code:

#undef	DEBUG
/* #define	L	4 */
/* #define	C	5 */

/* #define	real	long double */
/* real M[L][C]; */
#ifndef	abs
#define abs(N) ((N>0)?(N):(-(N)))
#endif
{
	int i,j,k;
	int lead,maxlead;

for(i=lead=0;(i<L)&&(lead<C);i++,lead++)
{
/* Maximal by absolute value leading element search */
{	real max=0;

	for(j=i;j<L;j++)
	if(abs(M[j][lead])>max)
	{	maxlead=j;	max=abs(M[j][lead]);	};
	/* If no maximal leading element */
	if(max==0)	{	i--;	continue;	};
};
/* Swap lines #i and #maxlead */
if(i!=maxlead)
{	real t;
	for(j=lead;j<C;j++)
	{	t=M[i][j];
		M[i][j]=M[maxlead][j];
		M[maxlead][j]=t;
	};
};
/* Now subtract from each line below #i line #i */
for(j=i+1;j<L;j++)
{	real ratio=M[j][lead]/M[i][lead];
	for(k=lead+1;k<C;k++)
		M[j][k]-=M[i][k]*ratio;
	M[j][lead]=0;
};
/* Next line */
};

/* Backward pass */
for(i=L-1;i>0;i--)
{
/* Determine leading element */
for(lead=0;(lead<C)&&(M[i][lead]==0);lead++);
if(lead==C) continue;
/* Subtract current line from upper lines */
for(j=0;j<i;j++)
{	real ratio=M[j][lead]/M[i][lead];
	for(k=lead+1;k<C;k++)
		M[j][k]-=M[i][k]*ratio;
	M[j][lead]=0;
};
/* Next line */
};

/* Normalize */
for(i=0;i<L;i++)
{
/* Determine leading element */
	for(lead=0;(lead<C)&&(M[i][lead]==0);lead++);
	if(lead==C) continue;
/* Divide line on leading element */
	for(j=lead+1;j<C;j++)	M[i][j]/=M[i][lead];
	M[i][lead]=1;
};

/* Finished */
};