[i] -= p[i + n];
        }
        for (int i = 0; i < m - n; i++)
        {
            z[i + n] -= p[i];
        }
    }
    
    /// Solve the tridiagonal preconditioning matrix problem for z.
    public void  Preconditioner(double[] r, double[] z)
    {
        int n = z.Length;
        double[] w = new double[n];
        double[] v = new double[n];
        double[] u = new double[n];
        w[0] = precondA[0];
        v[0] = precondC[0] / w[0];
        u[0] = r[0] / w[0];
        for (int i = 1; i < n; i++)
        {
            w[i] = precondA[i] - precondB[i] * v[i - 1];
            v[i] = precondC[i] / w[i];
            u[i] = (r[i] - precondB[i] * u[i - 1]) / w[i];
        }
        z[n - 1] = u[n - 1];
        for (int j = n - 2; j >= 0; j--)
        {
            z[j] = u[j] - v[j] * z[j + 1];
        }
    }
    
    public static void  Main(String[] args)
    {
        int n = 400;
        double[] b = new double[n];
        double[] xguess = new double[n];
        
        GenMinResEx6 atp = new GenMinResEx6(n);
        
        // 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 = 60
The final residual norm is 2.84141491724154E-07

Link to C# source.