    {-# OPTIONS_GHC -fno-warn-missing-signatures #-}

    {- |
    Module      :  Numeric.GSL.Integration
    Copyright   :  (c) Alberto Ruiz 2006
    License     :  GPL
    Maintainer  :  Alberto Ruiz
    Stability   :  provisional

    Numerical integration routines.

    <http://www.gnu.org/software/gsl/manual/html_node/Numerical-Integration.html#Numerical-Integration>
    -}

    module Numeric.GSL.Integration (
        integrateQNG,
        integrateQAGS,
        integrateQAGI,
        integrateQAGIU,
        integrateQAGIL,
        integrateCQUAD
    ) where

    import Foreign.C.Types
    import Foreign.Marshal.Alloc(malloc, free)
    import Foreign.Ptr(Ptr, FunPtr, freeHaskellFunPtr)
    import Foreign.Storable(peek)
    import Numeric.GSL.Internal
    import System.IO.Unsafe(unsafePerformIO)

    eps :: Double
    eps = Double
    1e-12

    {- | conversion of Haskell functions into function pointers that can be used in the C side
    -}
    foreign import ccall safe "wrapper" mkfun:: (Double -> Ptr() -> Double) -> IO( FunPtr (Double -> Ptr() -> Double))

    --------------------------------------------------------------------
    {- | Numerical integration using /gsl_integration_qags/ (adaptive integration with singularities). For example:

    >>> let quad = integrateQAGS 1E-9 1000
    >>> let f a x = x**(-0.5) * log (a*x)
    >>> quad (f 1) 0 1
    (-3.999999999999974,4.871658632055187e-13)

    -}

    integrateQAGS :: Double               -- ^ precision (e.g. 1E-9)
                     -> Int               -- ^ size of auxiliary workspace (e.g. 1000)
                     -> (Double -> Double) -- ^ function to be integrated on the interval (a,b)
                     -> Double           -- ^ a
                     -> Double           -- ^ b
                     -> (Double, Double) -- ^ result of the integration and error
    integrateQAGS :: Double
    -> Int
    -> (Double -> Double)
    -> Double
    -> Double
    -> (Double, Double)
    integrateQAGS Double
    prec Int
    n Double -> Double
    f Double
    a Double
    b = IO (Double, Double) -> (Double, Double)
    forall a. IO a -> a
    unsafePerformIO (IO (Double, Double) -> (Double, Double))
    -> IO (Double, Double) -> (Double, Double)
    forall a b. (a -> b) -> a -> b
    $ do
        r <- IO (Ptr Double)
    forall a. Storable a => IO (Ptr a)
    malloc
        e <- malloc
        fp <- mkfun (\Double
    x Ptr ()
    _ -> Double -> Double
    f Double
    x)
        c_integrate_qags fp a b eps prec (fromIntegral n) r e // check "integrate_qags"
        vr <- peek r
        ve <- peek e
        let result = (Double
    vr,Double
    ve)
        free r
        free e
        freeHaskellFunPtr fp
        return result

    foreign import ccall safe "integrate_qags" c_integrate_qags
        :: FunPtr (Double-> Ptr() -> Double) -> Double -> Double
        -> Double -> Double -> CInt -> Ptr Double -> Ptr Double -> IO CInt

    -----------------------------------------------------------------
    {- | Numerical integration using /gsl_integration_qng/ (useful for fast integration of smooth functions). For example:

    >>> let quad = integrateQNG 1E-6
    >>> quad (\x -> 4/(1+x*x)) 0 1
    (3.141592653589793,3.487868498008632e-14)

    -}

    integrateQNG :: Double               -- ^ precision (e.g. 1E-9)
                     -> (Double -> Double) -- ^ function to be integrated on the interval (a,b)
                     -> Double           -- ^ a
                     -> Double           -- ^ b
                     -> (Double, Double) -- ^ result of the integration and error
    integrateQNG :: Double
    -> (Double -> Double) -> Double -> Double -> (Double, Double)
    integrateQNG Double
    prec Double -> Double
    f Double
    a Double
    b = IO (Double, Double) -> (Double, Double)
    forall a. IO a -> a
    unsafePerformIO (IO (Double, Double) -> (Double, Double))
    -> IO (Double, Double) -> (Double, Double)
    forall a b. (a -> b) -> a -> b
    $ do
        r <- IO (Ptr Double)
    forall a. Storable a => IO (Ptr a)
    malloc
        e <- malloc
        fp <- mkfun (\Double
    x Ptr ()
    _ -> Double -> Double
    f Double
    x)
        c_integrate_qng fp a b eps prec r e  // check "integrate_qng"
        vr <- peek r
        ve <- peek e
        let result = (Double
    vr,Double
    ve)
        free r
        free e
        freeHaskellFunPtr fp
        return result

    foreign import ccall safe "integrate_qng" c_integrate_qng
        :: FunPtr (Double-> Ptr() -> Double) -> Double -> Double
        -> Double -> Double -> Ptr Double -> Ptr Double -> IO CInt

    --------------------------------------------------------------------
    {- | Numerical integration using /gsl_integration_qagi/ (integration over the infinite integral -Inf..Inf using QAGS).
    For example:

    >>> let quad = integrateQAGI 1E-9 1000
    >>> let f a x = exp(-a * x^2)
    >>> quad (f 0.5)
    (2.5066282746310002,6.229215880648858e-11)

    -}

    integrateQAGI :: Double               -- ^ precision (e.g. 1E-9)
                     -> Int               -- ^ size of auxiliary workspace (e.g. 1000)
                     -> (Double -> Double) -- ^ function to be integrated on the interval (-Inf,Inf)
                     -> (Double, Double) -- ^ result of the integration and error
    integrateQAGI :: Double -> Int -> (Double -> Double) -> (Double, Double)
    integrateQAGI Double
    prec Int
    n Double -> Double
    f = IO (Double, Double) -> (Double, Double)
    forall a. IO a -> a
    unsafePerformIO (IO (Double, Double) -> (Double, Double))
    -> IO (Double, Double) -> (Double, Double)
    forall a b. (a -> b) -> a -> b
    $ do
        r <- IO (Ptr Double)
    forall a. Storable a => IO (Ptr a)
    malloc
        e <- malloc
        fp <- mkfun (\Double
    x Ptr ()
    _ -> Double -> Double
    f Double
    x)
        c_integrate_qagi fp eps prec (fromIntegral n) r e // check "integrate_qagi"
        vr <- peek r
        ve <- peek e
        let result = (Double
    vr,Double
    ve)
        free r
        free e
        freeHaskellFunPtr fp
        return result

    foreign import ccall safe "integrate_qagi" c_integrate_qagi
        :: FunPtr (Double-> Ptr() -> Double) -> Double -> Double
        -> CInt -> Ptr Double -> Ptr Double -> IO CInt

    --------------------------------------------------------------------
    {- | Numerical integration using /gsl_integration_qagiu/ (integration over the semi-infinite integral a..Inf).
    For example:

    >>> let quad = integrateQAGIU 1E-9 1000
    >>> let f a x = exp(-a * x^2)
    >>> quad (f 0.5) 0
    (1.2533141373155001,3.114607940324429e-11)

    -}

    integrateQAGIU :: Double               -- ^ precision (e.g. 1E-9)
                     -> Int               -- ^ size of auxiliary workspace (e.g. 1000)
                     -> (Double -> Double) -- ^ function to be integrated on the interval (a,Inf)
                     -> Double             -- ^ a
                     -> (Double, Double) -- ^ result of the integration and error
    integrateQAGIU :: Double -> Int -> (Double -> Double) -> Double -> (Double, Double)
    integrateQAGIU Double
    prec Int
    n Double -> Double
    f Double
    a = IO (Double, Double) -> (Double, Double)
    forall a. IO a -> a
    unsafePerformIO (IO (Double, Double) -> (Double, Double))
    -> IO (Double, Double) -> (Double, Double)
    forall a b. (a -> b) -> a -> b
    $ do
        r <- IO (Ptr Double)
    forall a. Storable a => IO (Ptr a)
    malloc
        e <- malloc
        fp <- mkfun (\Double
    x Ptr ()
    _ -> Double -> Double
    f Double
    x)
        c_integrate_qagiu fp a eps prec (fromIntegral n) r e // check "integrate_qagiu"
        vr <- peek r
        ve <- peek e
        let result = (Double
    vr,Double
    ve)
        free r
        free e
        freeHaskellFunPtr fp
        return result

    foreign import ccall safe "integrate_qagiu" c_integrate_qagiu
        :: FunPtr (Double-> Ptr() -> Double) -> Double -> Double
        -> Double -> CInt -> Ptr Double -> Ptr Double -> IO CInt

    --------------------------------------------------------------------
    {- | Numerical integration using /gsl_integration_qagil/ (integration over the semi-infinite integral -Inf..b).
    For example:

    >>> let quad = integrateQAGIL 1E-9 1000
    >>> let f a x = exp(-a * x^2)
    >>> quad (f 0.5) 0
    (1.2533141373155001,3.114607940324429e-11)

    -}

    integrateQAGIL :: Double               -- ^ precision (e.g. 1E-9)
                     -> Int               -- ^ size of auxiliary workspace (e.g. 1000)
                     -> (Double -> Double) -- ^ function to be integrated on the interval (a,Inf)
                     -> Double             -- ^ b
                     -> (Double, Double) -- ^ result of the integration and error
    integrateQAGIL :: Double -> Int -> (Double -> Double) -> Double -> (Double, Double)
    integrateQAGIL Double
    prec Int
    n Double -> Double
    f Double
    b = IO (Double, Double) -> (Double, Double)
    forall a. IO a -> a
    unsafePerformIO (IO (Double, Double) -> (Double, Double))
    -> IO (Double, Double) -> (Double, Double)
    forall a b. (a -> b) -> a -> b
    $ do
        r <- IO (Ptr Double)
    forall a. Storable a => IO (Ptr a)
    malloc
        e <- malloc
        fp <- mkfun (\Double
    x Ptr ()
    _ -> Double -> Double
    f Double
    x)
        c_integrate_qagil fp b eps prec (fromIntegral n) r e // check "integrate_qagil"
        vr <- peek r
        ve <- peek e
        let result = (Double
    vr,Double
    ve)
        free r
        free e
        freeHaskellFunPtr fp
        return result

    foreign import ccall safe "gsl-aux.h integrate_qagil" c_integrate_qagil
        :: FunPtr (Double-> Ptr() -> Double) -> Double -> Double
        -> Double -> CInt -> Ptr Double -> Ptr Double -> IO CInt

    --------------------------------------------------------------------
    {- | Numerical integration using /gsl_integration_cquad/ (quadrature
    for general integrands).  From the GSL manual:

    @CQUAD is a new doubly-adaptive general-purpose quadrature routine
    which can handle most types of singularities, non-numerical function
    values such as Inf or NaN, as well as some divergent integrals. It
    generally requires more function evaluations than the integration
    routines in QUADPACK, yet fails less often for difficult integrands.@

    For example:

    >>> let quad = integrateCQUAD 1E-12 1000
    >>> let f a x = exp(-a * x^2)
    >>> quad (f 0.5) 2 5
    (5.7025405463957006e-2,9.678874441303705e-16,95)

    Unlike other quadrature methods, integrateCQUAD also returns the
    number of function evaluations required.

    -}

    integrateCQUAD :: Double               -- ^ precision (e.g. 1E-9)
                     -> Int               -- ^ size of auxiliary workspace (e.g. 1000)
                     -> (Double -> Double) -- ^ function to be integrated on the interval (a, b)
                     -> Double             -- ^ a
                     -> Double             -- ^ b
                     -> (Double, Double, Int) -- ^ result of the integration, error and number of function evaluations performed
    integrateCQUAD :: Double
    -> Int
    -> (Double -> Double)
    -> Double
    -> Double
    -> (Double, Double, Int)
    integrateCQUAD Double
    prec Int
    n Double -> Double
    f Double
    a Double
    b = IO (Double, Double, Int) -> (Double, Double, Int)
    forall a. IO a -> a
    unsafePerformIO (IO (Double, Double, Int) -> (Double, Double, Int))
    -> IO (Double, Double, Int) -> (Double, Double, Int)
    forall a b. (a -> b) -> a -> b
    $ do
        r <- IO (Ptr Double)
    forall a. Storable a => IO (Ptr a)
    malloc
        e <- malloc
        neval <- malloc
        fp <- mkfun (\Double
    x Ptr ()
    _ -> Double -> Double
    f Double
    x)
        c_integrate_cquad fp a b eps prec (fromIntegral n) r e neval // check "integrate_cquad"
        vr <- peek r
        ve <- peek e
        vneval <- peek neval
        let result = (Double
    vr,Double
    ve,Int
    vneval)
        free r
        free e
        free neval
        freeHaskellFunPtr fp
        return result

    foreign import ccall safe "integrate_cquad" c_integrate_cquad
        :: FunPtr (Double-> Ptr() -> Double) -> Double -> Double
        -> Double -> Double -> CInt -> Ptr Double -> Ptr Double -> Ptr Int -> IO CInt