# Crout's Reduction Algorithm
    
    if($messages eq "on"){
        print "$n<code>$n";
    }

    $beginning_time = localtime(time);

    if($matrix[1][1]!=0){
        for($k=2;$k<=$var_no;$k+=1){
            for($i=$k;$i<=$var_no;$i+=1){
                for($j=($var_no+1);$j>=($k-1);$j-=1){
                    if($messages eq "on"){
                        print $matrix[$i][$j];
                        print "  --->   ";
                    }
                    # Crout's Reduction Formula
                    $matrix[$i][$j] -= $matrix[$k-1][$j] * ( $matrix[$
+i][$k-1] / $matrix[$k-1][$k-1] );
                    if($messages eq "on"){
                        print $matrix[$i][$j];
                        print $n;
                    }
                }
            }
        }
    }
    
    if($messages eq "on"){
        for($i=1;$i<=($var_no+1);$i+=1){
            for($j=1;$j<=($var_no+1);$j+=1){
                print $matrix[$i][$j];
                print "&nbsp;&nbsp;&nbsp;";
            }
            print $n
        }
        print "&lt;/code&gt;$n";
    }

    # Solving the equations once the matrix is reduced.
    for($i=$var_no;$i>=1;$i-=1){
        $sum = 0;
        for($k=($i+1);$k<=$var_no;$k+=1){
            $sum += $x[$k] * $matrix[$i][$k];
        }
        $x[$i] = ( $matrix[$i][$var_no+1] - $sum) / $matrix[$i][$i]
    }
    
    $ending_time = localtime(time);

    if($messages eq "on"){
        print "$n<code>\n";
        for($i=1;$i<=$var_no;$i+=1){
            print $x[$i];
            print $n
        }
        print "$nThe whole thing done in only ".($ending_time-$beginni
+ng_time)." seconds.";
        print "&lt;/code&gt;$n";
    }
[download]