For the following two problems, create two codes for each. The first way is to u

Question

For the following two problems, create two codes for each. The first way is to use the simple Matlab matrix algebra rules, i.e. if you're multiplying a matrix L by a vector xi, it's just L*xi. For the second way, create for loops that perform the same multiplication old-school style

A = [2 3 -1; 1 -2 1; 1 -1 1];b = [3 1 1]';
A = [1 -2 1; 2 3 -1; 1 -1 1];b = [1 3 1]';
A
U = triu(A,1)L = tril(A,-1)D = diag(diag(A))
Di = inv(D);
eps = 1e-3;err = 1;i = 1;
x = [0 0 0]';
while err > eps    xn = Di*(b-(L+U)*x);    err = max(abs(xn-x));

    disp(['Iteration ' num2str(i) ' error ' num2str(err)]);    i = i + 1;    x = xn;end

Use the above code to solve a large matrix. Return the iteration count to solve this problem and a plot of the solution.

Modify the Jacobi method to implement the Gauss-Seidel method) and solve the same matrix from above. Again include the iteration count and solution plot.

Explanation / Answer

import java.io.*;
import java.util.Arrays;
import java.util.StringTokenizer;

public class Jacobi_Matrix
{

    public static final int MAX_ITERATIONS = 100;
    private double[][] M;

    public Jacobi_Matrix(double [][] matrix) { M = matrix; }

    public void print()
    {
        int n = M.length;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n + 1; j++)
                System.out.print(M[i][j] + " ");
            System.out.println();
        }
    }

    public boolean trans_domiant(int r, boolean[] V, int[] R)
    {
        int n = M.length;
        if (r == M.length) {
            double[][] T = new double[n][n+1];
            for (int i = 0; i < R.length; i++) {
                for (int j = 0; j < n + 1; j++)
                    T[i][j] = M[R[i]][j];
            }

            M = T;

            return true;
        }

        for (int i = 0; i < n; i++) {
            if (V[i]) continue;

            double sum = 0;

            for (int j = 0; j < n; j++)
                sum += Math.abs(M[i][j]);

            if (2 * Math.abs(M[i][r]) > sum)
            {
                V[i] = true;
                R[r] = i;

                if (trans_domiant(r + 1, V, R))
                    return true;

                V[i] = false;
            }
        }

        return false;
    }

    public boolean Dominant()
    {
        boolean[] visited = new boolean[M.length];
        int[] rows = new int[M.length];

        Arrays.fill(visited, false);

        return trans_domiant(0, visited, rows);
    }

  
    public void solve()
    {
        int iterations = 0;
        int n = M.length;
        double epsilon = 1e-15;
        double[] X = new double[n];
        double[] P = new double[n];
        Arrays.fill(X, 0);
        Arrays.fill(P, 0);

        while (true) {
            for (int i = 0; i < n; i++) {
                double sum = M[i][n]; // b_n

                for (int j = 0; j < n; j++)
                    if (j != i)
                        sum -= M[i][j] * P[j];

                X[i] = 1/M[i][i] * sum;
            }

            System.out.print("X_" + iterations + " = {");
            for (int i = 0; i < n; i++)
                System.out.print(X[i] + " ");
            System.out.println("}");

            iterations++;
            if (iterations == 1) continue;

            boolean stop = true;
            for (int i = 0; i < n && stop; i++)
                if (Math.abs(X[i] - P[i]) > epsilon)
                    stop = false;

            if (stop || iterations == MAX_ITERATIONS) break;
            P = (double[])X.clone();
        }
    }

    public static void main(String[] args) throws IOException
    {
        int n;
        double[][] M;

        BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter writer = new PrintWriter(System.out, true);

        n = Integer.parseInt(reader.readLine());
        M = new double[n][n+1];

        for (int i = 0; i < n; i++) {
            StringTokenizer strtk = new StringTokenizer(reader.readLine());

            while (strtk.hasMoreTokens())
                for (int j = 0; j < n + 1 && strtk.hasMoreTokens(); j++)
                    M[i][j] = Integer.parseInt(strtk.nextToken());
        }

        Jacobi_Matrix Jacobi_Matrix = new Jacobi_Matrix(M);

        if (!Jacobi_Matrix.Dominant()) {
            writer.println("The system isn't diagonally dominant: " +
                    "The method cannot guarantee convergence.");
        }

        writer.println();
        Jacobi_Matrix.print();
        Jacobi_Matrix.solve();
    }
}

In the Jacobi method the values of the approximations xi are used up to the next iteration of the method while in the Gauss-Seidel version each approximation xi is used

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