Add some commentary to Gauss-Seidel function

brush-strokes/src/lib/Math/Interval.hs

@@ -115,7 +115,7 @@ bisect x@( 𝕀 x_inf x_sup )
   where x_mid = midpoint x
 
 infixl 6 ⊖
-(⊖) :: ( Ring a, Ord a ) => 𝕀 a -> 𝕀 a -> 𝕀 a
+(⊖) :: ( AbelianGroup a, Ord a ) => 𝕀 a -> 𝕀 a -> 𝕀 a
 (⊖) a@( 𝕀 lo1 hi1 ) b@( 𝕀 lo2 hi2 )
   | width a >= width b
   = 𝕀 ( lo1 - lo2 ) ( hi1 - hi2 )

brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs

@@ -202,7 +202,13 @@ gaussSeidelStep
   -> [ ( T ( 𝕀ℝ n ), Bool ) ]
 gaussSeidelStep as b ( T x0 ) = coerce $
   forEachCoord @n ( x0, True ) $ \ i ( x, contraction ) -> do
-    -- x_i' = ( b_i - sum { j /= i } a_ij * x_j ) / a_ii
+    -- For each i, we have an equation: sum_j a_ij * x_j = b_i
+    --
+    -- Re-arrange this with x_i on the left to get an iteration:
+    --   x_i' = ( b_i - sum { j /= i } a_ij * x_j ) / a_ii
+    --
+    -- We perform each iteration in turn (for i = 1, ..., n),
+    -- **using the latest updated value of each x_j** in each iteration.
     let s = b `index` i - sum [ ( as ! j ) `index` i * x `index` j
                               | j <- toList ( universe @n ), j /= i ]
         x_i = x `index` i