{-# OPTIONS_GHC -fglasgow-exts #-} ----------------------------------------------------------------------------- {- | Module : Numeric.GSL.Minimization Copyright : (c) Alberto Ruiz 2006-9 License : GPL-style Maintainer : Alberto Ruiz (aruiz at um dot es) Stability : provisional Portability : uses ffi Minimization of a multidimensional function using some of the algorithms described in: <http://www.gnu.org/software/gsl/manual/html_node/Multidimensional-Minimization.html> -} ----------------------------------------------------------------------------- module Numeric.GSL.Minimization ( minimizeConjugateGradient, minimizeVectorBFGS2, minimizeNMSimplex ) where import Data.Packed.Internal import Data.Packed.Matrix import Foreign import Foreign.C.Types(CInt) ------------------------------------------------------------------------- {- | The method of Nelder and Mead, implemented by /gsl_multimin_fminimizer_nmsimplex/. The gradient of the function is not required. This is the example in the GSL manual: @minimize f xi = minimizeNMSimplex f xi (replicate (length xi) 1) 1e-2 100 \ -- f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30 \ -- main = do let (s,p) = minimize f [5,7] print s print p \ -- \> main [0.9920430849306285,1.9969168063253164] 0. 512.500 1.082 6.500 5. 1. 290.625 1.372 5.250 4. 2. 290.625 1.372 5.250 4. 3. 252.500 1.372 5.500 1. 4. 101.406 1.823 2.625 3.500 5. 101.406 1.823 2.625 3.500 6. 60. 1.823 0. 3. 7. 42.275 1.303 2.094 1.875 8. 42.275 1.303 2.094 1.875 9. 35.684 1.026 0.258 1.906 10. 35.664 0.804 0.588 2.445 11. 30.680 0.467 1.258 2.025 12. 30.680 0.356 1.258 2.025 13. 30.539 0.285 1.093 1.849 14. 30.137 0.168 0.883 2.004 15. 30.137 0.123 0.883 2.004 16. 30.090 0.100 0.958 2.060 17. 30.005 6.051e-2 1.022 2.004 18. 30.005 4.249e-2 1.022 2.004 19. 30.005 4.249e-2 1.022 2.004 20. 30.005 2.742e-2 1.022 2.004 21. 30.005 2.119e-2 1.022 2.004 22. 30.001 1.530e-2 0.992 1.997 23. 30.001 1.259e-2 0.992 1.997 24. 30.001 7.663e-3 0.992 1.997@ The path to the solution can be graphically shown by means of: @'Graphics.Plot.mplot' $ drop 3 ('toColumns' p)@ -} minimizeNMSimplex :: ([Double] -> Double) -- ^ function to minimize -> [Double] -- ^ starting point -> [Double] -- ^ sizes of the initial search box -> Double -- ^ desired precision of the solution -> Int -- ^ maximum number of iterations allowed -> ([Double], Matrix Double) -- ^ solution vector, and the optimization trajectory followed by the algorithm minimizeNMSimplex f xi sz tol maxit = unsafePerformIO $ do let xiv = fromList xi szv = fromList sz n = dim xiv fp <- mkVecfun (iv (f.toList)) rawpath <- ww2 withVector xiv withVector szv $ \xiv' szv' -> createMIO maxit (n+3) (c_minimizeNMSimplex fp tol (fi maxit) // xiv' // szv') "minimizeNMSimplex" let it = round (rawpath @@> (maxit-1,0)) path = takeRows it rawpath [sol] = toLists $ dropRows (it-1) path freeHaskellFunPtr fp return (drop 3 sol, path) foreign import ccall "gsl-aux.h minimize" c_minimizeNMSimplex:: FunPtr (CInt -> Ptr Double -> Double) -> Double -> CInt -> TVVM ---------------------------------------------------------------------------------- {- | The Fletcher-Reeves conjugate gradient algorithm /gsl_multimin_fminimizer_conjugate_fr/. This is the example in the GSL manual: @minimize = minimizeConjugateGradient 1E-2 1E-4 1E-3 30 f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30 \ -- df [x,y] = [20*(x-1), 40*(y-2)] \ -- main = do let (s,p) = minimize f df [5,7] print s print p \ -- \> main [1.0,2.0] 0. 687.848 4.996 6.991 1. 683.555 4.989 6.972 2. 675.013 4.974 6.935 3. 658.108 4.944 6.861 4. 625.013 4.885 6.712 5. 561.684 4.766 6.415 6. 446.467 4.528 5.821 7. 261.794 4.053 4.632 8. 75.498 3.102 2.255 9. 67.037 2.852 1.630 10. 45.316 2.191 1.762 11. 30.186 0.869 2.026 12. 30. 1. 2.@ The path to the solution can be graphically shown by means of: @'Graphics.Plot.mplot' $ drop 2 ('toColumns' p)@ -} minimizeConjugateGradient :: Double -- ^ initial step size -> Double -- ^ minimization parameter -> Double -- ^ desired precision of the solution (gradient test) -> Int -- ^ maximum number of iterations allowed -> ([Double] -> Double) -- ^ function to minimize -> ([Double] -> [Double]) -- ^ gradient -> [Double] -- ^ starting point -> ([Double], Matrix Double) -- ^ solution vector, and the optimization trajectory followed by the algorithm minimizeConjugateGradient = minimizeWithDeriv 0 {- | Taken from the GSL manual: The vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region. The bfgs2 version of this minimizer is the most efficient version available, and is a faithful implementation of the line minimization scheme described in Fletcher's Practical Methods of Optimization, Algorithms 2.6.2 and 2.6.4. It supercedes the original bfgs routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance tol corresponds to the parameter \sigma used by Fletcher. A value of 0.1 is recommended for typical use (larger values correspond to less accurate line searches). -} minimizeVectorBFGS2 :: Double -- ^ initial step size -> Double -- ^ minimization parameter tol -> Double -- ^ desired precision of the solution (gradient test) -> Int -- ^ maximum number of iterations allowed -> ([Double] -> Double) -- ^ function to minimize -> ([Double] -> [Double]) -- ^ gradient -> [Double] -- ^ starting point -> ([Double], Matrix Double) -- ^ solution vector, and the optimization trajectory followed by the algorithm minimizeVectorBFGS2 = minimizeWithDeriv 1 minimizeWithDeriv method istep minimpar tol maxit f df xi = unsafePerformIO $ do let xiv = fromList xi n = dim xiv f' = f . toList df' = (fromList . df . toList) fp <- mkVecfun (iv f') dfp <- mkVecVecfun (aux_vTov df') rawpath <- withVector xiv $ \xiv' -> createMIO maxit (n+2) (c_minimizeWithDeriv method fp dfp istep minimpar tol (fi maxit) // xiv') "minimizeDerivV" let it = round (rawpath @@> (maxit-1,0)) path = takeRows it rawpath sol = toList $ cdat $ dropColumns 2 $ dropRows (it-1) path freeHaskellFunPtr fp freeHaskellFunPtr dfp return (sol,path) foreign import ccall "gsl-aux.h minimizeWithDeriv" c_minimizeWithDeriv :: CInt -> FunPtr (CInt -> Ptr Double -> Double) -> FunPtr (CInt -> Ptr Double -> Ptr Double -> IO ()) -> Double -> Double -> Double -> CInt -> TVM --------------------------------------------------------------------- iv :: (Vector Double -> Double) -> (CInt -> Ptr Double -> Double) iv f n p = f (createV (fromIntegral n) copy "iv") where copy n' q = do copyArray q p (fromIntegral n') return 0 -- | conversion of Haskell functions into function pointers that can be used in the C side foreign import ccall "wrapper" mkVecfun :: (CInt -> Ptr Double -> Double) -> IO( FunPtr (CInt -> Ptr Double -> Double)) -- | another required conversion foreign import ccall "wrapper" mkVecVecfun :: (CInt -> Ptr Double -> Ptr Double -> IO ()) -> IO (FunPtr (CInt -> Ptr Double -> Ptr Double->IO())) aux_vTov :: (Vector Double -> Vector Double) -> (CInt -> Ptr Double -> Ptr Double -> IO()) aux_vTov f n p r = g where V {fptr = pr} = f x x = createV (fromIntegral n) copy "aux_vTov" copy n' q = do copyArray q p (fromIntegral n') return 0 g = withForeignPtr pr $ \p' -> copyArray r p' (fromIntegral n) -------------------------------------------------------------------- createV n fun msg = unsafePerformIO $ do r <- createVector n app1 fun vec r msg return r createMIO r c fun msg = do res <- createMatrix RowMajor r c app1 fun mat res msg return res