Below is the syntax highlighted version of GaussJordanElimination.java
from §9.9 Scientific Computing.
/****************************************************************************** * Compilation: javac GaussJordanElimination.java * Execution: java GaussJordanElimination n * Dependencies: StdOut.java * * Finds a solutions to Ax = b using Gauss-Jordan elimination with partial * pivoting. If no solution exists, find a solution to yA = 0, yb != 0, * which serves as a certificate of infeasibility. * * % java GaussJordanElimination * -1.000000 * 2.000000 * 2.000000 * * 3.000000 * -1.000000 * -2.000000 * * System is infeasible * * -6.250000 * -4.500000 * 0.000000 * 0.000000 * 1.000000 * * System is infeasible * * -1.375000 * 1.625000 * 0.000000 * * ******************************************************************************/ /** * The {@code GaussJordanElimination} data type provides methods * to solve a linear system of equations <em>Ax</em> = <em>b</em>, * where <em>A</em> is an <em>n</em>-by-<em>n</em> matrix * and <em>b</em> is a length <em>n</em> vector. * If no solution exists, it finds a solution <em>y</em> to * <em>yA</em> = 0, <em>yb</em> ≠ 0, * which serves as a certificate of infeasibility. * <p> * This implementation uses Gauss-Jordan elimination with partial pivoting. * See {@link GaussianElimination} for an implementation that uses * Gaussian elimination (but does not provide the certificate of infeasibility). * For an industrial-strength numerical linear algebra library, * see <a href = "http://math.nist.gov/javanumerics/jama/">JAMA</a>. * <p> * This computes correct results if all arithmetic performed is * without floating-point rounding error or arithmetic overflow. * In practice, there will be floating-point rounding error; * partial pivoting helps prevent accumulated floating-point rounding * errors from growing out of control (though it does not * provide any guarantees). * <p> * For additional documentation, see * <a href="https://algs4.cs.princeton.edu/99scientific">Section 9.9</a> * <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne. * * @author Robert Sedgewick * @author Kevin Wayne */ public class GaussJordanElimination { private static final double EPSILON = 1e-8; private final int n; // n-by-n system private double[][] a; // n-by-(n+1) augmented matrix // Gauss-Jordan elimination with partial pivoting /** * Solves the linear system of equations <em>Ax</em> = <em>b</em>, * where <em>A</em> is an <em>n</em>-by-<em>n</em> matrix and <em>b</em> * is a length <em>n</em> vector. * * @param A the <em>n</em>-by-<em>n</em> constraint matrix * @param b the length <em>n</em> right-hand-side vector */ public GaussJordanElimination(double[][] A, double[] b) { n = b.length; // build augmented matrix a = new double[n][n+n+1]; for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) a[i][j] = A[i][j]; // only needed if you want to find certificate of infeasibility (or compute inverse) for (int i = 0; i < n; i++) a[i][n+i] = 1.0; for (int i = 0; i < n; i++) a[i][n+n] = b[i]; solve(); assert certifySolution(A, b); } private void solve() { // Gauss-Jordan elimination for (int p = 0; p < n; p++) { // show(); // find pivot row using partial pivoting int max = p; for (int i = p+1; i < n; i++) { if (Math.abs(a[i][p]) > Math.abs(a[max][p])) { max = i; } } // exchange row p with row max swap(p, max); // singular or nearly singular if (Math.abs(a[p][p]) <= EPSILON) { continue; // throw new ArithmeticException("Matrix is singular or nearly singular"); } // pivot pivot(p, p); } // show(); } // swap row1 and row2 private void swap(int row1, int row2) { double[] temp = a[row1]; a[row1] = a[row2]; a[row2] = temp; } // pivot on entry (p, q) using Gauss-Jordan elimination private void pivot(int p, int q) { // everything but row p and column q for (int i = 0; i < n; i++) { double alpha = a[i][q] / a[p][q]; for (int j = 0; j <= n+n; j++) { if (i != p && j != q) a[i][j] -= alpha * a[p][j]; } } // zero out column q for (int i = 0; i < n; i++) if (i != p) a[i][q] = 0.0; // scale row p (ok to go from q+1 to n, but do this for consistency with simplex pivot) for (int j = 0; j <= n+n; j++) if (j != q) a[p][j] /= a[p][q]; a[p][q] = 1.0; } /** * Returns a solution to the linear system of equations <em>Ax</em> = <em>b</em>. * * @return a solution <em>x</em> to the linear system of equations * <em>Ax</em> = <em>b</em>; {@code null} if no such solution */ public double[] primal() { double[] x = new double[n]; for (int i = 0; i < n; i++) { if (Math.abs(a[i][i]) > EPSILON) x[i] = a[i][n+n] / a[i][i]; else if (Math.abs(a[i][n+n]) > EPSILON) return null; } return x; } /** * Returns a solution to the linear system of equations <em>yA</em> = 0, * <em>yb</em> ≠ 0. * * @return a solution <em>y</em> to the linear system of equations * <em>yA</em> = 0, <em>yb</em> ≠ 0; {@code null} if no such solution */ public double[] dual() { double[] y = new double[n]; for (int i = 0; i < n; i++) { if ((Math.abs(a[i][i]) <= EPSILON) && (Math.abs(a[i][n+n]) > EPSILON)) { for (int j = 0; j < n; j++) y[j] = a[i][n+j]; return y; } } return null; } /** * Returns true if there exists a solution to the linear system of * equations <em>Ax</em> = <em>b</em>. * * @return {@code true} if there exists a solution to the linear system * of equations <em>Ax</em> = <em>b</em>; {@code false} otherwise */ public boolean isFeasible() { return primal() != null; } // print the tableaux private void show() { for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { StdOut.printf("%8.3f ", a[i][j]); } StdOut.printf("| "); for (int j = n; j < n+n; j++) { StdOut.printf("%8.3f ", a[i][j]); } StdOut.printf("| %8.3f\n", a[i][n+n]); } StdOut.println(); } // check that Ax = b or yA = 0, yb != 0 private boolean certifySolution(double[][] A, double[] b) { // check that Ax = b if (isFeasible()) { double[] x = primal(); for (int i = 0; i < n; i++) { double sum = 0.0; for (int j = 0; j < n; j++) { sum += A[i][j] * x[j]; } if (Math.abs(sum - b[i]) > EPSILON) { StdOut.println("not feasible"); StdOut.printf("b[%d] = %8.3f, sum = %8.3f\n", i, b[i], sum); return false; } } return true; } // or that yA = 0, yb != 0 else { double[] y = dual(); for (int j = 0; j < n; j++) { double sum = 0.0; for (int i = 0; i < n; i++) { sum += A[i][j] * y[i]; } if (Math.abs(sum) > EPSILON) { StdOut.println("invalid certificate of infeasibility"); StdOut.printf("sum = %8.3f\n", sum); return false; } } double sum = 0.0; for (int i = 0; i < n; i++) { sum += y[i] * b[i]; } if (Math.abs(sum) < EPSILON) { StdOut.println("invalid certificate of infeasibility"); StdOut.printf("yb = %8.3f\n", sum); return false; } return true; } } private static void test(String name, double[][] A, double[] b) { StdOut.println("----------------------------------------------------"); StdOut.println(name); StdOut.println("----------------------------------------------------"); GaussJordanElimination gaussian = new GaussJordanElimination(A, b); if (gaussian.isFeasible()) { StdOut.println("Solution to Ax = b"); double[] x = gaussian.primal(); for (int i = 0; i < x.length; i++) { StdOut.printf("%10.6f\n", x[i]); } } else { StdOut.println("Certificate of infeasibility"); double[] y = gaussian.dual(); for (int j = 0; j < y.length; j++) { StdOut.printf("%10.6f\n", y[j]); } } StdOut.println(); StdOut.println(); } // 3-by-3 nonsingular system private static void test1() { double[][] A = { { 0, 1, 1 }, { 2, 4, -2 }, { 0, 3, 15 } }; double[] b = { 4, 2, 36 }; test("test 1", A, b); } // 3-by-3 nonsingular system private static void test2() { double[][] A = { { 1, -3, 1 }, { 2, -8, 8 }, { -6, 3, -15 } }; double[] b = { 4, -2, 9 }; test("test 2", A, b); } // 5-by-5 singular: no solutions // y = [ -1, 0, 1, 1, 0 ] private static void test3() { double[][] A = { { 2, -3, -1, 2, 3 }, { 4, -4, -1, 4, 11 }, { 2, -5, -2, 2, -1 }, { 0, 2, 1, 0, 4 }, { -4, 6, 0, 0, 7 }, }; double[] b = { 4, 4, 9, -6, 5 }; test("test 3", A, b); } // 5-by-5 singular: infinitely many solutions private static void test4() { double[][] A = { { 2, -3, -1, 2, 3 }, { 4, -4, -1, 4, 11 }, { 2, -5, -2, 2, -1 }, { 0, 2, 1, 0, 4 }, { -4, 6, 0, 0, 7 }, }; double[] b = { 4, 4, 9, -5, 5 }; test("test 4", A, b); } // 3-by-3 singular: no solutions // y = [ 1, 0, 1/3 ] private static void test5() { double[][] A = { { 2, -1, 1 }, { 3, 2, -4 }, { -6, 3, -3 }, }; double[] b = { 1, 4, 2 }; test("test 5", A, b); } // 3-by-3 singular: infinitely many solutions private static void test6() { double[][] A = { { 1, -1, 2 }, { 4, 4, -2 }, { -2, 2, -4 }, }; double[] b = { -3, 1, 6 }; test("test 6 (infinitely many solutions)", A, b); } /** * Unit tests the {@code GaussJordanElimination} data type. * * @param args the command-line arguments */ public static void main(String[] args) { test1(); test2(); test3(); test4(); test5(); test6(); // n-by-n random system (likely full rank) int n = Integer.parseInt(args[0]); double[][] A = new double[n][n]; for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) A[i][j] = StdRandom.uniformInt(1000); double[] b = new double[n]; for (int i = 0; i < n; i++) b[i] = StdRandom.uniformInt(1000); test("random " + n + "-by-" + n + " (likely full rank)", A, b); // n-by-n random system (likely infeasible) A = new double[n][n]; for (int i = 0; i < n-1; i++) for (int j = 0; j < n; j++) A[i][j] = StdRandom.uniformInt(1000); for (int i = 0; i < n-1; i++) { double alpha = StdRandom.uniformInt(-5, 5); for (int j = 0; j < n; j++) { A[n-1][j] += alpha * A[i][j]; } } b = new double[n]; for (int i = 0; i < n; i++) b[i] = StdRandom.uniformInt(1000); test("random " + n + "-by-" + n + " (likely infeasible)", A, b); } }