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. 
import com.imsl.math.*;
public class GenMinResEx6 implements GenMinRes.Preconditioner {
    private double precondA[], precondB[], precondC[];
        
    // Creates a new instance of GenMinResEx6 
    public GenMinResEx6(int n) throws Exception {
        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[]) throws Exception {
        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.setMethod(gnmnrs.SECOND_HOUSEHOLDER);
        gnmnrs.setResidualUpdating(gnmnrs.DIRECT_AT_RESTART_ONLY);
        // Solve Ax = b
        gnmnrs.solve(b);
        int iterations = gnmnrs.getIterations();
        System.out.println("The number of iterations used = "+iterations);
        double resnorm = gnmnrs.getResidualNorm();
        System.out.println("The final residual norm is "+resnorm);   
    }
}

Output

The number of iterations used = 60
The final residual norm is 2.841414917241539E-7
Link to Java source.