notebook/code/ReducedRowEchelonForm.cc

// Reduced row echelon form via Gauss-Jordan elimination 
// with partial pivoting.  This can be used for computing
// the rank of a matrix.
//
// Running time: O(n^3)
//
// INPUT:    a[][] = an nxn matrix
//
// OUTPUT:   rref[][] = an nxm matrix (stored in a[][])
//           returns rank of a[][]

#include <iostream>
#include <vector>
#include <cmath>

using namespace std;

const double EPSILON = 1e-10;

typedef double T;
typedef vector<T> VT;
typedef vector<VT> VVT;

int rref(VVT &a) {
  int n = a.size();
  int m = a[0].size();
  int r = 0;
  for (int c = 0; c < m; c++) {
    int j = r;
    for (int i = r+1; i < n; i++) 
      if (fabs(a[i][c]) > fabs(a[j][c])) j = i;
    if (fabs(a[j][c]) < EPSILON) continue;
    swap(a[j], a[r]);
   
    T s = 1.0 / a[r][c];
    for (int j = 0; j < m; j++) a[r][j] *= s;
    for (int i = 0; i < n; i++) if (i != r) {
      T t = a[i][c];
      for (int j = 0; j < m; j++) a[i][j] -= t * a[r][j];
    }
    r++;
  }
  return r;
}

int main(){
  const int n = 5;
  const int m = 4;
  double A[n][m] = { {16,2,3,13},{5,11,10,8},{9,7,6,12},{4,14,15,1},{13,21,21,13} };
  VVT a(n);
  for (int i = 0; i < n; i++)
    a[i] = VT(A[i], A[i] + n);
  
  int rank = rref (a);
  
  // expected: 4
  cout << "Rank: " << rank << endl;
  
  // expected: 1 0 0 1 
  //           0 1 0 3 
  //           0 0 1 -3 
  //           0 0 0 2.78206e-15 
  //           0 0 0 3.22398e-15
  cout << "rref: " << endl;
  for (int i = 0; i < 5; i++){
    for (int j = 0; j < 4; j++)
      cout << a[i][j] << ' ';
    cout << endl;
  }
  
}