{-# LANGUAGE BangPatterns, RecursiveDo #-}

-- |
-- Module     : Simulation.Aivika.SystemDynamics
-- Copyright  : Copyright (c) 2009-2013, David Sorokin <david.sorokin@gmail.com>
-- License    : BSD3
-- Maintainer : David Sorokin <david.sorokin@gmail.com>
-- Stability  : experimental
-- Tested with: GHC 7.6.3
--
-- This module defines integrals and other functions of System Dynamics.
--

module Simulation.Aivika.SystemDynamics
       (-- * Equality and Ordering
        (.==.),
        (./=.),
        (.<.),
        (.>=.),
        (.>.),
        (.<=.),
        maxDynamics,
        minDynamics,
        ifDynamics,
        -- * Ordinary Differential Equations
        integ,
        smoothI,
        smooth,
        smooth3I,
        smooth3,
        smoothNI,
        smoothN,
        delay1I,
        delay1,
        delay3I,
        delay3,
        delayNI,
        delayN,
        forecast,
        trend,
        -- * Difference Equations
        sumDynamics,
        -- * Table Functions
        lookupDynamics,
        lookupStepwiseDynamics,
        -- * Discrete Functions
        delay,
        -- * Financial Functions
        npv,
        npve) where

import Data.Array
import Data.Array.IO.Safe
import Data.IORef
import Control.Monad
import Control.Monad.Trans

import Simulation.Aivika.Internal.Specs
import Simulation.Aivika.Internal.Simulation
import Simulation.Aivika.Internal.Dynamics
import Simulation.Aivika.Dynamics.Interpolate
import Simulation.Aivika.Dynamics.Memo.Unboxed
import Simulation.Aivika.Unboxed

--
-- Equality and Ordering
--

-- | Compare for equality.
(.==.) :: (Eq a) => Dynamics a -> Dynamics a -> Dynamics Bool
(.==.) = liftM2 (==)

-- | Compare for inequality.
(./=.) :: (Eq a) => Dynamics a -> Dynamics a -> Dynamics Bool
(./=.) = liftM2 (/=)

-- | Compare for ordering.
(.<.) :: (Ord a) => Dynamics a -> Dynamics a -> Dynamics Bool
(.<.) = liftM2 (<)

-- | Compare for ordering.
(.>=.) :: (Ord a) => Dynamics a -> Dynamics a -> Dynamics Bool
(.>=.) = liftM2 (>=)

-- | Compare for ordering.
(.>.) :: (Ord a) => Dynamics a -> Dynamics a -> Dynamics Bool
(.>.) = liftM2 (>)

-- | Compare for ordering.
(.<=.) :: (Ord a) => Dynamics a -> Dynamics a -> Dynamics Bool
(.<=.) = liftM2 (<=)

-- | Return the maximum.
maxDynamics :: (Ord a) => Dynamics a -> Dynamics a -> Dynamics a
maxDynamics = liftM2 max

-- | Return the minimum.
minDynamics :: (Ord a) => Dynamics a -> Dynamics a -> Dynamics a
minDynamics = liftM2 min

-- | Implement the if-then-else operator.
ifDynamics :: Dynamics Bool -> Dynamics a -> Dynamics a -> Dynamics a
ifDynamics cond x y =
  do a <- cond
     if a then x else y

--
-- Ordinary Differential Equations
--

integEuler :: Dynamics Double
             -> Dynamics Double 
             -> Dynamics Double 
             -> Point -> IO Double
integEuler (Dynamics f) (Dynamics i) (Dynamics y) p = 
  case pointIteration p of
    0 -> 
      i p
    n -> do 
      let sc = pointSpecs p
          ty = basicTime sc (n - 1) 0
          py = p { pointTime = ty, pointIteration = n - 1, pointPhase = 0 }
      a <- y py
      b <- f py
      let !v = a + spcDT (pointSpecs p) * b
      return v

integRK2 :: Dynamics Double
           -> Dynamics Double
           -> Dynamics Double
           -> Point -> IO Double
integRK2 (Dynamics f) (Dynamics i) (Dynamics y) p =
  case pointPhase p of
    0 -> case pointIteration p of
      0 ->
        i p
      n -> do
        let sc = pointSpecs p
            ty = basicTime sc (n - 1) 0
            t1 = ty
            t2 = basicTime sc (n - 1) 1
            py = p { pointTime = ty, pointIteration = n - 1, pointPhase = 0 }
            p1 = py
            p2 = p { pointTime = t2, pointIteration = n - 1, pointPhase = 1 }
        vy <- y py
        k1 <- f p1
        k2 <- f p2
        let !v = vy + spcDT sc / 2.0 * (k1 + k2)
        return v
    1 -> do
      let sc = pointSpecs p
          n  = pointIteration p
          ty = basicTime sc n 0
          t1 = ty
          py = p { pointTime = ty, pointIteration = n, pointPhase = 0 }
          p1 = py
      vy <- y py
      k1 <- f p1
      let !v = vy + spcDT sc * k1
      return v
    _ -> 
      error "Incorrect phase: integRK2"

integRK4 :: Dynamics Double
           -> Dynamics Double
           -> Dynamics Double
           -> Point -> IO Double
integRK4 (Dynamics f) (Dynamics i) (Dynamics y) p =
  case pointPhase p of
    0 -> case pointIteration p of
      0 -> 
        i p
      n -> do
        let sc = pointSpecs p
            ty = basicTime sc (n - 1) 0
            t1 = ty
            t2 = basicTime sc (n - 1) 1
            t3 = basicTime sc (n - 1) 2
            t4 = basicTime sc (n - 1) 3
            py = p { pointTime = ty, pointIteration = n - 1, pointPhase = 0 }
            p1 = py
            p2 = p { pointTime = t2, pointIteration = n - 1, pointPhase = 1 }
            p3 = p { pointTime = t3, pointIteration = n - 1, pointPhase = 2 }
            p4 = p { pointTime = t4, pointIteration = n - 1, pointPhase = 3 }
        vy <- y py
        k1 <- f p1
        k2 <- f p2
        k3 <- f p3
        k4 <- f p4
        let !v = vy + spcDT sc / 6.0 * (k1 + 2.0 * k2 + 2.0 * k3 + k4)
        return v
    1 -> do
      let sc = pointSpecs p
          n  = pointIteration p
          ty = basicTime sc n 0
          t1 = ty
          py = p { pointTime = ty, pointIteration = n, pointPhase = 0 }
          p1 = py
      vy <- y py
      k1 <- f p1
      let !v = vy + spcDT sc / 2.0 * k1
      return v
    2 -> do
      let sc = pointSpecs p
          n  = pointIteration p
          ty = basicTime sc n 0
          t2 = basicTime sc n 1
          py = p { pointTime = ty, pointIteration = n, pointPhase = 0 }
          p2 = p { pointTime = t2, pointIteration = n, pointPhase = 1 }
      vy <- y py
      k2 <- f p2
      let !v = vy + spcDT sc / 2.0 * k2
      return v
    3 -> do
      let sc = pointSpecs p
          n  = pointIteration p
          ty = basicTime sc n 0
          t3 = basicTime sc n 2
          py = p { pointTime = ty, pointIteration = n, pointPhase = 0 }
          p3 = p { pointTime = t3, pointIteration = n, pointPhase = 2 }
      vy <- y py
      k3 <- f p3
      let !v = vy + spcDT sc * k3
      return v
    _ -> 
      error "Incorrect phase: integRK4"

-- | Return an integral with the specified derivative and initial value.
--
-- To create a loopback, you should use the recursive do-notation.
-- It allows defining the differential equations unordered as
-- in mathematics:
--
-- @
-- model :: Simulation [Double]
-- model = 
--   mdo a <- integ (- ka * a) 100
--       b <- integ (ka * a - kb * b) 0
--       c <- integ (kb * b) 0
--       let ka = 1
--           kb = 1
--       runDynamicsInStopTime $ sequence [a, b, c]
-- @
integ :: Dynamics Double                  -- ^ the derivative
         -> Dynamics Double               -- ^ the initial value
         -> Simulation (Dynamics Double)  -- ^ the integral
integ diff i =
  mdo y <- memoDynamics z
      z <- Simulation $ \r ->
        case spcMethod (runSpecs r) of
          Euler -> return $ Dynamics $ integEuler diff i y
          RungeKutta2 -> return $ Dynamics $ integRK2 diff i y
          RungeKutta4 -> return $ Dynamics $ integRK4 diff i y
      return y

-- | Return the first order exponential smooth.
--
-- To create a loopback, you should use the recursive do-notation
-- with help of which the function itself is defined:
--
-- @
-- smoothI x t i =
--   mdo y <- integ ((x - y) \/ t) i
--       return y
-- @     
smoothI :: Dynamics Double                  -- ^ the value to smooth over time
           -> Dynamics Double               -- ^ time
           -> Dynamics Double               -- ^ the initial value
           -> Simulation (Dynamics Double)  -- ^ the first order exponential smooth
smoothI x t i =
  mdo y <- integ ((x - y) / t) i
      return y

-- | Return the first order exponential smooth.
--
-- This is a simplified version of the 'smoothI' function
-- without specifing the initial value.
smooth :: Dynamics Double                  -- ^ the value to smooth over time
          -> Dynamics Double               -- ^ time
          -> Simulation (Dynamics Double)  -- ^ the first order exponential smooth
smooth x t = smoothI x t x

-- | Return the third order exponential smooth.
--
-- To create a loopback, you should use the recursive do-notation
-- with help of which the function itself is defined:
--
-- @
-- smooth3I x t i =
--   mdo y  <- integ ((s2 - y) \/ t') i
--       s2 <- integ ((s1 - s2) \/ t') i
--       s1 <- integ ((x - s1) \/ t') i
--       let t' = t \/ 3.0
--       return y
-- @     
smooth3I :: Dynamics Double                  -- ^ the value to smooth over time
            -> Dynamics Double               -- ^ time
            -> Dynamics Double               -- ^ the initial value
            -> Simulation (Dynamics Double)  -- ^ the third order exponential smooth
smooth3I x t i =
  mdo y  <- integ ((s2 - y) / t') i
      s2 <- integ ((s1 - s2) / t') i
      s1 <- integ ((x - s1) / t') i
      let t' = t / 3.0
      return y

-- | Return the third order exponential smooth.
-- 
-- This is a simplified version of the 'smooth3I' function
-- without specifying the initial value.
smooth3 :: Dynamics Double                  -- ^ the value to smooth over time
           -> Dynamics Double               -- ^ time
           -> Simulation (Dynamics Double)  -- ^ the third order exponential smooth
smooth3 x t = smooth3I x t x

-- | Return the n'th order exponential smooth.
--
-- The result is not discrete in that sense that it may change within the integration time
-- interval depending on the integration method used. Probably, you should apply
-- the 'discreteDynamics' function to the result if you want to achieve an effect when
-- the value is not changed within the time interval, which is used sometimes.
smoothNI :: Dynamics Double                  -- ^ the value to smooth over time
            -> Dynamics Double               -- ^ time
            -> Int                           -- ^ the order
            -> Dynamics Double               -- ^ the initial value
            -> Simulation (Dynamics Double)  -- ^ the n'th order exponential smooth
smoothNI x t n i =
  mdo s <- forM [1 .. n] $ \k ->
        if k == 1
        then integ ((x - a ! 1) / t') i
        else integ ((a ! (k - 1) - a ! k) / t') i
      let a  = listArray (1, n) s 
          t' = t / fromIntegral n
      return $ a ! n

-- | Return the n'th order exponential smooth.
--
-- This is a simplified version of the 'smoothNI' function
-- without specifying the initial value.
smoothN :: Dynamics Double                  -- ^ the value to smooth over time
           -> Dynamics Double               -- ^ time
           -> Int                           -- ^ the order
           -> Simulation (Dynamics Double)  -- ^ the n'th order exponential smooth
smoothN x t n = smoothNI x t n x

-- | Return the first order exponential delay.
--
-- To create a loopback, you should use the recursive do-notation
-- with help of which the function itself is defined:
--
-- @
-- delay1I x t i =
--   mdo y <- integ (x - y \/ t) (i * t)
--       return $ y \/ t
-- @     
delay1I :: Dynamics Double                  -- ^ the value to conserve
           -> Dynamics Double               -- ^ time
           -> Dynamics Double               -- ^ the initial value
           -> Simulation (Dynamics Double)  -- ^ the first order exponential delay
delay1I x t i =
  mdo y <- integ (x - y / t) (i * t)
      return $ y / t

-- | Return the first order exponential delay.
--
-- This is a simplified version of the 'delay1I' function
-- without specifying the initial value.
delay1 :: Dynamics Double                  -- ^ the value to conserve
          -> Dynamics Double               -- ^ time
          -> Simulation (Dynamics Double)  -- ^ the first order exponential delay
delay1 x t = delay1I x t x

-- | Return the third order exponential delay.
delay3I :: Dynamics Double                  -- ^ the value to conserve
           -> Dynamics Double               -- ^ time
           -> Dynamics Double               -- ^ the initial value
           -> Simulation (Dynamics Double)  -- ^ the third order exponential delay
delay3I x t i =
  mdo y  <- integ (s2 / t' - y / t') (i * t')
      s2 <- integ (s1 / t' - s2 / t') (i * t')
      s1 <- integ (x - s1 / t') (i * t')
      let t' = t / 3.0
      return $ y / t'         

-- | Return the third order exponential delay.
--
-- This is a simplified version of the 'delay3I' function
-- without specifying the initial value.
delay3 :: Dynamics Double                  -- ^ the value to conserve
          -> Dynamics Double               -- ^ time
          -> Simulation (Dynamics Double)  -- ^ the third order exponential delay
delay3 x t = delay3I x t x

-- | Return the n'th order exponential delay.
delayNI :: Dynamics Double                  -- ^ the value to conserve
           -> Dynamics Double               -- ^ time
           -> Int                           -- ^ the order
           -> Dynamics Double               -- ^ the initial value
           -> Simulation (Dynamics Double)  -- ^ the n'th order exponential delay
delayNI x t n i =
  mdo s <- forM [1 .. n] $ \k ->
        if k == 1
        then integ (x - (a ! 1) / t') (i * t')
        else integ ((a ! (k - 1)) / t' - (a ! k) / t') (i * t')
      let a  = listArray (1, n) s
          t' = t / fromIntegral n
      return $ (a ! n) / t'

-- | Return the n'th order exponential delay.
--
-- This is a simplified version of the 'delayNI' function
-- without specifying the initial value.
delayN :: Dynamics Double                  -- ^ the value to conserve
          -> Dynamics Double               -- ^ time
          -> Int                           -- ^ the order
          -> Simulation (Dynamics Double)  -- ^ the n'th order exponential delay
delayN x t n = delayNI x t n x

-- | Return the forecast.
--
-- The function has the following definition:
--
-- @
-- forecast x at hz =
--   do y <- smooth x at
--      return $ x * (1.0 + (x \/ y - 1.0) \/ at * hz)
-- @
forecast :: Dynamics Double                  -- ^ the value to forecast
            -> Dynamics Double               -- ^ the average time
            -> Dynamics Double               -- ^ the time horizon
            -> Simulation (Dynamics Double)  -- ^ the forecast
forecast x at hz =
  do y <- smooth x at
     return $ x * (1.0 + (x / y - 1.0) / at * hz)

-- | Return the trend.
--
-- The function has the following definition:
--
-- @
-- trend x at i =
--   do y <- smoothI x at (x \/ (1.0 + i * at))
--      return $ (x \/ y - 1.0) \/ at
-- @
trend :: Dynamics Double                  -- ^ the value for which the trend is calculated
         -> Dynamics Double               -- ^ the average time
         -> Dynamics Double               -- ^ the initial value
         -> Simulation (Dynamics Double)  -- ^ the fractional change rate
trend x at i =
  do y <- smoothI x at (x / (1.0 + i * at))
     return $ (x / y - 1.0) / at

--
-- Difference Equations
--

-- | Retun the sum for the difference equation.
-- It is like an integral returned by the 'integ' function, only now
-- the difference is used instead of derivative.
--
-- As usual, to create a loopback, you should use the recursive do-notation.
sumDynamics :: (Num a, Unboxed a)
               => Dynamics a               -- ^ the difference
               -> Dynamics a               -- ^ the initial value
               -> Simulation (Dynamics a)  -- ^ the sum
sumDynamics (Dynamics diff) (Dynamics i) =
  mdo y <- memoDynamics z
      z <- Simulation $ \r ->
        return $ Dynamics $ \p ->
        case pointIteration p of
          0 -> i p
          n -> do 
            let Dynamics m = y
                sc = pointSpecs p
                ty = basicTime sc (n - 1) 0
                py = p { pointTime = ty, 
                         pointIteration = n - 1, 
                         pointPhase = 0 }
            a <- m py
            b <- diff py
            let !v = a + b
            return v
      return y

--
-- Table Functions
--

-- | Lookup @x@ in a table of pairs @(x, y)@ using linear interpolation.
lookupDynamics :: Dynamics Double -> Array Int (Double, Double) -> Dynamics Double
lookupDynamics (Dynamics m) tbl =
  Dynamics (\p -> do a <- m p; return $ find first last a) where
    (first, last) = bounds tbl
    find left right x =
      if left > right then
        error "Incorrect index: table"
      else
        let index = (left + 1 + right) `div` 2
            x1    = fst $ tbl ! index
        in if x1 <= x then 
             let y | index < right = find index right x
                   | right == last  = snd $ tbl ! right
                   | otherwise     = 
                     let x2 = fst $ tbl ! (index + 1)
                         y1 = snd $ tbl ! index
                         y2 = snd $ tbl ! (index + 1)
                     in y1 + (y2 - y1) * (x - x1) / (x2 - x1) 
             in y
           else
             let y | left < index = find left (index - 1) x
                   | left == first = snd $ tbl ! left
                   | otherwise    = error "Incorrect index: table"
             in y

-- | Lookup @x@ in a table of pairs @(x, y)@ using stepwise function.
lookupStepwiseDynamics :: Dynamics Double
                          -> Array Int (Double, Double)
                          -> Dynamics Double
lookupStepwiseDynamics (Dynamics m) tbl =
  Dynamics (\p -> do a <- m p; return $ find first last a) where
    (first, last) = bounds tbl
    find left right x =
      if left > right then
        error "Incorrect index: table"
      else
        let index = (left + 1 + right) `div` 2
            x1    = fst $ tbl ! index
        in if x1 <= x then 
             let y | index < right = find index right x
                   | right == last  = snd $ tbl ! right
                   | otherwise     = snd $ tbl ! right
             in y
           else
             let y | left < index = find left (index - 1) x
                   | left == first = snd $ tbl ! left
                   | otherwise    = error "Incorrect index: table"
             in y

--
-- Discrete Functions
--

-- | Return the delayed value.
--
-- If you want to apply the result recursively in some loopback then you
-- should use one of the memoization functions such as 'memoDynamics'
-- and 'memo0Dynamics'.    
delay :: Dynamics a          -- ^ the value to delay
         -> Dynamics Double  -- ^ the lag time
         -> Dynamics a       -- ^ the initial value
         -> Dynamics a       -- ^ the delayed value
delay (Dynamics x) (Dynamics d) (Dynamics i) = Dynamics r 
  where
    r p = do 
      let t  = pointTime p
          sc = pointSpecs p
          n  = pointIteration p
      a <- d p
      let t' = t - a
          n' = fromIntegral $ floor $ (t' - spcStartTime sc) / spcDT sc
          y | n' < 0    = i $ p { pointTime = spcStartTime sc,
                                  pointIteration = 0, 
                                  pointPhase = 0 }
            | n' < n    = x $ p { pointTime = t',
                                  pointIteration = n',
                                  pointPhase = -1 }
            | n' > n    = error $
                          "Cannot return the future data: delay. " ++
                          "The lag time cannot be negative."
            | otherwise = error $
                          "Cannot return the current data: delay. " ++
                          "The lag time is too small."
      y

--
-- Financial Functions
--

-- | Return the Net Present Value (NPV) of the stream computed using the specified
-- discount rate, the initial value and some factor (usually 1).
--
-- It is defined in the following way:
--
-- @
-- npv stream rate init factor =
--   mdo df <- integ (- df * rate) 1
--       accum <- integ (stream * df) init
--       return $ (accum + dt * stream * df) * factor
-- @
npv :: Dynamics Double                  -- ^ the stream
       -> Dynamics Double               -- ^ the discount rate
       -> Dynamics Double               -- ^ the initial value
       -> Dynamics Double               -- ^ factor
       -> Simulation (Dynamics Double)  -- ^ the Net Present Value (NPV)
npv stream rate init factor =
  mdo df <- integ (- df * rate) 1
      accum <- integ (stream * df) init
      return $ (accum + dt * stream * df) * factor

-- | Return the Net Present Value End of period (NPVE) of the stream computed
-- using the specified discount rate, the initial value and some factor.
--
-- It is defined in the following way:
--
-- @
-- npve stream rate init factor =
--   mdo df <- integ (- df * rate \/ (1 + rate * dt)) (1 \/ (1 + rate * dt))
--       accum <- integ (stream * df) init
--       return $ (accum + dt * stream * df) * factor
-- @
npve :: Dynamics Double                  -- ^ the stream
        -> Dynamics Double               -- ^ the discount rate
        -> Dynamics Double               -- ^ the initial value
        -> Dynamics Double               -- ^ factor
        -> Simulation (Dynamics Double)  -- ^ the Net Present Value End (NPVE)
npve stream rate init factor =
  mdo df <- integ (- df * rate / (1 + rate * dt)) (1 / (1 + rate * dt))
      accum <- integ (stream * df) init
      return $ (accum + dt * stream * df) * factor