diff --git a/CHANGELOG.md b/CHANGELOG.md
new file mode 100644
--- /dev/null
+++ b/CHANGELOG.md
@@ -0,0 +1,3 @@
+1.0.0.0
+-------
+* initial release
diff --git a/LICENSE b/LICENSE
new file mode 100644
--- /dev/null
+++ b/LICENSE
@@ -0,0 +1,674 @@
+ GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
+
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+ 17. Interpretation of Sections 15 and 16.
+
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+above cannot be given local legal effect according to their terms,
+reviewing courts shall apply local law that most closely approximates
+an absolute waiver of all civil liability in connection with the
+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+
+ Copyright (C)
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see .
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ Copyright (C)
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+.
diff --git a/README.md b/README.md
new file mode 100644
--- /dev/null
+++ b/README.md
@@ -0,0 +1,101 @@
+# scubature
+
+Pure Haskell implementation of simplicial cubature (integration over a simplex).
+
+```haskell
+integrateOnSimplex
+ :: (VectorD -> VectorD) -- integrand
+ -> Simplices -- domain of integration (union of the simplices)
+ -> Int -- number of components of the integrand
+ -> Int -- maximum number of evaluations
+ -> Double -- desired absolute error
+ -> Double -- desired relative error
+ -> Int -- integration rule: 1, 2, 3 or 4
+ -> IO Results -- values, error estimates, evaluations, success
+```
+
+## Example
+
+
+
+Define the integrand:
+
+```haskell
+import Data.Vector.Unboxed as V
+:{
+f :: Vector Double -> Vector Double
+f v = singleton $ exp (V.sum v)
+:}
+```
+
+Define the simplex:
+
+```haskell
+simplex = [[0, 0, 0], [1, 1, 1], [0, 1, 1], [0, 0, 1]]
+```
+
+Integrate:
+
+```haskell
+import Numeric.Integration.SimplexCubature
+integrateOnSimplex f [simplex] 1 100000 0 1e-10 3
+-- Results { values = [0.8455356852954488]
+-- , errorEstimates = [8.082378899762402e-11]
+-- , evaluations = 8700
+-- , success = True }
+```
+
+For a scalar-valued integrand, it's more convenient to define... a scalar-valued
+integrand! That is:
+
+```haskell
+:{
+f :: Vector Double -> Double
+f v = exp (V.sum v)
+:}
+```
+
+and then to use `integrateOnSimplex'`:
+
+```haskell
+integrateOnSimplex' f [simplex] 100000 0 1e-10 3
+-- Result { value = 0.8455356852954488
+-- , errorEstimate = 8.082378899762402e-11
+-- , evaluations = 8700
+-- , success = True }
+```
+
+## Integration on a spherical triangle
+
+The library also allows to evaluate an integral on a spherical simplex on the
+unit sphere (in dimension 3, a spherical triangle).
+
+For example take the first orthant in dimension 3:
+
+```haskell
+import Numeric.Integration.SphericalSimplexCubature
+o1 = orthants 3 !! 0
+o1
+-- [ [1.0, 0.0, 0.0]
+-- , [0.0, 1.0, 0.0]
+-- , [0.0, 0.0, 1.0] ]
+```
+
+And this integrand:
+
+```haskell
+:{
+integrand :: [Double] -> Double
+integrand x = (x!!0 * x!!0 * x!!2) + (x!!1 * x!!1 * x!!2) + (x!!2 * x!!2 * x!!2)
+:}
+```
+
+Compute the integral (the exact result is `pi/4 ≈ 0.7853981634`):
+
+```haskell
+integrateOnSphericalSimplex integrand o1 20000 0 1e-7 3
+-- Result { value = 0.7853981641913279
+-- , errorEstimate = 7.71579524444753e-8
+-- , evaluations = 17065
+-- , success = True }
+```
diff --git a/Setup.hs b/Setup.hs
new file mode 100644
--- /dev/null
+++ b/Setup.hs
@@ -0,0 +1,2 @@
+import Distribution.Simple
+main = defaultMain
diff --git a/scubature.cabal b/scubature.cabal
new file mode 100644
--- /dev/null
+++ b/scubature.cabal
@@ -0,0 +1,37 @@
+name: scubature
+version: 1.0.0.0
+synopsis: Multidimensional integration over simplices
+description: This library allows to evaluate integrals over Euclidean and spherical simplices.
+homepage: https://github.com/stla/scubature#readme
+license: GPL-3
+license-file: LICENSE
+author: Stéphane Laurent
+maintainer: laurent_step@outlook.fr
+copyright: 2022 Stéphane Laurent
+category: Numeric, Integration
+build-type: Simple
+extra-source-files: README.md
+ CHANGELOG.md
+cabal-version: >=1.10
+
+library
+ hs-source-dirs: src
+ exposed-modules: Numeric.Integration.SimplexCubature
+ , Numeric.Integration.SphericalSimplexCubature
+ other-modules: Numeric.Integration.SimplexCubature.Internal
+ , Numeric.Integration.Simplex.Simplex
+ , Numeric.Integration.SphericalSimplexCubature.Internal
+ other-extensions: BangPatterns
+ , DuplicateRecordFields
+ build-depends: base >= 4.7 && < 5
+ , array >= 0.5.4.0 && < 0.6
+ , vector >= 0.12.3.1 && < 0.13
+ , matrix >= 0.3.6.1 && < 0.4
+ , containers >= 0.6.4.1 && < 0.7
+ , ilist >= 0.4.0.1 && < 0.5
+ default-language: Haskell2010
+ ghc-options: -Wall
+
+source-repository head
+ type: git
+ location: https://github.com/stla/scubature
diff --git a/src/Numeric/Integration/Simplex/Simplex.hs b/src/Numeric/Integration/Simplex/Simplex.hs
new file mode 100644
--- /dev/null
+++ b/src/Numeric/Integration/Simplex/Simplex.hs
@@ -0,0 +1,35 @@
+module Numeric.Integration.Simplex.Simplex
+ where
+import Data.Matrix (detLU, elementwiseUnsafe, fromLists)
+
+type Simplex = [[Double]]
+type Simplices = [Simplex]
+
+isValidSimplex :: Simplex -> Bool
+isValidSimplex simplex =
+ (length simplex == dim + 1) &&
+ all ((== dim) . length) (tail simplex)
+ where dim = length (head simplex)
+
+isValidSimplices :: Simplices -> Bool
+isValidSimplices simplices =
+ all isValidSimplex simplices &&
+ all ((== spaceDim (head simplices)) . spaceDim) (tail simplices)
+ where spaceDim simplex = length (head simplex)
+
+canonicalSimplex :: Int -> Simplex
+canonicalSimplex dim =
+ replicate dim 0 :
+ map (\v -> map (fromIntegral.fromEnum.(== v)) [1..dim]) [1..dim]
+
+simplexVolume :: Simplex -> Double -- rq: tu calcules le fact à chaque fois
+simplexVolume s = abs (detLU v) / fromIntegral (product [1..n])
+ where n = length s - 1
+ m1 = fromLists (tail s)
+ m2 = fromLists $ replicate n (head s)
+ v = elementwiseUnsafe (-) m1 m2
+
+jacobian :: Simplex -> Double -- not used
+jacobian s = abs (detLU (elementwiseUnsafe (-) m1 m2))
+ where m1 = fromLists (tail s)
+ m2 = fromLists $ replicate (length s - 1) (head s)
diff --git a/src/Numeric/Integration/SimplexCubature.hs b/src/Numeric/Integration/SimplexCubature.hs
new file mode 100644
--- /dev/null
+++ b/src/Numeric/Integration/SimplexCubature.hs
@@ -0,0 +1,72 @@
+{-# LANGUAGE DuplicateRecordFields #-}
+module Numeric.Integration.SimplexCubature
+ (Result(..), Results(..), integrateOnSimplex, integrateOnSimplex')
+ where
+import Data.Array.Unboxed (UArray, array)
+import Data.Array.Unsafe (unsafeThaw)
+import qualified Data.Vector.Unboxed as UV
+import Numeric.Integration.Simplex.Simplex ( isValidSimplices, Simplices )
+import Numeric.Integration.SimplexCubature.Internal ( VectorD, IO3dArray, adsimp )
+
+data Results = Results
+ { values :: [Double]
+ , errorEstimates :: [Double]
+ , evaluations :: Int
+ , success :: Bool
+ } deriving Show
+
+data Result = Result
+ { value :: Double
+ , errorEstimate :: Double
+ , evaluations :: Int
+ , success :: Bool
+ } deriving Show
+
+simplicesToArray :: Simplices -> IO IO3dArray
+simplicesToArray simplices = do
+ let dim = length (head (head simplices))
+ nsimplices = length simplices
+ assocList = map (\[i, j, k] -> ((i, j, k), (simplices!!(k-1))!!(j-1)!!(i-1)))
+ (sequence [[1 .. dim], [1 .. (dim+1)], [1 .. nsimplices]])
+ arr = array ((1, 1, 1), (dim, dim+1, nsimplices)) assocList
+ :: UArray (Int, Int, Int) Double
+ unsafeThaw arr
+
+-- | Integral of a vector-valued function over an union of simplices.
+integrateOnSimplex
+ :: (VectorD -> VectorD) -- ^ integrand
+ -> Simplices -- ^ domain of integration
+ -> Int -- ^ number of components
+ -> Int -- ^ maximum number of evaluations
+ -> Double -- ^ desired absolute error
+ -> Double -- ^ desired relative error
+ -> Int -- ^ integration rule: 1, 2, 3 or 4
+ -> IO Results -- ^ integral, error, evaluations, success
+integrateOnSimplex f s ncomp maxevals absError relError rule = do
+ let n = length (head s) - 1
+ if isValidSimplices s
+ then do
+ v <- simplicesToArray s
+ (vals, errors, nevals, fl) <-
+ adsimp n ncomp maxevals f absError relError rule v
+ return $ Results (UV.toList vals) (UV.toList errors) nevals (not fl)
+ else error "invalid simplices"
+
+-- | Integral of a real-valued function over an union of simplices.
+integrateOnSimplex'
+ :: (VectorD -> Double) -- ^ integrand
+ -> Simplices -- ^ domain of integration
+ -> Int -- ^ maximum number of evaluations
+ -> Double -- ^ desired absolute error
+ -> Double -- ^ desired relative error
+ -> Int -- ^ integration rule: 1, 2, 3 or 4
+ -> IO Result -- ^ integral, error, evaluations, success
+integrateOnSimplex' f s maxevals absError relError rule = do
+ let n = length (head s) - 1
+ if isValidSimplices s
+ then do
+ v <- simplicesToArray s
+ (val, err, nevals, fl) <-
+ adsimp n 1 maxevals (UV.singleton . f) absError relError rule v
+ return $ Result (UV.head val) (UV.head err) nevals (not fl)
+ else error "invalid simplices"
diff --git a/src/Numeric/Integration/SimplexCubature/Internal.hs b/src/Numeric/Integration/SimplexCubature/Internal.hs
new file mode 100644
--- /dev/null
+++ b/src/Numeric/Integration/SimplexCubature/Internal.hs
@@ -0,0 +1,765 @@
+{-# LANGUAGE BangPatterns #-}
+module Numeric.Integration.SimplexCubature.Internal
+ where
+import Data.Array.IO (IOUArray, getBounds, getElems
+ , mapIndices, newArray, newArray_
+ , newListArray, readArray
+ , writeArray)
+import Data.List (foldl', foldl1', sort)
+import Control.Monad ((<$!>), when)
+import qualified Data.Foldable as DF
+import Data.Maybe (fromJust)
+import Data.Sequence (Seq, index, update, (><))
+import qualified Data.Sequence as S
+import Data.Vector.Unboxed (Vector, fromList, toList
+ , unsafeFreeze)
+import qualified Data.Vector.Unboxed as UV
+import Data.Vector.Unboxed.Mutable (IOVector, new, unsafeRead
+ , unsafeWrite)
+import qualified Data.Vector.Unboxed.Mutable as UMV
+import Numeric.Integration.Simplex.Simplex (Simplex, simplexVolume)
+
+type IOMatrix = IOUArray (Int, Int) Double
+type IO1dArray = IOUArray Int Double
+type IO3dArray = IOUArray (Int, Int, Int) Double
+type IOVectorD = IOVector Double
+type IOVectorI = IOVector Int
+type VectorD = Vector Double
+
+toDbl :: Int -> Double
+toDbl = fromIntegral
+
+pow :: Double -> Int -> Double
+pow x n = product (replicate n x)
+
+permuteMultiset :: (Eq a, Ord a) => [a] -> [[a]]
+permuteMultiset list = unfold1 next (sort list) where
+ unfold1 :: (a -> Maybe a) -> a -> [a]
+ unfold1 f x = case f x of
+ Nothing -> [x]
+ Just y -> x : unfold1 f y
+ -- next :: [a] -> Maybe [a]
+ next xs = case findj (reverse xs, []) of
+ Nothing -> Nothing
+ Just ( l:ls, rs) -> Just $ inc l ls (reverse rs, [])
+ Just ( [], _ ) -> error "permute: should not happen"
+ -- we use simple list zippers: (left,right)
+ -- findj :: ([a],[a]) -> Maybe ([a],[a])
+ findj ( xxs@(x:xs), yys@(y:_) ) = if x >= y
+ then findj ( xs, x : yys )
+ else Just ( xxs, yys )
+ findj ( x:xs , [] ) = findj ( xs , [x] )
+ findj ( [] , _ ) = Nothing
+ -- inc :: a -> [a] -> ([a],[a]) -> [a]
+ inc !u us ( x:xs, yys ) = if u >= x
+ then inc u us ( xs, x : yys )
+ else reverse (x:us) ++ reverse (u:yys) ++ xs
+ inc _ _ ( [], _ ) = error "permute: should not happen"
+
+square :: Double -> Double
+square x = x*x
+
+smprms :: Int -> Int -> IO (IOMatrix, Seq VectorD, [Int])
+smprms n key = do
+ let (rls, gms, wts) | key == 1 = (3, 2, 3) :: (Int, Int, Int)
+ | key == 2 = (5, 4, 6) :: (Int, Int, Int)
+ | key == 3 = (7, 7, 11) :: (Int, Int, Int)
+ | key == 4 = (if n == 2 then (7, 11, 20) else (7, 12, 21))
+ :: (Int, Int, Int)
+ | otherwise = error "this should not happen"
+ w <- newArray ((1, 1), (wts, rls)) 0 :: IO IOMatrix
+ pts <- UMV.replicate wts 0 :: IO IOVectorI
+ g <- newArray ((1, 1), (n+1, wts)) 0 :: IO IOMatrix
+ let np = n+1
+ n2 = np * (n+2)
+ n4 = n2 * (n+3) * (n+4)
+ n6 = n4 * (n+5) * (n+6)
+ n8 = n6 * (n+7) * (n+8)
+ o = div (n*np) 2
+ ndbl = toDbl n
+ sqrt15 = 3.872983346207416885179265399782399611
+ r1 = (ndbl + 4 - sqrt15) / (ndbl*ndbl + 8*ndbl + 1)
+ s1 = 1 - ndbl*r1
+ l1 = s1 - r1
+ mapM_ (\j -> writeArray g (j, 1) (1/(ndbl+1))) [1 .. np]
+ unsafeWrite pts 0 1
+ writeArray g (1, gms+1) s1
+ mapM_ (\j -> writeArray g (j, gms+1) r1) [2 .. np]
+ unsafeWrite pts gms np
+ when (key < 4) $ do
+ writeArray w (1, rls) 1
+ writeArray w (gms+1, rls-1) (1/(ndbl+1))
+ let iw = if key < 4 then rls-2 else rls
+ writeArray g (1, 2) (3/(ndbl+3))
+ mapM_ (\j -> writeArray g (j, 2) (1/(ndbl+3))) [2 .. np]
+ unsafeWrite pts 1 np
+ let n2double = toDbl n2
+ writeArray w (2, iw) (pow (ndbl+3) 2 / (4*n2double))
+ when (key > 1) $ do
+ if n == 2
+ then do
+ let l2 = 0.62054648267200632589046034361711
+ r1' = (1+sqrt(0.5 - l2*l2))/3
+ writeArray g (1, gms+1) (1 - 2*r1')
+ mapM_ (\j -> writeArray g (j, gms+1) r1') [2 .. np]
+ unsafeWrite pts gms 3
+ writeArray w (gms+1, iw-1) (1/6)
+ let r2 = (1-l2)/3
+ s2 = 1 - 2*r2
+ writeArray g (1, gms+2) s2
+ mapM_ (\j -> writeArray g (j, gms+2) r2) [2 .. np]
+ unsafeWrite pts (gms+1) 3
+ writeArray w (gms+2, iw-1) (1/6)
+ else do
+ let r2 = (ndbl+4+sqrt15) / (ndbl*ndbl+8*ndbl+1)
+ s2 = 1 - ndbl*r2
+ l2 = s2 - r2
+ writeArray g (1, gms+2) s2
+ mapM_ (\j -> writeArray g (j, gms+2) r2) [2 .. np]
+ unsafeWrite pts (gms+1) np
+ writeArray w (gms+2, iw-1) ((2/(ndbl+3)-l1)/(n2double*(l2-l1)*l2*l2))
+ writeArray w (gms+1, iw-1) ((2/(ndbl+3)-l2)/(n2double*(l1-l2)*l1*l1))
+ writeArray g (1, 3) (5/(ndbl+5))
+ mapM_ (\j -> writeArray g (j, 3) (1/(ndbl+5))) [2 .. np]
+ unsafeWrite pts 2 np
+ writeArray g (1, 4) (3/(ndbl+5))
+ writeArray g (2, 4) (3/(ndbl+5))
+ mapM_ (\j -> writeArray g (j, 4) (1/(ndbl+5))) [3 .. np]
+ unsafeWrite pts 3 o
+ let tmp = toDbl (16*n4)
+ writeArray w (2, iw-2) (- pow (ndbl+3) 5 / tmp)
+ writeArray w (3, iw-2) (pow (ndbl+5) 4 / tmp)
+ writeArray w (4, iw-2) (pow (ndbl+5) 4 / tmp)
+ when (key > 2) $ do
+ let tmp' = ndbl*ndbl + 14*ndbl - 11
+ u1 = (ndbl+7+2*sqrt15) / tmp'
+ v1 = 0.5*(1-(ndbl-1)*u1)
+ d1 = v1 - u1
+ writeArray g (1, gms+3) v1
+ writeArray g (2, gms+3) v1
+ mapM_ (\j -> writeArray g (j, gms+3) u1) [3 .. np]
+ unsafeWrite pts (gms+2) o
+ let u2 = (ndbl + 7 - 2*sqrt15) / tmp'
+ v2 = (1-(ndbl-1)*u2)/2
+ d2 = v2 - u2
+ writeArray g (1, gms+4) v2
+ writeArray g (2, gms+4) v2
+ mapM_ (\j -> writeArray g (j, gms+4) u2) [3 .. np]
+ unsafeWrite pts (gms+3) o
+ if n == 2
+ then do
+ writeArray w (gms+3, iw-3) ((155-sqrt15)/1200)
+ writeArray w (gms+4, iw-3) ((155+sqrt15)/1200)
+ writeArray w (1, iw-3) (9/40)
+ else
+ if n == 3
+ then do
+ writeArray w (gms+1, iw-3) ((2665+14*sqrt15)/37800)
+ writeArray w (gms+2, iw-3) ((2665-14*sqrt15)/37800)
+ writeArray w (gms+3, iw-3) (10/189)
+ unsafeWrite pts (gms+3) 0
+ else do
+ let r2 = (ndbl+4+sqrt15) / (ndbl*ndbl+8*ndbl+1)
+ l2 = 1 - (ndbl+1)*r2
+ n4dbl = toDbl n4
+ writeArray w (gms+1, iw-3)
+ ((2*(27-ndbl)/(ndbl+5)-l2*(13-ndbl)) /
+ (pow l1 4 * (l1-l2) * n4dbl))
+ writeArray w (gms+2, iw-3)
+ ((2*(27-ndbl)/(ndbl+5)-l1*(13-ndbl)) /
+ (pow l2 4 *(l2-l1)*n4dbl))
+ writeArray w (gms+3, iw-3)
+ ((2/(ndbl+5)-d2) / (n4dbl * (d1-d2) * pow d1 4))
+ writeArray w (gms+4, iw-3)
+ ((2/(ndbl+5)-d1) / (n4dbl * (d2-d1) * pow d2 4))
+ writeArray g (1, 5) (7/(ndbl+7))
+ mapM_ (\i -> writeArray g (i, 5) (1/(ndbl+7))) [2 .. np]
+ unsafeWrite pts 4 np
+ let invnp7 = 1/(ndbl+7)
+ writeArray g (1, 6) (5*invnp7)
+ writeArray g (2, 6) (3*invnp7)
+ mapM_ (\i -> writeArray g (i, 6) invnp7) [3 .. np]
+ unsafeWrite pts 5 (np*n)
+ mapM_ (\i -> writeArray g (i, 7) (3*invnp7)) [1, 2, 3]
+ when (np > 3) $
+ mapM_ (\i -> writeArray g (i, 7) invnp7) [4..np]
+ unsafeWrite pts 6 (div ((n-1)*n*np) 6)
+ writeArray w (2, iw-4) (pow (ndbl+3) 7 / toDbl (128*n4*(n+5)))
+ let tmp'' = toDbl (64*n6)
+ mapM_ (\i -> writeArray w (i, iw-4) (-pow (ndbl+5) 7 / tmp'')) [3, 4]
+ mapM_ (\i -> writeArray w (i, iw-4) (pow (ndbl+7) 6 / tmp'')) [5, 6, 7]
+ when (key == 4) $ do
+ let sg = 1 / toDbl (23328*n6)
+ u5 = -216 * sg * toDbl (52212 - n*(6353 + n*(1934-n*27)))
+ u6 = 1296 * sg * toDbl (7884 - n*(1541 - n*9))
+ u7 = -7776 * sg * toDbl (8292 - n*(1139 - n*3))/(ndbl + 7)
+ p0 = toDbl $ -144 * (142528 + n*(23073 - n*115))
+ p1 = toDbl $ -12 * (6690556 + n*(2641189 + n*(245378 - n*1495)))
+ p2 = toDbl $ -16 * (6503401 + n*(4020794+n*(787281+n*(47323-n*385))))
+ p3 = toDbl $ -(6386660 + n*(4411997+n*(951821+n*(61659-n*665))))*(n+7)
+ a = p2/(3*p3)
+ p = a*(p1/p2 - a)
+ q = a*(2*a*a - p1/p3) + p0/p3
+ th = acos(-q/(2 * sqrt(-p*p*p))) / 3
+ r = 2*sqrt(-p)
+ tp = 2*pi/3
+ a1 = -a + r* cos th
+ a3 = -a + r* cos(th+tp)
+ a2 = -3*a - a1 - a3 -- a3 = -a + r*(cos(th+tp))
+ npdbl = toDbl np
+ writeArray g (1, gms+5) ((1-ndbl*a1)/npdbl)
+ mapM_ (\i -> writeArray g (i, gms+5) ((1+a1)/npdbl)) [2..np]
+ unsafeWrite pts (gms+4) np
+ writeArray g (1, gms+6) ((1-ndbl*a2)/npdbl)
+ mapM_ (\i -> writeArray g (i, gms+6) ((1+a2)/npdbl)) [2..np]
+ unsafeWrite pts (gms+5) np
+ writeArray g (1, gms+7) ((1-ndbl*a3)/npdbl)
+ mapM_ (\i -> writeArray g (i, gms+7) ((1+a3)/npdbl)) [2..np]
+ unsafeWrite pts (gms+6) np
+ writeArray w (gms+5, iw-5)
+ ((u7-(a2+a3)*u6+a2*a3*u5)/(a1*a1-(a2+a3)*a1+a2*a3) / pow a1 5)
+ writeArray w (gms+6, iw-5)
+ ((u7-(a1+a3)*u6+a1*a3*u5)/(a2*a2-(a1+a3)*a2+a1*a3) / pow a2 5)
+ writeArray w (gms+7, iw-5)
+ ((u7-(a2+a1)*u6+a2*a1*u5)/(a3*a3-(a2+a1)*a3+a2*a1) / pow a3 5)
+ let invnp7' = 1/(ndbl+7)
+ writeArray g (1, gms+8) (4*invnp7')
+ writeArray g (2, gms+8) (4*invnp7')
+ mapM_ (\i -> writeArray g (i, gms+8) invnp7') [3..np]
+ unsafeWrite pts (gms+7) o
+ writeArray w (gms+8, iw-5) (10 * pow (ndbl+7) 6 / toDbl (729*n6))
+ writeArray g (1, gms+9) (5.5*invnp7')
+ writeArray g (2, gms+9) (2.5*invnp7')
+ mapM_ (\i -> writeArray g (i, gms+9) invnp7') [3..np]
+ unsafeWrite pts (gms+8) (np*n)
+ writeArray w (gms+9, iw-5) (64 * pow (ndbl+7) 6 / toDbl (6561*n6))
+ let tmp''' = toDbl (64*n6)
+ writeArray w (4, iw-5) (-pow (ndbl+5) 7 / tmp''')
+ writeArray w (7, iw-5) (pow (ndbl+7) 6 / tmp''')
+ let invnp9 = 1/(ndbl+9)
+ writeArray g (1, 8) (9*invnp9)
+ mapM_ (\i -> writeArray g (i, 8) invnp9) [2..np]
+ unsafeWrite pts 7 np
+ writeArray g (1, 9) (7*invnp9)
+ writeArray g (2, 9) (3*invnp9)
+ mapM_ (\i -> writeArray g (i, 9) invnp9) [3..np]
+ unsafeWrite pts 8 (np*n)
+ mapM_ (\i -> writeArray g (i, 10) (5*invnp9)) [1, 2]
+ mapM_ (\i -> writeArray g (i, 10) invnp9) [3..np]
+ unsafeWrite pts 9 o
+ writeArray g (1, 11) (5*invnp9)
+ mapM_ (\i -> writeArray g (i, 11) (3*invnp9)) [2, 3]
+ when (np > 3) $
+ mapM_ (\i -> writeArray g (i, 11) invnp9) [4..np]
+ unsafeWrite pts 10 (o*(n-1))
+ writeArray w (2, iw-6) (-pow (ndbl+3) 9 / toDbl (1536*n6))
+ mapM_ (\i -> writeArray w (i, iw-6)
+ (pow (ndbl+5) 9 / toDbl (512*n6*(n+7)))) [3, 4]
+ let tmp'''' = toDbl $ 256*n8
+ mapM_ (\i -> writeArray w (i, iw-6) (-pow (ndbl+7) 9 / tmp'''')) [5, 6, 7]
+ mapM_ (\i -> writeArray w (i, iw-6) (pow (ndbl+9) 8 / tmp'''')) [8..11]
+ when (n > 2) $ do
+ mapM_ (\i -> writeArray g (i, 12) (3*invnp9)) [1..4]
+ mapM_ (\i -> writeArray g (i, 12) invnp9) [5..np]
+ unsafeWrite pts 11 (div (np*n*(n-1)*(n-2)) 24)
+ writeArray w (12, iw-6) (pow (ndbl+9) 8 / toDbl (256*n8))
+ rowsIO <- mapM (extractRow w) (S.fromList [2..wts])
+ rows <- mapM array1dToVectorD rowsIO
+ let cols = transpose rows
+ pts_out <- UV.unsafeFreeze pts
+ let ptsU = UV.map toDbl pts_out
+ row1 = fmap (\col -> 1 - UV.foldr (+) 0
+ (UV.zipWith (*) (UV.tail ptsU) col))
+ (S.take rls cols)
+ wcols = fmap (\j -> UV.cons (index row1 j) (index cols j))
+ (S.fromList [0..(rls-1)])
+ col1 = index wcols 0
+ wcols2 = (S.<|) col1
+ (fmap (\col -> UV.zipWith (-) col col1) (S.drop 1 wcols))
+ col2 = index wcols2 1
+ nb = UV.foldr (+) 0 (UV.zipWith (*) ptsU (UV.map square col1))
+ ratio = nb / UV.foldr (+) 0 (UV.zipWith (*) ptsU (UV.map square col2))
+ wcol2 = UV.map (* sqrt ratio) col2
+ wcols3 = (S.<|) col1 ((S.<|) wcol2 (S.drop 2 wcols2))
+ let updateW :: Seq VectorD -> Int -> Seq VectorD
+ updateW cols' k = update (k-1) wknew cols'
+ where
+ ptsW = UV.map (/nb) (UV.zipWith (*) ptsU (index cols' (k-1)))
+ slice = S.drop 1 (S.take (k-1) cols')
+ prod1 = (fromList . DF.toList) $ -- ou alors pas de vecteurs
+ fmap (UV.foldr (+) 0 . UV.zipWith (*) ptsW) slice
+ rows' = transpose slice
+ prod2 = (fromList . DF.toList) $
+ fmap (UV.foldr (+) 0 . UV.zipWith (*) prod1) rows'
+ wk = UV.zipWith (-) (index cols' (k-1)) prod2
+ ratio' = nb /
+ UV.foldr (+) 0 (UV.zipWith (*) ptsU (UV.map square wk))
+ wknew = UV.map (* sqrt ratio') wk
+ wcolsnew = foldl' updateW wcols3 [3..rls]
+ return (g, transpose wcolsnew, toList $ UV.findIndices (/= 0) pts_out)
+
+transpose :: Seq VectorD -> Seq VectorD
+transpose cols =
+ fmap (\i -> (fromList . DF.toList) $ fmap (UV.!i) cols)
+ (S.fromList [0..(UV.length (index cols 0) - 1)])
+
+matprod :: IOMatrix -> VectorD -> IO VectorD
+matprod mat x = do
+ (_, (m, n)) <- getBounds mat
+ out <- UMV.new m :: IO IOVectorD
+ let step i | i == m+1 = unsafeFreeze out
+ | otherwise = do
+ !coef <- innerstep 1 0
+ unsafeWrite out (i-1) coef
+ step (i+1)
+ where
+ innerstep :: Int -> Double -> IO Double
+ innerstep j !s | j == n+1 = return s
+ | otherwise = do
+ mat_ij <- readArray mat (i, j)
+ innerstep (j+1) (s + mat_ij * (x UV.! (j-1)))
+ step 1
+
+smpsms :: IOMatrix -> Int -> (VectorD -> VectorD) -> IO1dArray
+ -> Double -> IO IO1dArray
+smpsms vertex nf f g scalar = do
+ gAsList <- getElems g
+ f_gPermuts <- mapM (fmap f . matprod vertex . fromList)
+ (permuteMultiset gAsList)
+ newListArray (1, nf)
+ (toList (UV.map (*scalar) (foldl1' (UV.zipWith (+)) f_gPermuts)))
+
+extractColumn :: IOMatrix -> Int -> IO IO1dArray
+extractColumn m j = do
+ (_, (nrow, _)) <- getBounds m
+ mapIndices (1, nrow) (\i -> (i, j)) m
+
+extractRow :: IOMatrix -> Int -> IO IO1dArray
+extractRow m i = do
+ (_, (_, ncol)) <- getBounds m
+ mapIndices (1, ncol) (\j -> (i, j)) m
+
+outerProduct :: IO1dArray -> VectorD -> IO IOMatrix
+outerProduct x1 x2 = do
+ (_, n1) <- getBounds x1
+ let n2 = UV.length x2
+ out <- newArray_ ((1, 1), (n1, n2)) :: IO IOMatrix
+ let step :: Int -> IO IOMatrix
+ step i | i == n1+1 = return out
+ | otherwise = do
+ x1_i <- readArray x1 i
+ inner x1_i 0
+ where
+ inner !x j | j == n2 = step (i+1)
+ | otherwise = do
+ writeArray out (i, j+1) (x * (x2 UV.! j))
+ inner x (j+1)
+ step 1
+
+sumMatrices :: [IOMatrix] -> IO IOMatrix
+sumMatrices matrices = do
+ (_, (n1, n2)) <- getBounds (head matrices)
+ out <- newArray_ ((1, 1), (n1, n2)) :: IO IOMatrix
+ let step :: Int -> IO IOMatrix
+ step i | i == n1+1 = return out
+ | otherwise = inner 1
+ where
+ inner :: Int -> IO IOMatrix
+ inner j | j == n2+1 = step (i+1)
+ | otherwise = do
+ coefs <- mapM (\m -> readArray m (i, j)) matrices
+ writeArray out (i, j) (sum coefs)
+ inner (j+1)
+ step 1
+
+smprul :: IOMatrix -> Int -> (VectorD -> VectorD) -> Double -> IOMatrix
+ -> Seq VectorD -> [Int] -> IO (IO1dArray, IO1dArray)
+smprul vrts nf f vol g w pospts = do
+ let rtmn = 0.1
+ small = 1e-12
+ errcof = 8
+ rls = UV.length (index w 0)
+ toSum <- mapM (\k -> do
+ g_colk <- extractColumn g (k+1)
+ sms <- smpsms vrts nf f g_colk vol
+ outerProduct sms (index w k))
+ pospts
+ rule <- sumMatrices toSum
+ basval <- extractColumn rule 1
+ rgnerr <- newArray (1, nf) 0 :: IO IO1dArray
+ let step :: Int -> IO ()
+ step i | i == nf+1 = return ()
+ | otherwise = do
+ basval_i <- readArray basval i
+ let nmbs = abs basval_i
+ (rt, nmcp) <- inner rls rtmn 0 nmbs
+ when (rt < 1 && rls > 3) $
+ writeArray rgnerr i (rt*nmcp)
+ rgnerr_i <- readArray rgnerr i
+ writeArray rgnerr i (max (errcof*rgnerr_i) (small*nmbs))
+ step (i+1)
+ where
+ inner :: Int -> Double -> Double -> Double -> IO (Double, Double)
+ inner k !x !y !z | k == 1 = return (x, y)
+ | otherwise = do
+ rule_ik <- readArray rule (i, k)
+ rule_ikm1 <- readArray rule (i, k-1)
+ let nmrl = max (abs rule_ik) (abs rule_ikm1)
+ rgnerr_i <- readArray rgnerr i
+ writeArray rgnerr i (max nmrl rgnerr_i)
+ if nmrl > small*z && k < rls
+ then inner (k-2) (max (nmrl/y) x) nmrl z
+ else inner (k-2) x nmrl z
+ step 1
+ return (basval, rgnerr)
+
+rowMeans :: IOMatrix -> IO VectorD
+rowMeans m = do
+ (_, (nrow, ncol)) <- getBounds m
+ outIO <- new nrow :: IO IOVectorD
+ let step :: Int -> IO ()
+ step i | i == nrow+1 = return ()
+ | otherwise = do
+ !sum_i <- inner 1 0
+ unsafeWrite outIO (i-1) sum_i
+ step (i+1)
+ where
+ inner :: Int -> Double -> IO Double
+ inner j !x | j == ncol+1 = return (x / toDbl ncol)
+ | otherwise = do
+ coef <- readArray m (i, j)
+ inner (j+1) (x + coef)
+ step 1
+ unsafeFreeze outIO
+
+array1dToVectorD :: IO1dArray -> IO VectorD
+array1dToVectorD array = (<$!>) fromList (getElems array)
+
+getVectors :: Int -> IO3dArray -> Int -> Int -> Int -> IO (VectorD, VectorD) -- pas de gain
+getVectors n m k j1 j2 = do
+ out1U <- (=<<) array1dToVectorD (mapIndices (1, n) (\i -> (i, j1, k)) m)
+ out2U <- (=<<) array1dToVectorD (mapIndices (1, n) (\i -> (i, j2, k)) m)
+ -- out1 <- new n :: IO IOVectorD
+ -- out2 <- new n :: IO IOVectorD
+ -- let loop :: Int -> IO ()
+ -- loop i | i == n+1 = return ()
+ -- | otherwise = do
+ -- coef1 <- readArray m (i, j1, k)
+ -- unsafeWrite out1 (i-1) coef1
+ -- coef2 <- readArray m (i, j2, k)
+ -- unsafeWrite out2 (i-1) coef2
+ -- loop (i+1)
+ -- loop 1
+ -- out1U <- UV.unsafeFreeze out1
+ -- out2U <- UV.unsafeFreeze out2
+ return (out1U, out2U)
+
+
+smpdfs :: Int -> (VectorD -> VectorD) -> Int -> Int
+ -> IO3dArray -> IO (Int, IO3dArray)
+smpdfs nd f top sbs vrts = do
+ let cuttf = 2.0
+ cuttb = 8.0
+ v <- mapIndices ((1, 1), (nd, nd+1)) (\(i, j) -> (i, j, top)) vrts
+ cn <- rowMeans v
+ let fc = f cn
+ dfmd = UV.foldr (+) 0 (UV.map abs fc)
+ frthdf <- newArray ((1, 1), (nd, nd+1)) 0 :: IO IOMatrix
+ iejeitjtisjsls <- new 7 :: IO IOVectorI
+ unsafeWrite iejeitjtisjsls 4 1
+ unsafeWrite iejeitjtisjsls 5 2
+ dfmxdfnx <- UMV.replicate 2 0 :: IO IOVectorD
+ let step :: Int -> Double -> IO ()
+ step i x | i == nd+1 = return ()
+ | otherwise = do
+ !emx <- inner (i+1) x
+ step (i+1) emx
+ where
+ inner :: Int -> Double -> IO Double
+ inner j !y | j == nd+2 = return y
+ | otherwise = do
+ vi <- (=<<) array1dToVectorD (extractColumn v i)
+ vj <- (=<<) array1dToVectorD (extractColumn v j)
+ let h = UV.map (*(2/(5*(toDbl nd +1))))
+ (UV.zipWith (-) vi vj)
+ ewd = UV.foldr (+) 0 (UV.map abs h)
+ twoh = UV.map (*2) h
+ t1 = f (UV.zipWith (-) cn twoh)
+ t2 = f (UV.zipWith (+) cn twoh)
+ t3 = UV.map (*6) fc
+ t4 = f (UV.zipWith (-) cn h)
+ t5 = f (UV.zipWith (+) cn h)
+ t6 = UV.zipWith (((*(-4)).).(+)) t4 t5
+ tsum = foldl1' (UV.zipWith (+)) [t1, t2, t3, t6]
+ dfr1 = UV.foldr (+) 0 (UV.map abs tsum)
+ dfr2 = if dfmd+dfr1/8 == dfmd then 0 else dfr1
+ dfr3 = dfr2*ewd
+ dfmx <- unsafeRead dfmxdfnx 0
+ if dfr3 >= dfmx
+ then do
+ is <- unsafeRead iejeitjtisjsls 4
+ js <- unsafeRead iejeitjtisjsls 5
+ unsafeWrite iejeitjtisjsls 2 is
+ unsafeWrite iejeitjtisjsls 3 js
+ unsafeWrite iejeitjtisjsls 4 i
+ unsafeWrite iejeitjtisjsls 5 j
+ unsafeWrite dfmxdfnx 1 dfmx
+ unsafeWrite dfmxdfnx 0 dfr3
+ else do
+ dfnx <- unsafeRead dfmxdfnx 1
+ when (dfr3 >= dfnx) $ do
+ unsafeWrite iejeitjtisjsls 2 i
+ unsafeWrite iejeitjtisjsls 3 j
+ unsafeWrite dfmxdfnx 1 dfr3
+ writeArray frthdf (i, j) dfr3
+ if ewd >= y
+ then do
+ unsafeWrite iejeitjtisjsls 0 i
+ unsafeWrite iejeitjtisjsls 1 j
+ inner (j+1) ewd
+ else inner (j+1) y
+ step 1 0
+ dfmx <- unsafeRead dfmxdfnx 0
+ dfnx <- unsafeRead dfmxdfnx 1
+ let nregions = if dfnx > dfmx/cuttf then 4 else 3
+ if dfnx > dfmx/cuttf
+ then return ()
+ else
+ if dfmx == 0
+ then do
+ ie <- unsafeRead iejeitjtisjsls 0
+ je <- unsafeRead iejeitjtisjsls 1
+ unsafeWrite iejeitjtisjsls 4 ie
+ unsafeWrite iejeitjtisjsls 5 je
+ else do
+ let loop :: Int -> Double -> Int -> IO Int
+ loop l !x !ls | l == nd+2 = return ls
+ | otherwise = do
+ is <- unsafeRead iejeitjtisjsls 4
+ js <- unsafeRead iejeitjtisjsls 5
+ if (l /= is) && (l /= js)
+ then do
+ let it = minimum [l, is, js]
+ jt = maximum [l, is, js]
+ unsafeWrite iejeitjtisjsls 2 it
+ unsafeWrite iejeitjtisjsls 3 jt
+ let lt = is+js+l-it-jt
+ dfr1 <- readArray frthdf (it, lt)
+ dfr2 <- readArray frthdf (lt, jt)
+ let dfr = dfr1 + dfr2
+ if dfr >= x
+ then loop (l+1) dfr l
+ else loop (l+1) x ls
+ else loop (l+1) x ls
+ !ls <- loop 1 0 0
+ unsafeWrite iejeitjtisjsls 6 ls
+ is <- unsafeRead iejeitjtisjsls 4
+ js <- unsafeRead iejeitjtisjsls 5
+ difil <- readArray frthdf (min is ls, max is ls)
+ diflj <- readArray frthdf (min js ls, max js ls)
+ let dfnx' = max difil diflj
+ unsafeWrite dfmxdfnx 1 dfnx'
+ when (dfmx/cuttb < dfnx' && difil > diflj) $ do
+ is' <- unsafeRead iejeitjtisjsls 4
+ js' <- unsafeRead iejeitjtisjsls 5
+ it <- unsafeRead iejeitjtisjsls 2
+ unsafeWrite iejeitjtisjsls 2 is'
+ unsafeWrite iejeitjtisjsls 4 js'
+ unsafeWrite iejeitjtisjsls 5 it
+ vrts2 <- mapIndices ((1, 1, 1), (nd, nd+1, sbs+nregions-1))
+ (\(i, j, k) -> (i, j, if k <= sbs then k else top)) vrts
+ -- vrts2 <- newArray_ ((1, 1, 1), (nd, nd+1, sbs+nregions-1)) :: IO IO3dArray
+ -- let go1 :: Int -> IO ()
+ -- go1 i | i == nd+1 = return ()
+ -- | otherwise = do
+ -- let go2 j | j == nd+2 = go1 (i+1)
+ -- | otherwise = do
+ -- let go3 k | k == sbs+nregions = go2 (j+1)
+ -- | otherwise = do
+ -- case k <= sbs of
+ -- True -> do
+ -- coef <- readArray vrts (i, j, k)
+ -- writeArray vrts2 (i, j, k) coef
+ -- False -> do
+ -- coef <- readArray v (i, j)
+ -- writeArray vrts2 (i, j, k) coef
+ -- go3 (k+1)
+ -- go3 1
+ -- go2 1
+ -- go1 1
+ is <- unsafeRead iejeitjtisjsls 4
+ js <- unsafeRead iejeitjtisjsls 5
+ vti <- (=<<) array1dToVectorD (extractColumn v is)
+ vtj <- (=<<) array1dToVectorD (extractColumn v js)
+ if nregions == 4
+ then do
+ let vt = UV.zipWith (((*0.5).).(+)) vti vtj
+ replaceDimensions vrts2 (js, top) (js, sbs+1) vt
+ replaceDimension vrts2 (is, sbs+1) vti
+ replaceDimensions vrts2 (is, sbs+2) (is, sbs+3) vt
+ replaceDimensions vrts2 (js, sbs+2) (js, sbs+3) vtj
+ it <- unsafeRead iejeitjtisjsls 2
+ jt <- unsafeRead iejeitjtisjsls 3
+ (vtit, vtjt) <- getVectors nd vrts2 top it jt
+ -- vtit <- (=<<) array1dToVectorD (mapIndices (1, nd) (\i -> (i, it, top)) vrts2)
+ -- vtjt <- (=<<) array1dToVectorD (mapIndices (1, nd) (\i -> (i, jt, top)) vrts2)
+ -- vtit <- array1dToVectorD vtitIO -- (=<<) (return . fromList) (getElems vtitIO)
+ -- vtjt <- array1dToVectorD vtjtIO -- (=<<) (return . fromList) (getElems vtjtIO)
+ let vtt = UV.zipWith (((*0.5).).(+)) vtit vtjt
+ replaceDimensions vrts2 (jt, top) (it, sbs+1) vtt
+ replaceDimension vrts2 (jt, sbs+1) vtjt
+ (vti2, vtj2) <- getVectors nd vrts2 (sbs+2) it jt
+ -- vti2 <- (=<<) array1dToVectorD (mapIndices (1, nd) (\i -> (i, it, sbs+2)) vrts2)
+ -- vtj2 <- (=<<) array1dToVectorD (mapIndices (1, nd) (\i -> (i, jt, sbs+2)) vrts2) -- :: IO IO1dArray
+ -- vti2 <- (=<<) (return . fromList) (getElems vti2IO)
+ -- vtj2 <- (=<<) (return . fromList) (getElems vtj2IO)
+ let vt2 = UV.zipWith (((*0.5).).(+)) vti2 vtj2
+ replaceDimensions vrts2 (jt, sbs+2) (it, sbs+3) vt2
+ replaceDimension vrts2 (jt, sbs+3) vtj2
+ else do
+ let vt = UV.zipWith (\x y -> (2*x+y)/3) vti vtj
+ replaceDimensions vrts2 (js, top) (is, sbs+1) vt
+ if dfmx/cuttf < dfnx
+ then do
+ replaceDimensions vrts2 (js, sbs+1) (is, sbs+2) vtj
+ replaceDimension vrts2 (js, sbs+2) vtj
+ ls <- unsafeRead iejeitjtisjsls 6
+ (vtj1, vtl1) <- getVectors nd vrts2 (sbs+1) js ls
+ -- vtj1 <- (=<<) array1dToVectorD (mapIndices (1, nd) (\i -> (i, js, sbs+1)) vrts2)
+ -- vtl1 <- (=<<) array1dToVectorD (mapIndices (1, nd) (\i -> (i, ls, sbs+1)) vrts2)
+ -- vtj1 <- (=<<) (return . fromList) (getElems vtj1IO)
+ -- vtl1 <- (=<<) (return . fromList) (getElems vtl1IO)
+ let vt1 = UV.zipWith (((*0.5).).(+)) vtj1 vtl1
+ replaceDimensions vrts2 (ls, sbs+1) (js, sbs+2) vt1
+ replaceDimension vrts2 (ls, sbs+2) vtl1
+ else do
+ let vv = UV.zipWith (\x y -> (x+2*y)/3) vti vtj
+ replaceDimensions vrts2 (js, sbs+1) (is, sbs+2) vv
+ replaceDimension vrts2 (js, sbs+2) vtj
+ return (nregions, vrts2)
+ where
+ replaceDimension :: IO3dArray -> (Int, Int) -> VectorD -> IO ()
+ replaceDimension m (j, k) v = do
+ let loop :: Int -> IO ()
+ loop i | i == nd+1 = return ()
+ | otherwise = do
+ writeArray m (i, j, k) ((UV.!) v (i-1))
+ loop (i+1)
+ loop 1
+ replaceDimensions :: IO3dArray -> (Int, Int) -> (Int, Int) -> VectorD -> IO ()
+ replaceDimensions m (j1, k1) (j2, k2) v = do
+ let loop :: Int -> IO ()
+ loop i | i == nd+1 = return ()
+ | otherwise = do
+ writeArray m (i, j1, k1) ((UV.!) v (i-1))
+ writeArray m (i, j2, k2) ((UV.!) v (i-1))
+ loop (i+1)
+ loop 1
+
+-- | Number of evaluations for each subregion
+smpchc :: Int -> Int -> Int
+smpchc nd key
+ | key == 3
+ = div ((nd+4)*(nd+3)*(nd+2)) 6 + (nd+2)*(nd+1)
+ | key == 1
+ = 2*nd + 3
+ | key == 2
+ = div ((nd+3)*(nd+2)) 2 + 2*(nd+1)
+ | otherwise
+ = div ((nd+5)*(nd+4)*(nd+3)*(nd+2)) 24 + 5 * div ((nd+2)*(nd+1)) 2
+
+-- | Checks validity of parameters
+check :: Int -> Int -> Int -> Double -> Double -> Int -> Int -> Int
+check nd nf mxfs ea er sbs key
+ | ea < 0 || er < 0
+ = 5
+ | nf < 1
+ = 4
+ | nd < 2
+ = 3
+ | key < 1 || key > 4
+ = 2
+ | mxfs < sbs * smpchc nd key
+ = 1
+ | otherwise
+ = 0
+
+adsimp :: Int -> Int -> Int -> (VectorD -> VectorD) -> Double -> Double
+ -> Int -> IO3dArray -> IO (VectorD, VectorD, Int, Bool)
+adsimp nd nf mxfs f ea er key vrts = do
+ (_, (_, _, sbs)) <- getBounds vrts
+ case check nd nf mxfs ea er sbs key of
+ 0 -> smpsad nd nf f mxfs ea er key (smpchc nd key) sbs vrts
+ 1 -> smpsad nd nf f (sbs * smpchc nd key) ea er key (smpchc nd key) sbs vrts
+ 2 -> error "integration rule must be between 1 and 4"
+ 3 -> error "dimension must be at least 2"
+ 4 -> error "number of components must be at least 1"
+ 5 -> error "requested errors must be positive"
+ _ -> error "this should not happen"
+
+type Params = (Bool, Int, Int, Seq VectorD, Seq VectorD,
+ VectorD, VectorD, IO3dArray, Seq Double)
+
+smpsad :: Int -> Int -> (VectorD -> VectorD) -> Int -> Double -> Double -> Int
+ -> Int -> Int -> IO3dArray -> IO (VectorD, VectorD, Int, Bool)
+smpsad nd nf f mxfs ea er key rcls sbs vrts = do
+ let dfcost = 1 + 2*nd*(nd+1)
+ (g, w, pospts) <- smprms nd key
+ simplices <- mapM (toSimplex vrts (nd+1)) [1..sbs]
+ let vol = S.fromList $ map simplexVolume simplices
+ nv = sbs*rcls
+ matrices <- mapM
+ (\k -> mapIndices ((1, 1), (nd, nd+1)) (\(i, j) -> (i, j, k)) vrts)
+ (S.fromList [1..sbs])
+ br <- mapM (\(m, v) -> smprul m nf f v g w pospts) (S.zip matrices vol)
+ aes <- mapM (array1dToVectorD.snd) br
+ vls <- mapM (array1dToVectorD.fst) br
+ let vl = foldl1 (UV.zipWith (+)) vls
+ ae = foldl1 (UV.zipWith (+)) aes
+ fl = getFL ae vl
+ let loop :: Params -> IO (VectorD, VectorD, Int, Bool)
+ loop !params | not fl' || nv'+dfcost+4*rcls > mxfs =
+ return (vl', ae', nv', fl')
+ | otherwise = do
+ let maxs = fmap UV.maximum aes'
+ imax = fromJust $ S.findIndexL (== maximum maxs) maxs
+ vl0 = UV.zipWith (-) vl' (index vls' imax)
+ ae0 = UV.zipWith (-) ae' (index aes' imax)
+ (nregions, vrts2) <- smpdfs nd f (imax+1) sbs' vrts'
+ let vi = index vol' imax / toDbl nregions
+ nv2 = nv' + nregions*rcls + dfcost -- nregions*rcls ?
+ sbs2 = sbs' + nregions-1
+ matrices2 <- mapM
+ (\k -> mapIndices ((1, 1), (nd, nd+1))
+ (\(i, j) -> (i, j, k)) vrts2)
+ (S.fromList ((imax+1):[(sbs'+1)..sbs2]))
+ br2 <- mapM (\m -> smprul m nf f vi g w pospts) matrices2
+ rgnerrs <- mapM (array1dToVectorD.snd) br2
+ basvals <- mapM (array1dToVectorD.fst) br2
+ let vl2 = UV.zipWith (+) vl0
+ (foldl1 (UV.zipWith (+)) basvals)
+ ae2 = UV.zipWith (+) ae0
+ (foldl1 (UV.zipWith (+)) rgnerrs)
+ aes2 = update imax (index rgnerrs 0) aes'
+ >< S.drop 1 rgnerrs
+ vls2 = update imax (index basvals 0) vls'
+ >< S.drop 1 basvals
+ fl2 = getFL ae2 vl2
+ vol2 = update imax vi vol'
+ >< S.replicate (nregions-1) vi
+ loop (fl2, nv2, sbs2, aes2, vls2, ae2, vl2, vrts2, vol2)
+ where
+ (fl', nv', sbs', aes', vls', ae', vl', vrts', vol') = params
+ -- dans le code R il fait rowSums mais ça me semble inutile
+ loop (fl, nv, sbs, aes, vls, ae, vl, vrts, vol)
+ where
+ getFL a v = UV.any (> max ea (UV.maximum (UV.map ((*er).abs) v))) a
+
+
+toSimplex :: IO3dArray -> Int -> Int -> IO Simplex
+toSimplex m n k = do
+ (_, (nrow, _, _)) <- getBounds m
+ let getColumn :: Int -> IO IO1dArray
+ getColumn col = mapIndices (1, nrow) (\i -> (i, col, k)) m
+ columns <- mapM getColumn [1..n]
+ mapM getElems columns
diff --git a/src/Numeric/Integration/SphericalSimplexCubature.hs b/src/Numeric/Integration/SphericalSimplexCubature.hs
new file mode 100644
--- /dev/null
+++ b/src/Numeric/Integration/SphericalSimplexCubature.hs
@@ -0,0 +1,23 @@
+module Numeric.Integration.SphericalSimplexCubature
+ (integrateOnSphericalSimplex, SphericalSimplex, orthants, Result (..))
+ where
+import Numeric.Integration.Simplex.Simplex ( canonicalSimplex )
+import Numeric.Integration.SimplexCubature ( integrateOnSimplex'
+ , Result(..) )
+import Numeric.Integration.SphericalSimplexCubature.Internal ( orthants
+ , transformedIntegrand
+ , SphericalSimplex )
+
+-- | Integral of a real-valued function over a spherical simplex.
+integrateOnSphericalSimplex
+ :: ([Double] -> Double) -- ^ integrand
+ -> SphericalSimplex -- ^ integration domain
+ -> Int -- ^ maximum number of evaluations
+ -> Double -- ^ desired absolute error
+ -> Double -- ^ desired relative error
+ -> Int -- ^ integration rule: 1, 2, 3 or 4
+ -> IO Result -- ^ integral, error, evaluations, success
+integrateOnSphericalSimplex f ssimplex = integrateOnSimplex' f' [simplex]
+ where
+ f' = transformedIntegrand ssimplex f
+ simplex = canonicalSimplex (length ssimplex - 1)
diff --git a/src/Numeric/Integration/SphericalSimplexCubature/Internal.hs b/src/Numeric/Integration/SphericalSimplexCubature/Internal.hs
new file mode 100644
--- /dev/null
+++ b/src/Numeric/Integration/SphericalSimplexCubature/Internal.hs
@@ -0,0 +1,62 @@
+module Numeric.Integration.SphericalSimplexCubature.Internal
+ (orthants, SphericalSimplex, transformedIntegrand)
+ where
+import Data.List.Index (imap)
+import Data.Matrix (detLU, diagonalList, fromLists,
+ minorMatrix, toLists, zero, (<->))
+import Data.Vector.Unboxed (Vector)
+import qualified Data.Vector.Unboxed as V
+
+type SphericalSimplex = [[Double]] -- square [v1, v2, v3, v4]
+
+orthants :: Int -> [SphericalSimplex]
+orthants n = reverse $ map (toLists . diagonalList n 0) (pm n)
+ where pm 2 = [[i, j] | i <- [-1, 1], j <- [-1, 1]]
+ pm k = [i : l | i <- [-1, 1], l <- pm (k-1)]
+
+norm2 :: [Double] -> Double
+norm2 v = sqrt $ sum $ zipWith (*) v v
+
+dotproduct :: [Double] -> [Double] -> Double
+dotproduct a b = sum $ zipWith (*) a b
+
+scalarTimesList :: Double -> [Double] -> [Double]
+scalarTimesList lambda = map (* lambda)
+
+f :: SphericalSimplex -> [Double] -> [Double]
+f vertices stu = foldr (zipWith (+)) (vertices!!0) terms
+ where
+ w = map (zipWith subtract (vertices!!0)) (tail vertices)
+ terms = zipWith scalarTimesList stu w
+
+g :: SphericalSimplex -> [Double] -> [Double]
+g vertices stu = scalarTimesList (1 / norm2 fstu) fstu
+ where fstu = f vertices stu
+
+dg :: SphericalSimplex -> [Double] -> [[Double]]
+dg vertices stu = zipWith (zipWith subtract) fviv1 nviv1
+ where
+ fstu = f vertices stu
+ invn = 1 / norm2 fstu
+ invn3 = invn*invn*invn
+ viv1 = map (zipWith subtract (head vertices)) (tail vertices)
+ nviv1 = map (scalarTimesList invn) viv1
+ dpi = map ((*invn3) . dotproduct fstu) viv1
+ fviv1 = map (`scalarTimesList` fstu) dpi
+
+extProduct :: [[Double]] -> [Double]
+extProduct vectors =
+ imap (\i mat -> (if even i then 1 else -1) * detLU mat) minorMatrices
+ where
+ dim = length vectors + 1
+ matrix = zero 1 dim <-> fromLists vectors
+ minorMatrices = map (\j -> minorMatrix 1 j matrix) [1 .. dim]
+
+sigma :: SphericalSimplex -> [Double] -> Double
+sigma ssimplex stu = norm2 $ extProduct (dg ssimplex stu)
+
+transformedIntegrand :: SphericalSimplex -> ([Double] -> Double)
+ -> (Vector Double -> Double)
+transformedIntegrand ssimplex integrand stu =
+ let stul = V.toList stu in
+ sigma ssimplex stul * integrand (g ssimplex stul)