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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 +1,30 @@ +# jackpolynomials + +Schur polynomials have applications in combinatorics and zonal polynomials have +applications in multivariate statistics. They are particular cases of +[Jack polynomials](https://en.wikipedia.org/wiki/Jack_function). This package +allows to evaluate these polynomials. It can also compute their symbolic form. + +```haskell +import Math.Algebra.Jack +import Data.Ratio +jackPol [1, 1] [3, 1] (2%1) +-- 48 % 1 +``` + +```haskell +import Math.Algebra.JackPol +import Data.Ratio +import Math.Algebra.MultiPol +jp = jackPol 2 [3, 1] (2%1) +jp +-- (M (Monomial {coefficient = 18 % 1, powers = fromList [1,3]}) +-- :+: +-- M (Monomial {coefficient = 12 % 1, powers = fromList [2,2]})) +-- :+: +-- M (Monomial {coefficient = 18 % 1, powers = fromList [3,1]}) +prettyPol show "x" jp +-- "(18 % 1) * x^(1, 3) + (12 % 1) * x^(2, 2) + (18 % 1) * x^(3, 1)" +evalPoly jp [1, 1] +-- 48 % 1 +``` 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/jackpolynomials.cabal b/jackpolynomials.cabal new file mode 100644 --- /dev/null +++ b/jackpolynomials.cabal @@ -0,0 +1,35 @@ +name: jackpolynomials +version: 1.0.0.0 +synopsis: Jack, zonal, and Schur polynomials +description: This library can evaluate Jack polynomials, zonal polynomials and Schur polynomials. It is also able to compute them in symbolic form. +homepage: https://github.com/stla/jackpolynomials#readme +license: GPL-3 +license-file: LICENSE +author: Stéphane Laurent +maintainer: laurent_step@outlook.fr +copyright: 2022 Stéphane Laurent +category: Math, Algebra +build-type: Simple +extra-source-files: README.md + CHANGELOG.md +cabal-version: >=1.10 + +library + hs-source-dirs: src + exposed-modules: Math.Algebra.Jack.GPochhammer + , Math.Algebra.Jack + , Math.Algebra.JackPol + other-modules: Math.Algebra.Jack.Internal + build-depends: base >= 4.7 && < 5 + , ilist >= 0.4.0.1 && < 0.5 + , array >= 0.5.4.0 && < 0.6 + , lens >= 5.0.1 && < 5.1 + , math-functions >= 0.3.4.2 && < 0.4 + , mpolynomials >= 0.1.0.0 && < 0.2 + , numeric-prelude >= 0.4.4 && < 0.5 + default-language: Haskell2010 + ghc-options: -Wall + +source-repository head + type: git + location: https://github.com/stla/jack diff --git a/src/Math/Algebra/Jack.hs b/src/Math/Algebra/Jack.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Jack.hs @@ -0,0 +1,121 @@ +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE ScopedTypeVariables #-} +module Math.Algebra.Jack + (schur, jack, zonal) + where +import Control.Lens ( (.~), element ) +import Data.Array ( Array, (!), (//), listArray ) +import Data.Maybe ( fromJust, isJust ) +import Math.Algebra.Jack.Internal ( _N, hookLengths, _betaratio, _isPartition, Partition ) +import Numeric.SpecFunctions ( factorial ) + +-- | Evaluation of Jack polynomial +jack :: forall a. (Fractional a, Ord a) + => [a] -- ^ values of the variables + -> Partition -- ^ partition of integers + -> a -- ^ alpha parameter + -> a +jack x lambda alpha = + case _isPartition lambda && alpha > 0 of + False -> if _isPartition lambda + then error "alpha must be strictly positive" + else error "lambda is not a valid integer partition" + True -> jac (length x) 0 lambda lambda arr0 1 + where + nll = _N lambda lambda + n = length x + arr0 = listArray ((1, 1), (nll, n)) (replicate (nll * n) Nothing) + theproduct :: Int -> a + theproduct nu0 = if nu0 <= 1 + then 1 + else product $ map (\i -> alpha * fromIntegral i + 1) [1 .. nu0-1] + jac :: Int -> Int -> [Int] -> [Int] -> Array (Int,Int) (Maybe a) -> a -> a + jac m k mu nu arr beta + | null nu || head nu == 0 || m == 0 = 1 + | length nu > m && nu!!m > 0 = 0 + | m == 1 = head x ^ head nu * theproduct (head nu) + | k == 0 && isJust (arr ! (_N lambda nu, m)) = + fromJust $ arr ! (_N lambda nu, m) + | otherwise = s + where + s = go (jac (m-1) 0 nu nu arr 1 * beta * x!!(m-1) ^ (sum mu - sum nu)) + (max 1 k) + go :: a -> Int -> a + go !ss ii + | length nu < ii || nu!!(ii-1) == 0 = ss + | otherwise = + let u = nu!!(ii-1) in + if length nu == ii && u > 0 || u > nu!!ii + then + let nu' = (element (ii-1) .~ u-1) nu in + let gamma = beta * _betaratio mu nu ii alpha in + if u > 1 + then + go (ss + jac m ii mu nu' arr gamma) (ii + 1) + else + if head nu' == 0 + then + go (ss + gamma * x!!(m-1)^ sum mu) (ii + 1) + else + let arr' = arr // [((_N lambda nu, m), Just ss)] in + let jck = jac (m-1) 0 nu' nu' arr' 1 in + let jck' = jck * gamma * + x!!(m-1) ^ (sum mu - sum nu') in + go (ss+jck') (ii+1) + else + go ss (ii+1) + +-- | Evaluation of zonal polynomial +zonal :: (Fractional a, Ord a) + => [a] -- ^ values of the variables + -> Partition -- ^ partition of integers + -> a +zonal x lambda = c * jck + where + k = sum lambda + jlambda = product (hookLengths lambda 2) + c = 2^k * realToFrac (factorial k) / jlambda + jck = jack x lambda 2 + +-- | Evaluation of Schur polynomial +schur :: forall a. Fractional a + => [a] -- ^ values of the variables + -> Partition -- ^ partition of integers + -> a +schur x lambda = + case _isPartition lambda of + False -> error "lambda is not a valid integer partition" + True -> sch n 1 lambda arr0 + where + nll = _N lambda lambda + n = length x + arr0 = listArray ((1, 1), (nll, n)) (replicate (nll * n) Nothing) + sch :: Int -> Int -> [Int] -> Array (Int,Int) (Maybe a) -> a + sch m k nu arr + | null nu || head nu == 0 || m == 0 = 1 + | length nu > m && nu!!m > 0 = 0 + | m == 1 = head x ^ head nu + | isJust (arr ! (_N lambda nu, m)) = fromJust $ arr ! (_N lambda nu, m) + | otherwise = s + where + s = go (sch (m-1) 1 nu arr) k + go :: Fractional a => a -> Int -> a + go !ss ii + | length nu < ii || nu!!(ii-1) == 0 = ss + | otherwise = + let u = nu!!(ii-1) in + if length nu == ii && u > 0 || u > nu !! ii + then + let nu' = (element (ii-1) .~ u-1) nu in + if u > 1 + then + go (ss + x!!(m-1) * sch m ii nu' arr) (ii + 1) + else + if head nu' == 0 + then + go (ss + x!!(m-1)) (ii + 1) + else + let arr' = arr // [((_N lambda nu, m), Just ss)] in + go (ss + x!!(m-1) * sch (m-1) 1 nu' arr') (ii + 1) + else + go ss (ii+1) diff --git a/src/Math/Algebra/Jack/GPochhammer.hs b/src/Math/Algebra/Jack/GPochhammer.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Jack/GPochhammer.hs @@ -0,0 +1,40 @@ +module Math.Algebra.Jack.GPochhammer where +import Math.Algebra.Jack (zonal) + +gpochhammer :: Fractional a => a -> [Int] -> a -> a +gpochhammer a kappa alpha = + product $ + map (\i -> product $ + map (\j -> a - (fromIntegral i - 1)/alpha + fromIntegral j -1) + [1 .. kappa !! (i-1)]) + [1 .. length kappa] + +hcoeff :: Fractional a => [a] -> [a] -> [Int] -> a -> a +hcoeff a b kappa alpha = + numerator / denominator / fromIntegral (factorial (sum kappa)) + where + factorial n = product [1 .. n] + numerator = product $ map (\x -> gpochhammer x kappa alpha) a + denominator = product $ map (\x -> gpochhammer x kappa alpha) b + +testHypergeo :: Double +testHypergeo = + let a = [2,3] in + let b = [4] in + let coeff kappa = hcoeff a b kappa 2 in + let kappas = [[], [1], [1,1], [2]] in + let x = [5,6] in + sum $ map (\kappa -> coeff kappa * zonal x kappa) kappas + +_allPartitions :: Int -> [[Int]] +_allPartitions m = last ps + where + ps = [] : map parts [1..m] + parts n = [n] : [x : p | x <- [1..n], p <- ps !! (n - x), x <= head p] + +hypergeoPQ :: (Fractional a, Ord a) => Int -> [a] -> [a] -> [a] -> a +hypergeoPQ m a b x = + sum $ map (\kappa -> coeff kappa * zonal x kappa) kappas + where + kappas = filter (\kap -> length kap <= length x) (_allPartitions m) + coeff kappa = hcoeff a b kappa 2 diff --git a/src/Math/Algebra/Jack/Internal.hs b/src/Math/Algebra/Jack/Internal.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Jack/Internal.hs @@ -0,0 +1,77 @@ +{-# LANGUAGE BangPatterns #-} +module Math.Algebra.Jack.Internal + where +import qualified Algebra.Ring as AR +import Data.List.Index ( iconcatMap ) + +type Partition = [Int] + +_isPartition :: Partition -> Bool +_isPartition [] = True +_isPartition [x] = x > 0 +_isPartition (x:xs@(y:_)) = (x >= y) && _isPartition xs + +_diffSequence :: [Int] -> [Int] +_diffSequence = go where + go (x:ys@(y:_)) = (x-y) : go ys + go [x] = [x] + go [] = [] + +_dualPartition :: Partition -> Partition +_dualPartition [] = [] +_dualPartition xs = go 0 (_diffSequence xs) [] where + go !i (d:ds) acc = go (i+1) ds (d:acc) + go n [] acc = finish n acc + finish !j (k:ks) = replicate k j ++ finish (j-1) ks + finish _ [] = [] + +_ij :: Partition -> ([Int], [Int]) +_ij lambda = + ( + iconcatMap (\i a -> replicate a (i + 1)) lambda, + concatMap (\a -> [1 .. a]) (filter (>0) lambda) + ) + +_convParts :: Num b => [Int] -> ([b], [b]) +_convParts lambda = + (map fromIntegral lambda, map fromIntegral (_dualPartition lambda)) + +_N :: [Int] -> [Int] -> Int +_N lambda mu = sum $ zipWith (*) mu prods + where + prods = map (\i -> product $ drop i (map (+1) lambda)) [1 .. length lambda] + +hookLengths :: Fractional a => Partition -> a -> [a] +hookLengths lambda alpha = upper ++ lower + where + (i, j) = _ij lambda + (lambda', lambdaConj') = _convParts lambda + upper = zipWith (fup lambdaConj' lambda') i j + where + fup x y ii jj = + x!!(jj-1) - fromIntegral ii + alpha * (y!!(ii-1) - fromIntegral jj + 1) + lower = zipWith (flow lambdaConj' lambda') i j + where + flow x y ii jj = + x!!(jj-1) - fromIntegral ii + 1 + alpha * (y!!(ii-1) - fromIntegral jj) + +hookLengths' :: (Fractional a, AR.C a) => Partition -> a -> [a] +hookLengths' = hookLengths + +_betaratio :: Fractional a => Partition -> Partition -> Int -> a -> a +_betaratio kappa mu k alpha = alpha * prod1 * prod2 * prod3 + where + mukm1 = mu !! (k-1) + t = fromIntegral k - alpha * fromIntegral mukm1 + u = zipWith (\s kap -> t + 1 - fromIntegral s + alpha * fromIntegral kap) + [1 .. k] kappa + v = zipWith (\s m -> t - fromIntegral s + alpha * fromIntegral m) + [1 .. k-1] mu + w = zipWith (\s m -> fromIntegral m - t - alpha * fromIntegral s) + [1 .. mukm1-1] (_dualPartition mu) + prod1 = product $ map (\x -> x / (x + alpha - 1)) u + prod2 = product $ map (\x -> (x + alpha) / x) v + prod3 = product $ map (\x -> (x + alpha) / x) w + +_betaratio' :: (Fractional a, AR.C a) => [Int] -> [Int] -> Int -> a -> a +_betaratio' = _betaratio diff --git a/src/Math/Algebra/JackPol.hs b/src/Math/Algebra/JackPol.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/JackPol.hs @@ -0,0 +1,125 @@ +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE ScopedTypeVariables #-} +module Math.Algebra.JackPol + (schurPol, jackPol, zonalPol) + where +import qualified Algebra.Ring as AR +import Control.Lens ( (.~), element ) +import Data.Array ( Array, (!), (//), listArray ) +import Data.Maybe ( fromJust, isJust ) +import Math.Algebra.Jack.Internal ( _betaratio', hookLengths, _N + , _isPartition, Partition ) +import Math.Algebra.MultiPol ( (*^), (^**^), (^*^), (^+^) + , constant, lone, Polynomial ) +import Numeric.SpecFunctions ( factorial ) + +-- | Symbolic Jack polynomial +jackPol :: forall a. (Fractional a, Ord a, AR.C a) + => Int -- ^ number of variables + -> Partition -- ^ partition of integers + -> a -- ^ alpha parameter + -> Polynomial a +jackPol n lambda alpha = + case _isPartition lambda && alpha > 0 of + False -> if _isPartition lambda + then error "alpha must be strictly positive" + else error "lambda is not a valid integer partition" + True -> jac (length x) 0 lambda lambda arr0 1 + where + nll = _N lambda lambda + x = map lone [1 .. n] :: [Polynomial a] + arr0 = listArray ((1, 1), (nll, n)) (replicate (nll * n) Nothing) + theproduct :: Int -> a + theproduct nu0 = if nu0 <= 1 + then AR.one + else AR.product $ map (\i -> alpha * fromIntegral i + 1) [1 .. nu0-1] + jac :: Int -> Int -> Partition -> Partition -> Array (Int,Int) (Maybe (Polynomial a)) -> a -> Polynomial a + jac m k mu nu arr beta + | null nu || head nu == 0 || m == 0 = constant 1 + | length nu > m && nu!!m > 0 = constant 0 + | m == 1 = theproduct (head nu) *^ (head x ^**^ head nu) + | k == 0 && isJust (arr ! (_N lambda nu, m)) = + fromJust $ arr ! (_N lambda nu, m) + | otherwise = s + where + s = go (beta *^ (jac (m-1) 0 nu nu arr 1 ^*^ ((x!!(m-1)) ^**^ (sum mu - sum nu)))) + (max 1 k) + go :: Polynomial a -> Int -> Polynomial a + go !ss ii + | length nu < ii || nu!!(ii-1) == 0 = ss + | otherwise = + let u = nu!!(ii-1) in + if length nu == ii && u > 0 || u > nu!!ii + then + let nu' = (element (ii-1) .~ u-1) nu in + let gamma = beta * _betaratio' mu nu ii alpha in + if u > 1 + then + go (ss ^+^ jac m ii mu nu' arr gamma) (ii + 1) + else + if head nu' == 0 + then + go (ss ^+^ (gamma *^ (x!!(m-1) ^**^ sum mu))) (ii + 1) + else + let arr' = arr // [((_N lambda nu, m), Just ss)] in + let jck = jac (m-1) 0 nu' nu' arr' 1 in + let jck' = gamma *^ (jck ^*^ + (x!!(m-1) ^**^ (sum mu - sum nu'))) in + go (ss ^+^ jck') (ii+1) + else + go ss (ii+1) + +-- | Symbolic zonal polynomial +zonalPol :: (Fractional a, Ord a, AR.C a) + => Int -- ^ number of variables + -> Partition -- ^ partition of integers + -> Polynomial a +zonalPol n lambda = c *^ jck + where + k = sum lambda + jlambda = product (hookLengths lambda 2) + c = 2^k * realToFrac (factorial k) / jlambda + jck = jackPol n lambda 2 + +-- | Symbolic Schur polynomial +schurPol :: + Int -- ^ number of variables + -> Partition -- ^ partition of integers + -> Polynomial Int +schurPol n lambda = + case _isPartition lambda of + False -> error "lambda is not a valid integer partition" + True -> sch n 1 lambda arr0 + where + x = map lone [1 .. n] :: [Polynomial Int] + nll = _N lambda lambda + arr0 = listArray ((1, 1), (nll, n)) (replicate (nll * n) Nothing) + sch :: Int -> Int -> [Int] -> Array (Int,Int) (Maybe (Polynomial Int)) -> Polynomial Int + sch m k nu arr + | null nu || head nu == 0 || m == 0 = constant 1 + | length nu > m && nu!!m > 0 = constant 0 + | m == 1 = head x ^**^ head nu + | isJust (arr ! (_N lambda nu, m)) = fromJust $ arr ! (_N lambda nu, m) + | otherwise = s + where + s = go (sch (m-1) 1 nu arr) k + go :: Polynomial Int -> Int -> Polynomial Int + go !ss ii + | length nu < ii || nu!!(ii-1) == 0 = ss + | otherwise = + let u = nu!!(ii-1) in + if length nu == ii && u > 0 || u > nu !! ii + then + let nu' = (element (ii-1) .~ u-1) nu in + if u > 1 + then + go (ss ^+^ ((x!!(m-1)) ^*^ sch m ii nu' arr)) (ii + 1) + else + if head nu' == 0 + then + go (ss ^+^ (x!!(m-1))) (ii + 1) + else + let arr' = arr // [((_N lambda nu, m), Just ss)] in + go (ss ^+^ ((x!!(m-1)) ^*^ sch (m-1) 1 nu' arr')) (ii + 1) + else + go ss (ii+1)