jackpolynomials 1.1.0.1 → 1.1.1.0
raw patch · 10 files changed
+1253/−1196 lines, 10 filesdep ~arraydep ~hspraydep ~hypergeomatrixsetup-changed
Dependency ranges changed: array, hspray, hypergeomatrix, ilist, lens, math-functions, numeric-prelude, tasty, tasty-hunit
Files
- CHANGELOG.md +26/−17
- LICENSE +674/−674
- README.md +42/−35
- Setup.hs +2/−2
- jackpolynomials.cabal +57/−47
- src/Math/Algebra/Jack.hs +136/−121
- src/Math/Algebra/Jack/HypergeoPQ.hs +33/−33
- src/Math/Algebra/Jack/Internal.hs +71/−77
- src/Math/Algebra/JackPol.hs +137/−125
- tests/Main.hs +75/−65
CHANGELOG.md view
@@ -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
LICENSE view
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Use with the GNU Affero General Public License. + + Notwithstanding any other provision of this License, you have +permission to link or combine any covered work with a work licensed +under version 3 of the GNU Affero General Public License into a single +combined work, and to convey the resulting work. The terms of this +License will continue to apply to the part which is the covered work, +but the special requirements of the GNU Affero General Public License, +section 13, concerning interaction through a network will apply to the +combination as such. + + 14. Revised Versions of this License. + + The Free Software Foundation may publish revised and/or new versions of +the GNU General Public License from time to time. Such new versions will +be similar in spirit to the present version, but may differ in detail to +address new problems or concerns. + + Each version is given a distinguishing version number. If the +Program specifies that a certain numbered version of the GNU General +Public License "or any later version" applies to it, you have the +option of following the terms and conditions either of that numbered +version or of any later version published by the Free Software +Foundation. If the Program does not specify a version number of the +GNU General Public License, you may choose any version ever published +by the Free Software Foundation. + + If the Program specifies that a proxy can decide which future +versions of the GNU General Public License can be used, that proxy's +public statement of acceptance of a version permanently authorizes you +to choose that version for the Program. + + Later license versions may give you additional or different +permissions. However, no additional obligations are imposed on any +author or copyright holder as a result of your choosing to follow a +later version. + + 15. Disclaimer of Warranty. + + THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY +APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT +HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY +OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, +THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR +PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM +IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF +ALL NECESSARY SERVICING, REPAIR OR CORRECTION. + + 16. Limitation of Liability. + + IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING +WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS +THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY +GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE +USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF +DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD +PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), +EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF +SUCH DAMAGES. + + 17. Interpretation of Sections 15 and 16. + + If the disclaimer of warranty and limitation of liability provided +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. + + <one line to give the program's name and a brief idea of what it does.> + Copyright (C) <year> <name of author> + + 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 + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License + along with this program. If not, see <https://www.gnu.org/licenses/>. + +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: + + <program> Copyright (C) <year> <name of author> + 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 +<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
@@ -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. <https://www.symmetricfunctions.com/jack.htm>.+# jackpolynomials + +*Jack, zonal, and Schur polynomials.* + +<!-- badges: start --> +[](https://github.com/stla/jackpolynomials/actions/workflows/Stack-lts.yml) +[](https://github.com/stla/jackpolynomials/actions/workflows/Stack-nightly.yml) +<!-- badges: end --> + +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. <https://www.symmetricfunctions.com/jack.htm>.
Setup.hs view
@@ -1,2 +1,2 @@-import Distribution.Simple-main = defaultMain+import Distribution.Simple +main = defaultMain
jackpolynomials.cabal view
@@ -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
src/Math/Algebra/Jack.hs view
@@ -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)
src/Math/Algebra/Jack/HypergeoPQ.hs view
@@ -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
src/Math/Algebra/Jack/Internal.hs view
@@ -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
src/Math/Algebra/JackPol.hs view
@@ -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)
tests/Main.hs view
@@ -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 + ]