```{-# LANGUAGE ScopedTypeVariables #-}

--------------------------------------------------------------------------------
-- |
-- Module      :  LevMar.Fitting
-- Copyright   :  (c) 2009 Roel van Dijk & Bas van Dijk
--
-- Maintainer  :  vandijk.roel@gmail.com, v.dijk.bas@gmail.com
-- Stability   :  Experimental
--
-- This module provides the Levenberg-Marquardt algorithm specialised
-- for curve-fitting.
--
-- For additional documentation see the documentation of the levmar C
-- library which this library is based on:
-- <http://www.ics.forth.gr/~lourakis/levmar/>
--
--------------------------------------------------------------------------------

module LevMar.Fitting
( -- * Model & Jacobian.
Model
, SimpleModel
, Jacobian
, SimpleJacobian

-- * Levenberg-Marquardt algorithm.
, LMA_I.LevMarable
, levmar

, LinearConstraints
, noLinearConstraints
, Matrix

-- * Minimization options.
, LMA_I.Options(..)
, LMA_I.defaultOpts

-- * Output
, LMA_I.Info(..)
, LMA_I.StopReason(..)
, CovarMatrix

, LMA_I.LevMarError(..)

-- *Type-level machinery
, Z, S, Nat
, SizedList(..)
, NFunction
) where

import qualified LevMar.Intermediate.Fitting as LMA_I
import LevMar.Utils ( LinearConstraints
, noLinearConstraints
, convertLinearConstraints
, Matrix
, CovarMatrix
, convertResult
)

import TypeLevelNat ( Z, S, Nat )
import SizedList    ( SizedList(..), toList, unsafeFromList )
import NFunction    ( NFunction, (\$*) )

--------------------------------------------------------------------------------
-- Model & Jacobian.
--------------------------------------------------------------------------------

{- | A functional relation describing measurements represented as a function
from @m@ parameters and an x-value to an expected measurement.

For example, the quadratic function @f(x) = a*x^2 + b*x + c@ can be
written as:

@
type N3 = 'S' ('S' ('S' 'Z'))

quad :: 'Num' r => 'Model' N3 r r
quad a b c x = a*x^2 + b*x + c
@
-}
type Model m r a = NFunction m r (a -> r)

-- | This type synonym expresses that usually the @a@ in @'Model' m r a@
-- equals the type of the parameters.
type SimpleModel m r = Model m r r

{- | The jacobian of the 'Model' function. Expressed as a function from @n@
parameters and an x-value to the @m@ partial derivatives of the parameters.

See: <http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant>

For example, the jacobian of the quadratic function @f(x) = a*x^2 +
b*x + c@ can be written as:

@
type N3 = 'S' ('S' ('S' 'Z'))

quadJacob :: 'Num' r => 'Jacobian' N3 r r
quadJacob _ _ _ x =   x^2   -- with respect to a
::: x     -- with respect to b
::: 1     -- with respect to c
::: 'Nil'
@

Notice you don't have to differentiate for @x@.
-}
type Jacobian m r a = NFunction m r (a -> SizedList m r)

-- | This type synonym expresses that usually the @a@ in @'Jacobian' m r a@
-- equals the type of the parameters.
type SimpleJacobian m r = Jacobian m r r

--------------------------------------------------------------------------------
-- Levenberg-Marquardt algorithm.
--------------------------------------------------------------------------------

-- | The Levenberg-Marquardt algorithm specialised for curve-fitting.
levmar :: forall m k r a. (Nat m, Nat k, LMA_I.LevMarable r)
=> (Model m r a)                          -- ^ Model
-> Maybe (Jacobian m r a)                 -- ^ Optional jacobian
-> SizedList m r                          -- ^ Initial parameters
-> [(a, r)]                               -- ^ Samples
-> Integer                                -- ^ Maximum number of iterations
-> LMA_I.Options r                        -- ^ Minimization options
-> Maybe (SizedList m r)                  -- ^ Optional lower bounds
-> Maybe (SizedList m r)                  -- ^ Optional upper bounds
-> Maybe (LinearConstraints k m r)        -- ^ Optional linear constraints
-> Maybe (SizedList m r)                  -- ^ Optional weights
-> Either LMA_I.LevMarError (SizedList m r, LMA_I.Info r, CovarMatrix m r)
levmar model mJac params ys itMax opts mLowBs mUpBs mLinC mWghts =
fmap convertResult \$ LMA_I.levmar (convertModel model)
(fmap convertJacob mJac)
(toList params)
ys
itMax
opts
(fmap toList mLowBs)
(fmap toList mUpBs)
(fmap convertLinearConstraints mLinC)
(fmap toList mWghts)
where
convertModel mdl = \ps   ->          mdl \$* (unsafeFromList ps :: SizedList m r)
convertJacob jac = \ps x -> toList ((jac \$* (unsafeFromList ps :: SizedList m r)) x :: SizedList m r)

-- The End ---------------------------------------------------------------------
```