PowerSeries.hs

import Prelude

-- * Power series
--
-- treat power series representations of functions
-- like numbers and derivatives

-- ** Background
--
-- http://www.cs.dartmouth.edu/~doug/powser.html
-- there is a video too: https://youtu.be/jaHoYy2rnUc
-- This is very related to automatic differentiation
--   http://conal.net/papers/beautiful-differentiation/

data P a = P a (P a)

instance Num a => Num (P a) where
  P a xa + P b xb = (a + b) `P` (xa + xb)
  P a xa * P b xb = (a * b) `P` (xa * P b 0 + xb * P a 0)

  fromInteger = (`P` 0) . fromInteger
  signum (P a _) = signum a `P` 0 -- correct?
  abs (P a _) = abs a `P` 0 -- correct?
  negate (P a xa) = negate a `P` negate xa

-- * =toList=

toList (P a xa) = a : toList xa
fromList [] = 0
fromList (a:xa) = P a $ fromList xa

-- * =toFunction=
-- power series expansion at x to degree n

toFunction :: Num a => Int -> P a -> (a -> a)
toFunction 0 (P a _) _ = a
toFunction n (P a xa) x = a + x * toFunction (n-1) xa x


instance (Eq a, Show a, Num a) => Show (P a) where
  show (P a (P 0 _)) = show a
  show (P a (P xa _)) = show a ++ "+" ++ show xa ++ "x"

-- * TODO
--
-- - complete numerical classes
-- - make P =Symbol= tagged (to differentiate between /x/ and /y/)
-- - =ListLike=? https://hackage.haskell.org/package/ListLike/docs/Data-ListLike.html
-- - =IsLabel= to compute multivariate functions: https://www.reddit.com/r/haskell/comments/4x8tk8/overloadedlabels_considered_awesome/