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1{- |
2Module : Numeric.GSL.ODE
3Copyright : (c) Alberto Ruiz 2010
4License : GPL
5
6Maintainer : Alberto Ruiz (aruiz at um dot es)
7Stability : provisional
8Portability : uses ffi
9
10Solution of ordinary differential equation (ODE) initial value problems.
11
12<http://www.gnu.org/software/gsl/manual/html_node/Ordinary-Differential-Equations.html>
13
14A simple example:
15
16@
17import Numeric.GSL.ODE
18import Numeric.LinearAlgebra
19import Numeric.LinearAlgebra.Util(mplot)
20
21xdot t [x,v] = [v, -0.95*x - 0.1*v]
22
23ts = linspace 100 (0,20 :: Double)
24
25sol = odeSolve xdot [10,0] ts
26
27main = mplot (ts : toColumns sol)
28@
29
30-}
31-----------------------------------------------------------------------------
32
33module Numeric.GSL.ODE (
34 odeSolve, odeSolveV, ODEMethod(..), Jacobian
35) where
36
37import Data.Packed.Internal
38import Numeric.GSL.Internal
39
40import Foreign.Ptr(FunPtr, nullFunPtr, freeHaskellFunPtr)
41import Foreign.C.Types
42import System.IO.Unsafe(unsafePerformIO)
43
44-------------------------------------------------------------------------
45
46type Jacobian = Double -> Vector Double -> Matrix Double
47
48-- | Stepping functions
49data ODEMethod = RK2 -- ^ Embedded Runge-Kutta (2, 3) method.
50 | 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.
51 | RKf45 -- ^ Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
52 | RKck -- ^ Embedded Runge-Kutta Cash-Karp (4, 5) method.
53 | RK8pd -- ^ Embedded Runge-Kutta Prince-Dormand (8,9) method.
54 | RK2imp Jacobian -- ^ Implicit 2nd order Runge-Kutta at Gaussian points.
55 | RK4imp Jacobian -- ^ Implicit 4th order Runge-Kutta at Gaussian points.
56 | BSimp Jacobian -- ^ Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method is generally suitable for stiff problems.
57 | 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.
58 | 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.
59 | 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.
60
61
62-- | A version of 'odeSolveV' with reasonable default parameters and system of equations defined using lists.
63odeSolve
64 :: (Double -> [Double] -> [Double]) -- ^ xdot(t,x)
65 -> [Double] -- ^ initial conditions
66 -> Vector Double -- ^ desired solution times
67 -> Matrix Double -- ^ solution
68odeSolve xdot xi ts = odeSolveV RKf45 hi epsAbs epsRel (l2v xdot) (fromList xi) ts
69 where hi = (ts@>1 - ts@>0)/100
70 epsAbs = 1.49012e-08
71 epsRel = 1.49012e-08
72 l2v f = \t -> fromList . f t . toList
73
74-- | Evolution of the system with adaptive step-size control.
75odeSolveV
76 :: ODEMethod
77 -> Double -- ^ initial step size
78 -> Double -- ^ absolute tolerance for the state vector
79 -> Double -- ^ relative tolerance for the state vector
80 -> (Double -> Vector Double -> Vector Double) -- ^ xdot(t,x)
81 -> Vector Double -- ^ initial conditions
82 -> Vector Double -- ^ desired solution times
83 -> Matrix Double -- ^ solution
84odeSolveV RK2 = odeSolveV' 0 Nothing
85odeSolveV RK4 = odeSolveV' 1 Nothing
86odeSolveV RKf45 = odeSolveV' 2 Nothing
87odeSolveV RKck = odeSolveV' 3 Nothing
88odeSolveV RK8pd = odeSolveV' 4 Nothing
89odeSolveV (RK2imp jac) = odeSolveV' 5 (Just jac)
90odeSolveV (RK4imp jac) = odeSolveV' 6 (Just jac)
91odeSolveV (BSimp jac) = odeSolveV' 7 (Just jac)
92odeSolveV (RK1imp jac) = odeSolveV' 8 (Just jac)
93odeSolveV MSAdams = odeSolveV' 9 Nothing
94odeSolveV (MSBDF jac) = odeSolveV' 10 (Just jac)
95
96
97odeSolveV'
98 :: CInt
99 -> Maybe (Double -> Vector Double -> Matrix Double) -- ^ optional jacobian
100 -> Double -- ^ initial step size
101 -> Double -- ^ absolute tolerance for the state vector
102 -> Double -- ^ relative tolerance for the state vector
103 -> (Double -> Vector Double -> Vector Double) -- ^ xdot(t,x)
104 -> Vector Double -- ^ initial conditions
105 -> Vector Double -- ^ desired solution times
106 -> Matrix Double -- ^ solution
107odeSolveV' method mbjac h epsAbs epsRel f xiv ts = unsafePerformIO $ do
108 let n = dim xiv
109 fp <- mkDoubleVecVecfun (\t -> aux_vTov (checkdim1 n . f t))
110 jp <- case mbjac of
111 Just jac -> mkDoubleVecMatfun (\t -> aux_vTom (checkdim2 n . jac t))
112 Nothing -> return nullFunPtr
113 sol <- vec xiv $ \xiv' ->
114 vec (checkTimes ts) $ \ts' ->
115 createMIO (dim ts) n
116 (ode_c (method) h epsAbs epsRel fp jp // xiv' // ts' )
117 "ode"
118 freeHaskellFunPtr fp
119 return sol
120
121foreign import ccall safe "ode"
122 ode_c :: CInt -> Double -> Double -> Double -> FunPtr (Double -> TVV) -> FunPtr (Double -> TVM) -> TVVM
123
124-------------------------------------------------------
125
126checkdim1 n v
127 | dim v == n = v
128 | otherwise = error $ "Error: "++ show n
129 ++ " components expected in the result of the function supplied to odeSolve"
130
131checkdim2 n m
132 | rows m == n && cols m == n = m
133 | otherwise = error $ "Error: "++ show n ++ "x" ++ show n
134 ++ " Jacobian expected in odeSolve"
135
136checkTimes ts | dim ts > 1 && all (>0) (zipWith subtract ts' (tail ts')) = ts
137 | otherwise = error "odeSolve requires increasing times"
138 where ts' = toList ts
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