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+{- |
+Module      :  Numeric.GSL.ODE
+Copyright   :  (c) Alberto Ruiz 2010
+License     :  GPL
+Maintainer  :  Alberto Ruiz
+Stability   :  provisional
+
+Solution of ordinary differential equation (ODE) initial value problems.
+
+<http://www.gnu.org/software/gsl/manual/html_node/Ordinary-Differential-Equations.html>
+
+A simple example:
+
+@
+import Numeric.GSL.ODE
+import Numeric.LinearAlgebra
+import Numeric.LinearAlgebra.Util(mplot)
+
+xdot t [x,v] = [v, -0.95*x - 0.1*v]
+
+ts = linspace 100 (0,20 :: Double)
+
+sol = odeSolve xdot [10,0] ts
+
+main = mplot (ts : toColumns sol)
+@
+
+-}
+-----------------------------------------------------------------------------
+
+module Numeric.GSL.ODE (
+    odeSolve, odeSolveV, ODEMethod(..), Jacobian
+) where
+
+import Data.Packed
+import Numeric.GSL.Internal
+
+import Foreign.Ptr(FunPtr, nullFunPtr, freeHaskellFunPtr)
+import Foreign.C.Types
+import System.IO.Unsafe(unsafePerformIO)
+
+-------------------------------------------------------------------------
+
+type TVV  = TV (TV Res)
+type TVM  = TV (TM Res)
+type TVVM = TV (TV (TM Res))
+
+type Jacobian = Double -> Vector Double -> Matrix Double
+
+-- | Stepping functions
+data ODEMethod = RK2 -- ^ Embedded Runge-Kutta (2, 3) method.
+               | RK4 -- ^ 4th order (classical) Runge-Kutta. The error estimate is obtained by halving the step-size. For more efficient estimate of the error, use the embedded methods.
+               | RKf45 -- ^ Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
+               | RKck -- ^ Embedded Runge-Kutta Cash-Karp (4, 5) method.
+               | RK8pd -- ^ Embedded Runge-Kutta Prince-Dormand (8,9) method.
+               | RK2imp Jacobian -- ^ Implicit 2nd order Runge-Kutta at Gaussian points.
+               | RK4imp Jacobian -- ^ Implicit 4th order Runge-Kutta at Gaussian points.
+               | BSimp Jacobian -- ^ Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method is generally suitable for stiff problems.
+               | RK1imp Jacobian -- ^ Implicit Gaussian first order Runge-Kutta. Also known as implicit Euler or backward Euler method. Error estimation is carried out by the step doubling method.
+               | MSAdams -- ^ A variable-coefficient linear multistep Adams method in Nordsieck form. This stepper uses explicit Adams-Bashforth (predictor) and implicit Adams-Moulton (corrector) methods in P(EC)^m functional iteration mode. Method order varies dynamically between 1 and 12. 
+               | MSBDF Jacobian -- ^ A variable-coefficient linear multistep backward differentiation formula (BDF) method in Nordsieck form. This stepper uses the explicit BDF formula as predictor and implicit BDF formula as corrector. A modified Newton iteration method is used to solve the system of non-linear equations. Method order varies dynamically between 1 and 5. The method is generally suitable for stiff problems.
+
+
+-- | A version of 'odeSolveV' with reasonable default parameters and system of equations defined using lists.
+odeSolve
+    :: (Double -> [Double] -> [Double])        -- ^ xdot(t,x)
+    -> [Double]        -- ^ initial conditions
+    -> Vector Double   -- ^ desired solution times
+    -> Matrix Double   -- ^ solution
+odeSolve xdot xi ts = odeSolveV RKf45 hi epsAbs epsRel (l2v xdot) (fromList xi) ts
+    where hi = (ts@>1 - ts@>0)/100
+          epsAbs = 1.49012e-08
+          epsRel = 1.49012e-08
+          l2v f = \t -> fromList  . f t . toList
+
+-- | Evolution of the system with adaptive step-size control.
+odeSolveV
+    :: ODEMethod
+    -> Double -- ^ initial step size
+    -> Double -- ^ absolute tolerance for the state vector
+    -> Double -- ^ relative tolerance for the state vector
+    -> (Double -> Vector Double -> Vector Double)   -- ^ xdot(t,x)
+    -> Vector Double     -- ^ initial conditions
+    -> Vector Double     -- ^ desired solution times
+    -> Matrix Double     -- ^ solution
+odeSolveV RK2 = odeSolveV' 0 Nothing
+odeSolveV RK4 = odeSolveV' 1 Nothing
+odeSolveV RKf45 = odeSolveV' 2 Nothing
+odeSolveV RKck = odeSolveV' 3 Nothing
+odeSolveV RK8pd = odeSolveV' 4 Nothing
+odeSolveV (RK2imp jac) = odeSolveV' 5 (Just jac)
+odeSolveV (RK4imp jac) = odeSolveV' 6 (Just jac)
+odeSolveV (BSimp jac) = odeSolveV' 7 (Just jac)
+odeSolveV (RK1imp jac) = odeSolveV' 8 (Just jac)
+odeSolveV MSAdams = odeSolveV' 9 Nothing
+odeSolveV (MSBDF jac) = odeSolveV' 10 (Just jac)
+
+
+odeSolveV'
+    :: CInt
+    -> Maybe (Double -> Vector Double -> Matrix Double)   -- ^ optional jacobian
+    -> Double -- ^ initial step size
+    -> Double -- ^ absolute tolerance for the state vector
+    -> Double -- ^ relative tolerance for the state vector
+    -> (Double -> Vector Double -> Vector Double)   -- ^ xdot(t,x)
+    -> Vector Double     -- ^ initial conditions
+    -> Vector Double     -- ^ desired solution times
+    -> Matrix Double     -- ^ solution
+odeSolveV' method mbjac h epsAbs epsRel f  xiv ts = unsafePerformIO $ do
+    let n   = dim xiv
+    fp <- mkDoubleVecVecfun (\t -> aux_vTov (checkdim1 n . f t))
+    jp <- case mbjac of
+        Just jac -> mkDoubleVecMatfun (\t -> aux_vTom (checkdim2 n . jac t))
+        Nothing  -> return nullFunPtr
+    sol <- vec xiv $ \xiv' ->
+            vec (checkTimes ts) $ \ts' ->
+             createMIO (dim ts) n
+              (ode_c (method) h epsAbs epsRel fp jp // xiv' // ts' )
+              "ode"
+    freeHaskellFunPtr fp
+    return sol
+
+foreign import ccall safe "ode"
+    ode_c :: CInt -> Double -> Double -> Double -> FunPtr (Double -> TVV) -> FunPtr (Double -> TVM) -> TVVM
+
+-------------------------------------------------------
+
+checkdim1 n v
+    | dim v == n = v
+    | otherwise = error $ "Error: "++ show n
+                        ++ " components expected in the result of the function supplied to odeSolve"
+
+checkdim2 n m
+    | rows m == n && cols m == n = m
+    | otherwise = error $ "Error: "++ show n ++ "x" ++ show n
+                        ++ " Jacobian expected in odeSolve"
+
+checkTimes ts | dim ts > 1 && all (>0) (zipWith subtract ts' (tail ts')) = ts
+              | otherwise = error "odeSolve requires increasing times"
+    where ts' = toList ts