Question

/* Program to demonstrate gaussian <strong class="highlight">elimination</strong>
   on a set of linear simultaneous equations
 */

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

using namespace std;

const double eps = 1.e-15;

/*Preliminary pivoting strategy
  Pivoting function
 */
double pivot(vector<vector<double> > &a, vector<double> &b, int i)
 {
     int n = a.size();
     int j=i;
     double t=0;

     for(int k=i; k<n; k+=1)
     {
         double aki = fabs(a[k][i]);
         if(aki>t)
         {
             t=aki;
             j=k;
         }
     }
     if(j>i)
     {
         double dummy;
         for(int L=0; L<n; L+=1)
         {
             dummy = a[i][L];
             a[i][L]= a[j][L];
             a[j][L]= dummy;
         }
         double temp = b[j];
         b[i]=b[j];
         b[j]=temp;
     }
     return a[i][i];
 }        



/* Forward <strong class="highlight">elimination</strong> */
void triang(vector<vector<double> > &a, vector<double> &b) 
{
    int n = a.size();
    for(int i=0; i<n-1; i+=1)
    {
        double diag = pivot(a,b,i);
        if(fabs(diag)<eps)
        {
            cout<<"zero det"<<endl;
            return;
        }
        for(int j=i+1; j<n; j+=1)
        {
            double mult = a[j][i]/diag;
            for(int k = i+1; k<n; k+=1)
            {
                a[j][k]-=mult*a[i][k];
            }
            b[j]-=mult*b[i];
        }
    }
}

/*
DOT PRODUCT OF TWO VECTORS
*/
double dotProd(vector<double> &u, vector<double> &v, int k1,int k2)
{
  double sum = 0;
  for(int i = k1; i <= k2; i += 1)
  {
     sum += u[i] * v[i];
  }
  return sum;
}

/*
BACK SUBSTITUTION STEP
*/
void backSubst(vector<vector<double> > &a, vector<double> &b, vector<double> &x)
{
    int n = a.size();
  for(int i =  n-1; i >= 0; i -= 1)
  {
    x[i] = (b[i] - dotProd(a[i], x, i + 1,  n-1))/ a[i][i];
  }
}
/*
REFINED GAUSSIAN <strong class="highlight">ELIMINATION</strong> PROCEDURE
*/
void gauss(vector<vector<double> > &a, vector<double> &b, vector<double> &x)
{
   triang(a, b);
   backSubst(a, b, x);
}                               

// EXAMPLE MAIN PROGRAM
int main()
{
    int n;
    cin >> n;
    vector<vector<double> > a;
    vector<double> x;
    vector<double> b;
    for (int i = 0; i < n; i++) {
        vector<double> temp;
        for (int j = 0; j < n; j++) {
            int no;
            cin >> no;
            temp.push_back(no);
        }
        a.push_back(temp);
        b.push_back(0);
        x.push_back(0);
    }
    /* 
    for (int i = 0; i < n; i++) {
        int no;
        cin >> no;
        b.push_back(no);
        x.push_back(0);
    }
    */

  gauss(a, b, x);
  for (size_t i = 0; i < x.size(); i++) {
      cout << x[i] << endl;
  }
   return 0;
}

The above gaussian eleimination algorithm works fine on NxN matrices. But I need it to work on NxM matrix. Can anyone help me to do it? I am not very good at maths. I got this code on some website and i am stuck at it.

Answer 1

1. (optional) Understand this. Do some examples on paper.
2. Don’t write code for Gaussian elimination yourself. Without some care, the naive gauss pivoting is unstable. You have to scale the lines and take care of pivoting with the greatest element, a starting point is there. Note that this advice holds for most linear algebra algorithms.
3. If you want to solve systems of equations, LU decomposition, QR decomposition (stabler than LU, but slower), Cholesky decomposition (in the case the system is symmetric) or SVD (in the case the system is not square) are almost always better choices. Gaussian elimination is best for computing determinants however.
4. Use the algorithms from LAPACK for the problems which need Gaussian elimination (eg. solving systems, or computing determinants). Really. Don’t roll your own. Since you are doing C++, you may be interested in Armadillo which takes care of a lot of things for you.
5. If you must roll your own for pedagogical reasons, have a look first at Numerical Recipes, version 3. Version 2 can be found online for free if you’re low on budget / have no access to a library.
6. As a general advice, don’t code algorithms you don’t understand.