ProcessMatrix.hs

{-|
Module      : Math.ExpPairs.ProcessMatrix
Description : Monoidal wrapper for Matrix3
Copyright   : (c) Andrew Lelechenko, 2014-2015
License     : GPL-3
Maintainer  : andrew.lelechenko@gmail.com
Stability   : experimental
Portability : TemplateHaskell

Provides types for sequences of /A/- and /B/-processes of van der Corput. A good account on this topic can be found in /Graham S. W.,  Kolesnik G. A./ Van Der Corput's Method of Exponential Sums, Cambridge University Press, 1991, especially Ch. 5.
-}
{-# LANGUAGE TemplateHaskell, BangPatterns, GeneralizedNewtypeDeriving  #-}
module Math.ExpPairs.ProcessMatrix
	( Process (..)
	, ProcessMatrix ()
	, aMatrix
	, baMatrix
	, evalMatrix
	) where

import Data.Monoid           (Monoid, mempty, mappend)
import Data.Function.Memoize (deriveMemoizable)

import Math.ExpPairs.Matrix3

-- | Since B^2 = id, B 'Corput16' = 'Corput16', B 'Hux05' = 'Hux05' and B 'HuxW87b1' = ???, the sequence of /A/- and /B/-processes, applied to 'initPairs' can be rewritten as a sequence of 'A' and 'BA'.
data Process
	-- | /A/-process
	= A
	-- | /BA/-process
	| BA
	deriving (Eq, Show, Read, Ord, Enum)

deriveMemoizable ''Process

newtype ProcessMatrix = ProcessMatrix (Matrix3 Integer)
	deriving (Eq, Num, Show)

instance Monoid ProcessMatrix where
	mempty = 1
	mappend (ProcessMatrix a) (ProcessMatrix b) = ProcessMatrix $ normalize $ a * b

process2matrix :: Process -> ProcessMatrix
process2matrix  A = ProcessMatrix $ Matrix3 1 0 0 1 1 1  2 0 2
process2matrix BA = ProcessMatrix $ Matrix3 0 1 0 2 0 1  2 0 2

-- | Return process matrix for 'A'-process.
aMatrix :: ProcessMatrix
aMatrix = process2matrix A

-- | Return process matrix for 'BA'-process.
baMatrix :: ProcessMatrix
baMatrix = process2matrix BA

-- |Apply a projective transformation, defined by 'Path',
-- to a given point in two-dimensional projective space.
evalMatrix :: Num t => ProcessMatrix -> (t, t, t) -> (t, t, t)
evalMatrix (ProcessMatrix m) (a,b,c) = (a',b',c') where
	m' = fmap fromInteger m
	(Vector3 a' b' c') = multCol m' (Vector3 a b c)