-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Delay differential equations
--   
--   Please see the README on Github at
--   <a>https://github.com/masterdezign/dde#readme</a>
@package dde
@version 0.0.0

module Numeric.DDE.Types

-- | DDE right-hand side
type RHS = (State, HistorySnapshot, InputSnapshot) -> State

-- | Contains only the required snapshot of history to make steppers (e.g.
--   Heun) work. There could be several delay variables (as in rc-analog)
newtype HistorySnapshot
Hist :: Vector Double -> HistorySnapshot
[_histsnap] :: HistorySnapshot -> Vector Double

-- | Vector of input data points
newtype Input
Input :: Vector Double -> Input
[_input] :: Input -> Vector Double

-- | Input u(t) is one-dimensional
newtype InputSnapshot
Inp :: Double -> InputSnapshot
[_insnap] :: InputSnapshot -> Double

-- | State of a dynamical system, it can be a vector of any length (x(t),
--   y(t), ...).
newtype State
State :: Vector Double -> State
[_state] :: State -> Vector Double

-- | Stepper for DDEs with a single delay
--   
--   <pre>
--   &gt;&gt;&gt; _stepper stepSize rhs xyState xTau1_2 u1_2
--   </pre>
newtype Stepper1
Stepper1 :: (Double -> RHS -> State -> (Double, Double) -> (Double, Double) -> State) -> Stepper1
[_stepper] :: Stepper1 -> Double -> RHS -> State -> (Double, Double) -> (Double, Double) -> State


-- | <h1>Example DDE models</h1>
--   
--   These example models are defined by <a>rhs</a> function. Adding a new
--   example model requires adjusting <a>rhs</a> and <a>Par</a>. Switching
--   between models is performed by changing <a>Par</a>.
module Numeric.DDE.Model

-- | Model parameters
data Par

-- | Mackey-Glass model (no external input)
MackeyGlass :: Double -> Double -> Par
[_beta] :: Par -> Double
[_gamma] :: Par -> Double
RC :: (Double -> Double) -> Double -> BandpassFiltering -> Par

-- | Nonlinear transformation
[_fnl] :: Par -> Double -> Double

-- | External forcing weight
[_rho] :: Par -> Double
[_filt] :: Par -> BandpassFiltering

-- | Bandpass filter (linear) parameters
data BandpassFiltering
BandpassFiltering :: Double -> Double -> BandpassFiltering

-- | System response time, s (epsilon = tau / tau_D)
[_tau] :: BandpassFiltering -> Double

-- | Integration time, s (delta = tau_D / theta)
[_theta] :: BandpassFiltering -> Double

-- | DDE right-hand sides for example models
rhs :: Par -> RHS
instance GHC.Show.Show Numeric.DDE.Model.BandpassFiltering


-- | <h1>Delay differential equations (DDE)</h1>
--   
--   <h2>Example: Ikeda DDE</h2>
--   
--   Below is a complete example simulating the Ikeda DDE defined as:
--   <tt>tau * x(t)/dt = -x + beta * sin[x(t - tau_D)]</tt>.
--   
--   <pre>
--   import qualified Data.Vector.Storable as V
--   import qualified Numeric.DDE as DDE
--   
--   ikedaRhs beta ((DDE.State xs), (DDE.Hist hs), _) = DDE.State $ V.fromList [x']
--     where
--       -- Ikeda DDE definition
--       x' = (-x + beta * (sin x_tauD)) / tau
--   
--       -- Constants
--       tau = 0.01
--   
--       -- Dynamical variable x(t)
--       x = V.head xs
--   
--       -- Delay term x(t - tau_D)
--       x_tauD = V.head hs
--   
--   model beta hStep len1 totalIter = DDE.integ DDE.rk4 state0 hist0 len1 hStep (ikedaRhs beta) inp
--     where
--       -- Initial conditions:
--       -- dynamical state and delay history.
--       state0 = DDE.State $ V.fromList [pi/2]
--       hist0 = V.replicate len1 (pi/2)
--   
--       -- Input is ignored in ikedaRhs
--       inp = DDE.Input $ V.replicate (totalIter + 1) 0
--   
--   -- Control parameter
--   beta = 2.6
--   
--   main = do
--     let hStep = 0.001  -- Integration step
--         samplesPerDelay = round $ 1.0 / hStep
--         delays = 8
--         total = delays * samplesPerDelay
--   
--     let (state1, trace) = model beta hStep samplesPerDelay total
--   
--     mapM_ print $ V.toList trace
--   </pre>
module Numeric.DDE

-- | Generic integrator that records the whole time trace $x(t)$
integ :: Stepper1 -> State -> Vector Double -> Int -> Double -> RHS -> Input -> (State, Vector Double)

-- | Generic integrator for DDEs with a single delay
integ' :: (State -> (Double, Double) -> (Double, Double) -> State) -> Int -> Int -> Int -> (State, Vector Double, Input) -> (State, Vector Double)

-- | RK4 integrator shortcut for 2D DDEs with zero initial conditions
integRk4_2D :: Int -> Double -> RHS -> Input -> (State, Vector Double)

-- | Shortcut for Heun's 2nd order 2D DDEs with zero initial conditions
integHeun2_2D :: Int -> Double -> RHS -> Input -> (State, Vector Double)

-- | Vector of input data points
newtype Input
Input :: Vector Double -> Input
[_input] :: Input -> Vector Double

-- | State of a dynamical system, it can be a vector of any length (x(t),
--   y(t), ...).
newtype State
State :: Vector Double -> State
[_state] :: State -> Vector Double

-- | Contains only the required snapshot of history to make steppers (e.g.
--   Heun) work. There could be several delay variables (as in rc-analog)
newtype HistorySnapshot
Hist :: Vector Double -> HistorySnapshot
[_histsnap] :: HistorySnapshot -> Vector Double

-- | Stepper for DDEs with a single delay
--   
--   <pre>
--   &gt;&gt;&gt; _stepper stepSize rhs xyState xTau1_2 u1_2
--   </pre>
newtype Stepper1
Stepper1 :: (Double -> RHS -> State -> (Double, Double) -> (Double, Double) -> State) -> Stepper1
[_stepper] :: Stepper1 -> Double -> RHS -> State -> (Double, Double) -> (Double, Double) -> State

-- | Fourth order Runge-Kutta stepper
rk4 :: Stepper1

-- | Second order Heun's stepper
heun2 :: Stepper1