REF: scale by smallest non-zero element

Since we're technically manipulating a linear system, it seems that the property of the final result remaining the same despite scaling applies. If scaling by the inverse of the smallest element, the matrix error becomes relative to the magnitudes of the numbers rather than an absolute quantity.

This intuition seems true because all of the tests pass

Author: Aweptimum

src/LinearAlgebra/Reduction/RowEchelonForm.php

@@ -127,6 +127,19 @@ public static function gaussianElimination(NumericMatrix $A): array
         $swaps = 0;
         $ε     = $A->getError();
 
+        // Scale the matrix by its smallest non-zero element
+        $min = PHP_INT_MAX;
+        for ($i = 0; $i < $m; $i++) {
+            for ($j = 0; $j < $n; $j++) {
+                $elem = \abs($A[$i][$j]);
+                if (Support::isNotZero($elem, $ε) and $elem < $min) {
+                    $min = $elem;
+                }
+            }
+        }
+
+        $R = $A->scalarMultiply(1/$min)->getMatrix();
+
         for ($k = 0; $k < $size; $k++) {
             // Find column max
             $i_max = $k;