package algs91; // section 9.9
import stdlib.*;
/* ***********************************************************************
 *  Compilation:  javac GaussJordanElimination.java
 *  Execution:    java GaussJordanElimination N
 *
 *  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
 *
 *
 *************************************************************************/

public class XGaussJordanElimination {
  private static final double EPSILON = 1e-8;

  private final int N;      // N-by-N system
  private final double[][] a;     // N-by-N+1 augmented matrix

  // Gauss-Jordan elimination with partial pivoting
  public XGaussJordanElimination(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 need 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 check(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 Error("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;
  }

  // extract solution to Ax = b
  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;
  }

  // extract solution to yA = 0, yb != 0
  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;
  }

  // does the system have a solution?
  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.format("%8.3f ", a[i][j]);
      }
      StdOut.format("| ");
      for (int j = N; j < N+N; j++) {
        StdOut.format("%8.3f ", a[i][j]);
      }
      StdOut.format("| %8.3f\n", a[i][N+N]);
    }
    StdOut.println();
  }


  // check that Ax = b or yA = 0, yb != 0
  private boolean check(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.format("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.format("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.format("yb  = %8.3f\n", sum);
        return false;
      }
      return true;
    }
  }


  public static void test(double[][] A, double[] b) {
    XGaussJordanElimination gaussian = new XGaussJordanElimination(A, b);
    if (gaussian.isFeasible()) {
      StdOut.println("Solution to Ax = b");
      double[] x = gaussian.primal();
      for (double element : x) {
        StdOut.format("%10.6f\n", element);
      }
    }
    else {
      StdOut.println("Certificate of infeasibility");
      double[] y = gaussian.dual();
      for (double element : y) {
        StdOut.format("%10.6f\n", element);
      }
    }
    StdOut.println();
  }


  // 3-by-3 nonsingular system
  public static void test1() {
    double[][] A = {
        { 0, 1,  1 },
        { 2, 4, -2 },
        { 0, 3, 15 }
    };
    double[] b = { 4, 2, 36 };
    test(A, b);
  }

  // 3-by-3 nonsingular system
  public static void test2() {
    double[][] A = {
        {  1, -3,   1 },
        {  2, -8,   8 },
        { -6,  3, -15 }
    };
    double[] b = { 4, -2, 9 };
    test(A, b);
  }

  // 5-by-5 singular: no solutions
  // y = [ -1, 0, 1, 1, 0 ]
  public 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(A, b);
  }

  // 5-by-5 singluar: infinitely many solutions
  public 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(A, b);
  }

  // 3-by-3 singular: no solutions
  // y = [ 1, 0, 1/3 ]
  public static void test5() {
    double[][] A = {
        {  2, -1,  1 },
        {  3,  2, -4 },
        { -6,  3, -3 },
    };
    double[] b = { 1, 4, 2 };
    test(A, b);
  }

  // 3-by-3 singular: infinitely many solutions
  public static void test6() {
    double[][] A = {
        {  1, -1,  2 },
        {  4,  4, -2 },
        { -2,  2, -4 },
    };
    double[] b = { -3, 1, 6 };
    test(A, b);
  }

  // sample client
  public static void main(String[] args) {

    try                 { test1();             }
    catch (Exception e) { e.printStackTrace(); }
    StdOut.println("--------------------------------");

    try                 { test2();             }
    catch (Exception e) { e.printStackTrace(); }
    StdOut.println("--------------------------------");

    try                 { test3();             }
    catch (Exception e) { e.printStackTrace(); }
    StdOut.println("--------------------------------");

    try                 { test4();             }
    catch (Exception e) { e.printStackTrace(); }
    StdOut.println("--------------------------------");

    try                 { test5();             }
    catch (Exception e) { e.printStackTrace(); }
    StdOut.println("--------------------------------");

    try                 { test6();             }
    catch (Exception e) { e.printStackTrace(); }
    StdOut.println("--------------------------------");

    // 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.uniform(1000);
    double[] b = new double[N];
    for (int i = 0; i < N; i++)
      b[i] = StdRandom.uniform(1000);
    test(A, b);

    StdOut.println("--------------------------------");

    // 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.uniform(1000);
    for (int i = 0; i < N-1; i++) {
      double alpha = StdRandom.uniform(11) - 5.0;
      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.uniform(1000);
    test(A, b);


  }

}