```{- |
Module      :  Numeric.GSL.Integration
Copyright   :  (c) Alberto Ruiz 2006
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,
) 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 = 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 prec n f a b = unsafePerformIO \$ do
r <- malloc
e <- malloc
fp <- mkfun (\x _ -> f x)
c_integrate_qags fp a b eps prec (fromIntegral n) r e // check "integrate_qags"
vr <- peek r
ve <- peek e
let result = (vr,ve)
free r
free e
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 prec f a b = unsafePerformIO \$ do
r <- malloc
e <- malloc
fp <- mkfun (\x _ -> f x)
c_integrate_qng fp a b eps prec r e  // check "integrate_qng"
vr <- peek r
ve <- peek e
let result = (vr,ve)
free r
free e
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 prec n f = unsafePerformIO \$ do
r <- malloc
e <- malloc
fp <- mkfun (\x _ -> f x)
c_integrate_qagi fp eps prec (fromIntegral n) r e // check "integrate_qagi"
vr <- peek r
ve <- peek e
let result = (vr,ve)
free r
free e
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 prec n f a = unsafePerformIO \$ do
r <- malloc
e <- malloc
fp <- mkfun (\x _ -> f x)
c_integrate_qagiu fp a eps prec (fromIntegral n) r e // check "integrate_qagiu"
vr <- peek r
ve <- peek e
let result = (vr,ve)
free r
free e
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 prec n f b = unsafePerformIO \$ do
r <- malloc
e <- malloc
fp <- mkfun (\x _ -> f x)
c_integrate_qagil fp b eps prec (fromIntegral n) r e // check "integrate_qagil"
vr <- peek r
ve <- peek e
let result = (vr,ve)
free r
free e
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:

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 prec n f a b = unsafePerformIO \$ do
r <- malloc
e <- malloc
neval <- malloc
fp <- mkfun (\x _ -> f 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 = (vr,ve,vneval)
free r
free e
free neval