1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
|
{- |
Module : Numeric.GSL.ODE
Copyright : (c) Alberto Ruiz 2010
License : GPL
Maintainer : Alberto Ruiz (aruiz at um dot es)
Stability : provisional
Portability : uses ffi
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
import Numeric.LinearAlgebra
import Graphics.Plot
xdot t [x,v] = [v, -0.95*x - 0.1*v]
ts = linspace 100 (0,20)
sol = odeSolve xdot [10,0] ts
main = mplot (ts : toColumns sol)@
-}
-----------------------------------------------------------------------------
module Numeric.GSL.ODE (
odeSolve, odeSolveV, ODEMethod(..)
) where
import Data.Packed.Internal
import Data.Packed.Matrix
import Foreign
import Foreign.C.Types(CInt)
import Numeric.GSL.Internal
-------------------------------------------------------------------------
-- | 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 'RKf45'.
| 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 -- ^ Implicit 2nd order Runge-Kutta at Gaussian points.
| RK4imp -- ^ Implicit 4th order Runge-Kutta at Gaussian points.
| BSimp -- ^ Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian.
| Gear1 -- ^ M=1 implicit Gear method.
| Gear2 -- ^ M=2 implicit Gear method.
deriving (Enum,Eq,Show,Bounded)
-- | 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) Nothing (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
l2m f = \t -> fromLists . 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)
-> Maybe (Double -> Vector Double -> Matrix Double) -- ^ optional jacobian
-> Vector Double -- ^ initial conditions
-> Vector Double -- ^ desired solution times
-> Matrix Double -- ^ solution
odeSolveV method h epsAbs epsRel f mbjac 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 <- withVector xiv $ \xiv' ->
withVector (checkTimes ts) $ \ts' ->
createMIO (dim ts) n
(ode_c (fi (fromEnum method)) h epsAbs epsRel fp jp // xiv' // ts' )
"ode"
freeHaskellFunPtr fp
return sol
foreign import ccall "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
|
