[haskell] add newton-raphson square root program

haskell/newton-raphson.hs

@@ -0,0 +1,58 @@
+-- | Newton-Raphson algorithm to approximate square root of a number
+-- From 'Why functional programming matter' (1989) by John Hughes
+
+{-
+rtᵢ₊₁ = [ rtᵢ + n/rtᵢ] / 2
+-}
+
+-- | Given an approximation, find a better one
+next
+  -- :: (Num a, Ord a, Fractional a)
+  :: Fractional a
+  => a -- ^ Number whose square root is to be found
+  -> a -- ^ Old approximation of square root
+  -> a -- ^ New approximation of square root
+next n rt = (rt + (n/rt))/2
+
+-- | Starting from initial guess, keep finding better approximations
+rts
+  :: Fractional a
+  => a   -- ^ Number whose square root is to be found
+  -> a   -- ^ Initial guess of square root
+  -> [a] -- ^ List of approximations that keeps getting better
+rts n rt0 = rt0 : rts n (next n rt0)
+
+-- | Given a series of approximations, stop when error goes below `eps'
+within
+  :: (Ord a, Fractional a)
+  => a    -- ^ Error tolerance
+  -> [a]  -- ^ List of ascending values
+  -> a    -- ^ Value when error is toleratable
+within eps l = case l of
+  a0 : a1 : l' ->
+    if a1-a0 < eps then a1
+    else within eps (a1:l')
+
+nrSqrt
+  :: Float  -- ^ Number whose square root is to be found
+  -> Float  -- ^ Initial guess of square root
+  -> Float  -- ^ Error tolerance
+  -> Float  -- ^ Square root value
+nrSqrt n rt0 eps = within eps $ rts n rt0
+
+
+{-
+λ> sqrt 2
+1.4142135623730951
+
+λ> nrSqrt 2 1.5 0.0001
+1.4166666666666665
+λ> nrSqrt 2 1.5 0.000001
+1.4166666666666665
+
+λ> sqrt 3
+1.7320508075688772
+
+λ> nrSqrt 3 2 0.0001
+1.75
+-}