diff --git a/CHANGELOG.md b/CHANGELOG.md --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -1,17 +1,26 @@ -1.0.0.0 -------- -* initial release - -1.0.0.1 -------- -* removed the upper bounds of the dependencies - -1.1.0.0 -------- -* replaced the 'mpolynomials' dependency with 'hspray' -* unit tests - -1.1.0.1 -------- -* unexported some useless functions -* one more unit test +1.0.0.0 +------- +* initial release + +1.0.0.1 +------- +* removed the upper bounds of the dependencies + +1.1.0.0 +------- +* replaced the 'mpolynomials' dependency with 'hspray' +* unit tests + +1.1.0.1 +------- +* unexported some useless functions +* one more unit test + +1.1.1.0 +------- +* `schurPol` now returns a `Spray a` +* added package upper bounds in the cabal file +* increased the version of the dependencies **hspray** and **hypergeomatrix** +* cleaned the code +* tested with higher versions of GHC +* new unit tests diff --git a/LICENSE b/LICENSE --- a/LICENSE +++ b/LICENSE @@ -1,674 +1,674 @@ - GNU GENERAL PUBLIC LICENSE - Version 3, 29 June 2007 - - Copyright (C) 2007 Free Software Foundation, Inc. - 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. 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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 --- a/README.md +++ b/README.md @@ -1,35 +1,42 @@ -# 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 -jack [1, 1] [3, 1] (2%1) --- 48 % 1 -``` - -```haskell -import Math.Algebra.JackPol -import Data.Ratio -import Math.Algebra.Spray -jp = jackPol 2 [3, 1] (2%1) -prettySpray show "x" jp --- "(18 % 1) * x^(1, 3) + (12 % 1) * x^(2, 2) + (18 % 1) * x^(3, 1)" -evalSpray jp [1, 1] --- 48 % 1 -``` - - -## References - -* I.G. Macdonald. *Symmetric Functions and Hall Polynomials*. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995. - -* J. Demmel and P. Koev. *Accurate and efficient evaluation of Schur and Jack functions*. Mathematics of computations, vol. 75, n. 253, 223-229, 2005. - -* Jack polynomials. . +# jackpolynomials + +*Jack, zonal, and Schur polynomials.* + + +[![Stack-lts](https://github.com/stla/jackpolynomials/actions/workflows/Stack-lts.yml/badge.svg)](https://github.com/stla/jackpolynomials/actions/workflows/Stack-lts.yml) +[![Stack-nightly](https://github.com/stla/jackpolynomials/actions/workflows/Stack-nightly.yml/badge.svg)](https://github.com/stla/jackpolynomials/actions/workflows/Stack-nightly.yml) + + +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 +jack [1, 1] [3, 1] (2%1) +-- 48 % 1 +``` + +```haskell +import Math.Algebra.JackPol +import Data.Ratio +import Math.Algebra.Hspray +jp = jackPol 2 [3, 1] (2%1) +putStrLn $ prettySpray' jp +-- (18 % 1) x1^3x2 + (12 % 1) x1^2x2^2 + (18 % 1) x1x2^3 +evalSpray jp [1, 1] +-- 48 % 1 +``` + + +## References + +* I.G. Macdonald. *Symmetric Functions and Hall Polynomials*. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995. + +* J. Demmel and P. Koev. *Accurate and efficient evaluation of Schur and Jack functions*. Mathematics of computations, vol. 75, n. 253, 223-229, 2005. + +* Jack polynomials. . diff --git a/Setup.hs b/Setup.hs --- a/Setup.hs +++ b/Setup.hs @@ -1,2 +1,2 @@ -import Distribution.Simple -main = defaultMain +import Distribution.Simple +main = defaultMain diff --git a/jackpolynomials.cabal b/jackpolynomials.cabal --- a/jackpolynomials.cabal +++ b/jackpolynomials.cabal @@ -1,47 +1,57 @@ -name: jackpolynomials -version: 1.1.0.1 -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.HypergeoPQ - , Math.Algebra.Jack - , Math.Algebra.JackPol - other-modules: Math.Algebra.Jack.Internal - build-depends: base >= 4.7 && < 5 - , ilist >= 0.4.0.1 - , array >= 0.5.4.0 - , lens >= 5.0.1 - , math-functions >= 0.3.4.2 - , hspray >= 0.1.0.0 - , numeric-prelude >= 0.4.4 - default-language: Haskell2010 - ghc-options: -Wall - -test-suite unit-tests - type: exitcode-stdio-1.0 - main-is: Main.hs - hs-source-dirs: tests/ - Build-Depends: base >= 4.7 && < 5 - , tasty - , tasty-hunit - , jackpolynomials - , hspray - , hypergeomatrix - Default-Language: Haskell2010 - -source-repository head - type: git - location: https://github.com/stla/jackpolynomials +name: jackpolynomials +version: 1.1.1.0 +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.HypergeoPQ + , Math.Algebra.Jack + , Math.Algebra.JackPol + other-modules: Math.Algebra.Jack.Internal + build-depends: base >= 4.7 && < 5 + , ilist >= 0.4.0.1 && < 0.4.1 + , array >= 0.5.4.0 && < 0.6 + , lens >= 5.0.1 && < 5.3 + , math-functions >= 0.3.4.2 && < 0.3.5 + , hspray >= 0.2.2.0 && < 1 + , numeric-prelude >= 0.4.4 && < 0.5 + other-extensions: ScopedTypeVariables + , BangPatterns + default-language: Haskell2010 + ghc-options: -Wall + -Wcompat + -Widentities + -Wincomplete-record-updates + -Wincomplete-uni-patterns + -Wmissing-export-lists + -Wmissing-home-modules + -Wpartial-fields + -Wredundant-constraints + +test-suite unit-tests + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: tests/ + Build-Depends: base >= 4.7 && < 5 + , tasty >= 1.4 && < 1.6 + , tasty-hunit >= 0.10 && < 0.11 + , jackpolynomials + , hspray >= 0.2.2.0 && < 1 + , hypergeomatrix >= 1.1.0.2 && < 2 + Default-Language: Haskell2010 + +source-repository head + type: git + location: https://github.com/stla/jackpolynomials diff --git a/src/Math/Algebra/Jack.hs b/src/Math/Algebra/Jack.hs --- a/src/Math/Algebra/Jack.hs +++ b/src/Math/Algebra/Jack.hs @@ -1,121 +1,136 @@ -{-# 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) +{-| +Module : Math.Algebra.JackPol +Description : Evaluation of Jack polynomials. +Copyright : (c) Stéphane Laurent, 2024 +License : GPL-3 +Maintainer : laurent_step@outlook.fr + +Evaluation of Jack polynomials, zonal polynomials, and Schur polynomials. +See README for examples and references. +-} + +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE ScopedTypeVariables #-} +module Math.Algebra.Jack + (jack, zonal, schur) + where +import qualified Algebra.Additive as AA +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 ( _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 [] _ _ = error "jack: empty list of variables" +jack x@(x0:_) lambda alpha = + case _isPartition lambda && alpha > 0 of + False -> if _isPartition lambda + then error "jack: alpha must be strictly positive" + else error "jack: invalid 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 || nu!!0 == 0 || m == 0 = 1 + | length nu > m && nu!!m > 0 = 0 + | m == 1 = x0 ^ (nu!!0) * theproduct (nu!!0) + | 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 nu' !! 0 == 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. AR.C a + => [a] -- ^ values of the variables + -> Partition -- ^ partition of integers + -> a +schur [] _ = error "schur: empty list of variables" +schur x@(x0:_) lambda = + case _isPartition lambda of + False -> error "schur: invalid 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 || nu!!0 == 0 || m == 0 = AR.one + | length nu > m && nu!!m > 0 = AA.zero + | m == 1 = AR.product (replicate (nu!!0) x0) + | 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 :: 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 AA.+ x!!(m-1) AR.* sch m ii nu' arr) (ii + 1) + else + if nu' !! 0 == 0 + then + go (ss AA.+ x!!(m-1)) (ii + 1) + else + let arr' = arr // [((_N lambda nu, m), Just ss)] in + go (ss AA.+ x!!(m-1) AR.* sch (m-1) 1 nu' arr') (ii + 1) + else + go ss (ii+1) diff --git a/src/Math/Algebra/Jack/HypergeoPQ.hs b/src/Math/Algebra/Jack/HypergeoPQ.hs --- a/src/Math/Algebra/Jack/HypergeoPQ.hs +++ b/src/Math/Algebra/Jack/HypergeoPQ.hs @@ -1,33 +1,33 @@ -module Math.Algebra.Jack.HypergeoPQ - ( hypergeoPQ, _allPartitions - ) 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 - -_allPartitions :: Int -> [[Int]] -_allPartitions m = [[]] ++ (map reverse (concat ps)) - where - ps = [] : map parts [1 .. m] - parts n = [n] : [ x : p | x <- [1 .. n], p <- ps !! (n - x), x <= head p ] - --- | Inefficient hypergeometric function of a matrix argument -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 +module Math.Algebra.Jack.HypergeoPQ + ( hypergeoPQ + ) 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 + +_allPartitions :: Int -> [[Int]] +_allPartitions m = [] : map reverse (concat ps) + where + ps = [] : map parts [1 .. m] + parts n = [n] : [ x : p | x <- [1 .. n], p <- ps !! (n - x), x <= p!!0 ] + +-- | Inefficient hypergeometric function of a matrix argument (for testing purpose) +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 --- a/src/Math/Algebra/Jack/Internal.hs +++ b/src/Math/Algebra/Jack/Internal.hs @@ -1,77 +1,71 @@ -{-# 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 +{-# LANGUAGE BangPatterns #-} +module Math.Algebra.Jack.Internal + (Partition, hookLengths, _betaratio, _isPartition, _N) + where +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) + +_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 diff --git a/src/Math/Algebra/JackPol.hs b/src/Math/Algebra/JackPol.hs --- a/src/Math/Algebra/JackPol.hs +++ b/src/Math/Algebra/JackPol.hs @@ -1,125 +1,137 @@ -{-# 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.Hspray ( (*^), (^**^), (^*^), (^+^) - , constantSpray, lone, Spray ) -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 - -> Spray 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] :: [Spray 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 (Spray a)) -> a -> Spray a - jac m k mu nu arr beta - | null nu || head nu == 0 || m == 0 = constantSpray 1 - | length nu > m && nu!!m > 0 = constantSpray 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 :: Spray a -> Int -> Spray 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 - -> Spray 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 - -> Spray 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] :: [Spray Int] - nll = _N lambda lambda - arr0 = listArray ((1, 1), (nll, n)) (replicate (nll * n) Nothing) - sch :: Int -> Int -> [Int] -> Array (Int,Int) (Maybe (Spray Int)) -> Spray Int - sch m k nu arr - | null nu || head nu == 0 || m == 0 = constantSpray 1 - | length nu > m && nu!!m > 0 = constantSpray 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 :: Spray Int -> Int -> Spray 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) +{-| +Module : Math.Algebra.JackPol +Description : Symbolic Jack polynomials. +Copyright : (c) Stéphane Laurent, 2024 +License : GPL-3 +Maintainer : laurent_step@outlook.fr + +Computation of symbolic Jack polynomials, zonal polynomials, and Schur polynomials. +See README for examples and references. +-} + +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE ScopedTypeVariables #-} +module Math.Algebra.JackPol + (jackPol, zonalPol, schurPol) + 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.Hspray ( (*^), (^**^), (^*^), (^+^) + , lone, Spray + , zeroSpray, unitSpray ) +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 + -> Spray a +jackPol n lambda alpha = + case _isPartition lambda && alpha > 0 of + False -> if _isPartition lambda + then error "jackPol: alpha must be strictly positive" + else error "jackPol: invalid integer partition" + True -> jac (length x) 0 lambda lambda arr0 1 + where + nll = _N lambda lambda + x = map lone [1 .. n] :: [Spray a] + 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 -> Partition -> Partition -> Array (Int,Int) (Maybe (Spray a)) -> a -> Spray a + jac m k mu nu arr beta + | null nu || nu!!0 == 0 || m == 0 = unitSpray + | length nu > m && nu!!m > 0 = zeroSpray + | m == 1 = theproduct (nu!!0) *^ (x!!0 ^**^ nu!!0) + | 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 :: Spray a -> Int -> Spray 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 nu'!!0 == 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 + -> Spray 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 :: forall a. (Ord a, AR.C a) + => Int -- ^ number of variables + -> Partition -- ^ partition of integers + -> Spray a +schurPol n lambda = + case _isPartition lambda of + False -> error "schurPol: invalid integer partition" + True -> sch n 1 lambda arr0 + where + x = map lone [1 .. n] :: [Spray a] + nll = _N lambda lambda + arr0 = listArray ((1, 1), (nll, n)) (replicate (nll * n) Nothing) + sch :: Int -> Int -> [Int] -> Array (Int,Int) (Maybe (Spray a)) -> Spray a + sch m k nu arr + | null nu || nu!!0 == 0 || m == 0 = unitSpray + | length nu > m && nu!!m > 0 = zeroSpray + | m == 1 = x!!0 ^**^ nu!!0 + | 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 :: Spray a -> Int -> Spray 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 nu'!!0 == 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/tests/Main.hs b/tests/Main.hs --- a/tests/Main.hs +++ b/tests/Main.hs @@ -1,65 +1,75 @@ -module Main where -import Data.Ratio -import Math.Algebra.Hspray -import Math.Algebra.Jack -import Math.Algebra.Jack.HypergeoPQ -import Math.Algebra.JackPol -import Math.HypergeoMatrix -import Test.Tasty ( defaultMain - , testGroup - ) -import Test.Tasty.HUnit ( assertEqual - , testCase - ) - -main :: IO () -main = defaultMain $ testGroup - - "Tests" - - [ testCase "jackPol" $ do - let jp = jackPol 2 [3, 1] (2 % 1) - v = evalSpray jp [1, 1] - assertEqual "" v (48 % 1) - - , testCase "jack" $ do - assertEqual "" (jack [1, 1] [3, 1] (2 % 1)) (48 % 1) - - , testCase "schurPol" $ do - let sp1 = schurPol 4 [4] - sp2 = schurPol 4 [3, 1] - sp3 = schurPol 4 [2, 2] - sp4 = schurPol 4 [2, 1, 1] - sp5 = schurPol 4 [1, 1, 1, 1] - v = evalSpray (sp1 ^+^ 3 *^ sp2 ^+^ 2 *^ sp3 ^+^ 3 *^ sp4 ^+^ sp5) [2, 2, 2, 2] - assertEqual "" v 4096 - - , testCase "schur" $ do - let sp1 = schur [1, 1, 1, 1] [4] - sp2 = schur [1, 1, 1, 1] [3, 1] - sp3 = schur [1, 1, 1, 1] [2, 2] - sp4 = schur [1, 1, 1, 1] [2, 1, 1] - sp5 = schur [1, 1, 1, 1] [1, 1, 1, 1] - assertEqual "" (sp1 + 3 * sp2 + 2 * sp3 + 3 * sp4 + sp5) 256 - - , testCase "zonalPol" $ do - let zp1 = zonalPol 4 [3] :: Spray Rational - zp2 = zonalPol 4 [2, 1] :: Spray Rational - zp3 = zonalPol 4 [1, 1, 1] :: Spray Rational - v = evalSpray (zp1 ^+^ zp2 ^+^ zp3) [2, 2, 2, 2] - assertEqual "" v 512 - - , testCase "zonal" $ do - let zp1 = zonal [2 % 1, 2 % 1, 2 % 1, 2 % 1] [3] - zp2 = zonal [2 % 1, 2 % 1, 2 % 1, 2 % 1] [2, 1] - zp3 = zonal [2 % 1, 2 % 1, 2 % 1, 2 % 1] [1, 1, 1] - assertEqual "" (zp1 + zp2 + zp3) 512 - - , testCase "hypergeometric function" $ do - let a = [1 % 1, 2 % 1] - b = [3 % 1] - x = [1 % 5, 1 % 2] - h1 = hypergeoPQ 10 a b x - h2 <- hypergeomat 10 2 a b x - assertEqual "" h1 h2 - ] +module Main where +import Data.Ratio ( (%) ) +import Math.Algebra.Hspray ( (^+^), (*^), Spray + , evalSpray, isSymmetricSpray ) +import Math.Algebra.Jack ( jack, zonal, schur ) +import Math.Algebra.Jack.HypergeoPQ ( hypergeoPQ ) +import Math.Algebra.JackPol ( zonalPol, jackPol, schurPol ) +import Math.HypergeoMatrix ( hypergeomat ) +import Test.Tasty ( defaultMain + , testGroup + ) +import Test.Tasty.HUnit ( assertEqual + , assertBool + , testCase + ) + +main :: IO () +main = defaultMain $ testGroup + + "Tests" + + [ testCase "jackPol" $ do + let jp = jackPol 2 [3, 1] (2 % 1) :: Spray Rational + v = evalSpray jp [1, 1] + assertEqual "" v (48 % 1) + + , testCase "jackPol is symmetric" $ do + let jp = jackPol 3 [3, 2, 1] (2 % 1) :: Spray Rational + assertBool "" (isSymmetricSpray jp) + + , testCase "jack" $ do + assertEqual "" (jack [1, 1] [3, 1] (2 % 1)) (48 % 1 :: Rational) + + , testCase "schurPol" $ do + let sp1 = schurPol 4 [4] + sp2 = schurPol 4 [3, 1] + sp3 = schurPol 4 [2, 2] + sp4 = schurPol 4 [2, 1, 1] + sp5 = schurPol 4 [1, 1, 1, 1] :: Spray Int + v = evalSpray (sp1 ^+^ 3 *^ sp2 ^+^ 2 *^ sp3 ^+^ 3 *^ sp4 ^+^ sp5) [2, 2, 2, 2] + assertEqual "" v 4096 + + , testCase "schurPol is symmetric" $ do + let sp = schurPol 3 [3, 2, 1] :: Spray Rational + assertBool "" (isSymmetricSpray sp) + + , testCase "schur" $ do + let sp1 = schur [1, 1, 1, 1] [4] + sp2 = schur [1, 1, 1, 1] [3, 1] + sp3 = schur [1, 1, 1, 1] [2, 2] + sp4 = schur [1, 1, 1, 1] [2, 1, 1] + sp5 = schur [1, 1, 1, 1] [1, 1, 1, 1] :: Int + assertEqual "" (sp1 + 3 * sp2 + 2 * sp3 + 3 * sp4 + sp5) 256 + + , testCase "zonalPol" $ do + let zp1 = zonalPol 4 [3] :: Spray Rational + zp2 = zonalPol 4 [2, 1] :: Spray Rational + zp3 = zonalPol 4 [1, 1, 1] :: Spray Rational + v = evalSpray (zp1 ^+^ zp2 ^+^ zp3) [2, 2, 2, 2] + assertEqual "" v 512 + + , testCase "zonal" $ do + let zp1 = zonal [2 % 1, 2 % 1, 2 % 1, 2 % 1] [3] + zp2 = zonal [2 % 1, 2 % 1, 2 % 1, 2 % 1] [2, 1] + zp3 = zonal [2 % 1, 2 % 1, 2 % 1, 2 % 1] [1, 1, 1] :: Rational + assertEqual "" (zp1 + zp2 + zp3) 512 + + , testCase "hypergeometric function" $ do + let a = [1 % 1, 2 % 1] + b = [3 % 1] + x = [1 % 5, 1 % 2] + h1 = hypergeoPQ 10 a b x :: Rational + h2 <- hypergeomat 10 2 a b x + assertEqual "" h1 h2 + ]