+--
+-- Examples:
+--
+-- >>> import Linear.Matrix ( Col4, frobenius_norm, fromList )
+-- >>> import Naturals ( N7 )
+--
+-- >>> let f1 = [0.0418]
+-- >>> let f2 = [0.0805]
+-- >>> let f3 = [0.1007]
+-- >>> let f4 = [-0.0045]
+-- >>> let f5 = [-0.0332]
+-- >>> let f6 = [-0.0054]
+-- >>> let f7 = [-0.0267]
+-- >>> let big_F = fromList [f1,f2,f3,f4,f5,f6,f7] :: Col N7 Double
+--
+-- >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
+-- >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
+-- >>> let k3 = [0, -7.5, 12.5, 0, 0, 0, 0] :: [Double]
+-- >>> let k4 = [0, 0, 0, 6, 0, 0, 0] :: [Double]
+-- >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
+-- >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
+-- >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
+-- >>> let big_K = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
+--
+-- >>> let e1 = [1871/75000] :: [Double]
+-- >>> let e2 = [899/25000] :: [Double]
+-- >>> let e3 = [463/15625] :: [Double]
+-- >>> let e4 = [-3/4000] :: [Double]
+-- >>> let e5 = [-83/15000] :: [Double]
+-- >>> let e6 = [-9/10000] :: [Double]
+-- >>> let e7 = [-89/50000] :: [Double]
+-- >>> let expected = fromList [e1,e2,e3,e4,e5,e6,e7] :: Col N7 Double
+-- >>> let actual = solve_positive_definite big_K big_F
+-- >>> frobenius_norm (actual - expected) < 1e-12
+-- True
+--
+solve_positive_definite :: (Arity m, Algebraic.C a, Eq a, Field.C a)
+ => Mat m m a
+ -> Col m a
+ -> Col m a