scubature (empty) → 1.0.0.0
raw patch · 10 files changed
+1774/−0 lines, 10 filesdep +arraydep +basedep +containerssetup-changed
Dependencies added: array, base, containers, ilist, matrix, vector
Files
- CHANGELOG.md +3/−0
- LICENSE +674/−0
- README.md +101/−0
- Setup.hs +2/−0
- scubature.cabal +37/−0
- src/Numeric/Integration/Simplex/Simplex.hs +35/−0
- src/Numeric/Integration/SimplexCubature.hs +72/−0
- src/Numeric/Integration/SimplexCubature/Internal.hs +765/−0
- src/Numeric/Integration/SphericalSimplexCubature.hs +23/−0
- src/Numeric/Integration/SphericalSimplexCubature/Internal.hs +62/−0
+ CHANGELOG.md view
@@ -0,0 +1,3 @@+1.0.0.0+-------+* initial release
+ LICENSE view
@@ -0,0 +1,674 @@+ GNU GENERAL PUBLIC LICENSE+ Version 3, 29 June 2007++ Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>+ Everyone is permitted to copy and distribute verbatim copies+ of this license document, but changing it is not allowed.++ Preamble++ The GNU General Public License is a free, copyleft license for+software and other kinds of works.++ The licenses for most software and other practical works are designed+to take away your freedom to share and change the works. By contrast,+the GNU General Public License is intended to guarantee your freedom to+share and change all versions of a program--to make sure it remains free+software for all its users. We, the Free Software Foundation, use the+GNU General Public License for most of our software; it applies also to+any other work released this way by its authors. You can apply it to+your programs, too.++ When we speak of free software, we are referring to freedom, not+price. 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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+<https://www.gnu.org/licenses/>.++ 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+<https://www.gnu.org/licenses/why-not-lgpl.html>.
+ README.md view
@@ -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 }+```
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ scubature.cabal view
@@ -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
+ src/Numeric/Integration/Simplex/Simplex.hs view
@@ -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)
+ src/Numeric/Integration/SimplexCubature.hs view
@@ -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"
+ src/Numeric/Integration/SimplexCubature/Internal.hs view
@@ -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
+ src/Numeric/Integration/SphericalSimplexCubature.hs view
@@ -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)
+ src/Numeric/Integration/SphericalSimplexCubature/Internal.hs view
@@ -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)