How to solve a Linear Equation?
Let’s suppose we have to solve this:
x + y + z = 6 2y + 5z = -4 2x + 5y - z = 27
Now we can create matrix from this:
1 1 1 6 0 2 5 and -4 2 5 -1 27
Now we find Inverse of this matrix to get the solution
Now let’s write the program.
Java Program to Solve Linear Equations:
import java.util.Scanner; public class Solve_Linear_Equation { public static void main(String args[]) { char []var = {'x', 'y', 'z', 'w'}; System.out.println("Enter the number of variables in the equations: "); Scanner input = new Scanner(System.in); int n = input.nextInt(); System.out.println("Enter the coefficients of each variable for each equations"); System.out.println("ax + by + cz + ... = d"); double [][]mat = new double[n][n]; double [][]constants = new double[n][1]; //input for(int i=0; i<n; i++) { for(int j=0; j<n; j++) { mat[i][j] = input.nextDouble(); } constants[i][0] = input.nextDouble(); } //Matrix representation for(int i=0; i<n; i++) { for(int j=0; j<n; j++) { System.out.print(" "+mat[i][j]); } System.out.print(" "+ var[i]); System.out.print(" = "+ constants[i][0]); System.out.println(); } //inverse of matrix mat[][] double inverted_mat[][] = invert(mat); System.out.println("The inverse is: "); for (int i=0; i<n; ++i) { for (int j=0; j<n; ++j) { System.out.print(inverted_mat[i][j]+" "); } System.out.println(); } //Multiplication of mat inverse and constants double result[][] = new double[n][1]; for (int i = 0; i < n; i++) { for (int j = 0; j < 1; j++) { for (int k = 0; k < n; k++) { result[i][j] = result[i][j] + inverted_mat[i][k] * constants[k][j]; } } } System.out.println("The product is:"); for(int i=0; i<n; i++) { System.out.println(result[i][0] + " "); } input.close(); } public static double[][] invert(double a[][]) { int n = a.length; double x[][] = new double[n][n]; double b[][] = new double[n][n]; int index[] = new int[n]; for (int i=0; i<n; ++i) b[i][i] = 1; // Transform the matrix into an upper triangle gaussian(a, index); // Update the matrix b[i][j] with the ratios stored for (int i=0; i<n-1; ++i) for (int j=i+1; j<n; ++j) for (int k=0; k<n; ++k) b[index[j]][k] -= a[index[j]][i]*b[index[i]][k]; // Perform backward substitutions for (int i=0; i<n; ++i) { x[n-1][i] = b[index[n-1]][i]/a[index[n-1]][n-1]; for (int j=n-2; j>=0; --j) { x[j][i] = b[index[j]][i]; for (int k=j+1; k<n; ++k) { x[j][i] -= a[index[j]][k]*x[k][i]; } x[j][i] /= a[index[j]][j]; } } return x; } // Method to carry out the partial-pivoting Gaussian // elimination. Here index[] stores pivoting order. public static void gaussian(double a[][], int index[]) { int n = index.length; double c[] = new double[n]; // Initialize the index for (int i=0; i<n; ++i) index[i] = i; // Find the rescaling factors, one from each row for (int i=0; i<n; ++i) { double c1 = 0; for (int j=0; j<n; ++j) { double c0 = Math.abs(a[i][j]); if (c0 > c1) c1 = c0; } c[i] = c1; } // Search the pivoting element from each column int k = 0; for (int j=0; j<n-1; ++j) { double pi1 = 0; for (int i=j; i<n; ++i) { double pi0 = Math.abs(a[index[i]][j]); pi0 /= c[index[i]]; if (pi0 > pi1) { pi1 = pi0; k = i; } } // Interchange rows according to the pivoting order int itmp = index[j]; index[j] = index[k]; index[k] = itmp; for (int i=j+1; i<n; ++i) { double pj = a[index[i]][j]/a[index[j]][j]; // Record pivoting ratios below the diagonal a[index[i]][j] = pj; // Modify other elements accordingly for (int l=j+1; l<n; ++l) a[index[i]][l] -= pj*a[index[j]][l]; } } }
OUTPUT:
# javac Solve_Linear_Equation.java # java Solve_Linear_Equation Enter the number of variables in the equations: 2 Enter the coefficients of each variable for each equations ax + by + cz + ... = d 1 2 3 3 2 1 1.0 2.0 x = 3.0 3.0 2.0 y = 1.0 The inverse is: -0.49999999999999994 0.5 0.7499999999999999 -0.24999999999999997 The product is: -0.9999999999999998 1.9999999999999996