module Main ( main ) where
import qualified Data.Map.Strict as DM
import Data.Ratio ( (%) )
import Math.Algebra.Hspray ( FunctionLike (..)
, Spray, QSpray
, lone, qlone
, evalSpray
, evalParametricSpray'
, substituteParameters
, canCoerceToSimpleParametricSpray
, isHomogeneousSpray
, asRatioOfSprays
, (%//%)
, (/^)
)
import qualified Math.Algebra.Hspray as Hspray
import Math.Algebra.Jack ( schur, skewSchur
, jack', zonal' )
import Math.Algebra.Jack.HypergeoPQ ( hypergeoPQ )
import Math.Algebra.SymmetricPolynomials ( isSymmetricSpray
, prettySymmetricParametricQSpray
, laplaceBeltrami
, calogeroSutherland
, hallInnerProduct
, hallInnerProduct''
, symbolicHallInnerProduct
, symbolicHallInnerProduct''
, psPolynomial
, psCombination
)
import Math.Algebra.JackPol ( zonalPol, zonalPol', jackPol'
, schurPol, schurPol', skewSchurPol' )
import Math.Algebra.JackSymbolicPol ( jackSymbolicPol' )
import Math.Combinat.Classes ( HasDuality (..) )
import Math.Combinat.Partitions.Integer ( toPartition, fromPartition )
import Math.HypergeoMatrix ( hypergeomat )
import Test.Tasty ( defaultMain
, testGroup
)
import Test.Tasty.HUnit ( assertEqual
, assertBool
, testCase
)
main :: IO ()
main = defaultMain $ testGroup
"Tests"
[
testCase "jackSymbolicPol J" $ do
let jp = jackSymbolicPol' 3 [3, 1] 'J'
v = evalParametricSpray' jp [2] [-3, 4, 5]
assertEqual "" v 1488
, testCase "jackSymbolicPol J has polynomial coefficients only" $ do
let jp = jackSymbolicPol' 3 [3, 1] 'J'
assertBool "" (canCoerceToSimpleParametricSpray jp)
, testCase "jackSymbolicPol C" $ do
let jp = jackSymbolicPol' 4 [3, 1] 'C'
zp = zonalPol 4 [3, 1] :: Spray Rational
p = substituteParameters jp [2]
assertEqual "" zp p
, testCase "jackSymbolicPol Q is symmetric" $ do
let jp = jackSymbolicPol' 4 [3, 1] 'Q'
assertBool "" (isSymmetricSpray jp)
, testCase "jackSymbolicPol P is symmetric" $ do
let jp = jackSymbolicPol' 5 [3, 2, 1] 'P'
assertBool "" (isSymmetricSpray jp)
, testCase "prettySymmetricParametricQSpray - jack J" $ do
let jp = jackSymbolicPol' 3 [3, 1, 1] 'J'
assertEqual ""
(prettySymmetricParametricQSpray ["a"] jp)
("{ [ 4*a^2 + 10*a + 6 ] }*M[3,1,1] + { [ 8*a + 12 ] }*M[2,2,1]")
, testCase "prettySymmetricParametricQSpray - jack C" $ do
let jp = jackSymbolicPol' 3 [3, 1, 1] 'C'
assertEqual ""
(prettySymmetricParametricQSpray ["a"] jp)
("{ [ 20*a^2 ] %//% [ a^2 + (5/3)*a + (2/3) ] }*M[3,1,1] + { [ 40*a^2 ] %//% [ a^3 + (8/3)*a^2 + (7/3)*a + (2/3) ] }*M[2,2,1]")
, testCase "jackPol" $ do
let jp = jackPol' 2 [3, 1] (2 % 1) 'J'
v = evalSpray jp [1, 1]
assertEqual "" v 48
, testCase "jackPol is homogeneous" $ do
let jp = jackPol' 4 [3, 1] (2 % 1) 'J'
assertEqual "" (isHomogeneousSpray jp) (True, Just 4)
, testCase "jackPol is symmetric (Groebner)" $ do
let jp = jackPol' 3 [3, 2, 1] (2 % 1) 'J'
assertBool "" (Hspray.isSymmetricSpray jp)
, testCase "jack" $ do
assertEqual "" (jack' [1, 1] [3, 1] (2 % 1) 'J') 48
, testCase "Jack polynomial is eigenpolynomial for Laplace-Beltrami" $ do
let
alpha = 3 % 1
lambda = [2, 2]
b :: [Int] -> Rational
b mu = toRational $ sum $ zipWith (*) mu [0 .. ]
eigenvalue :: Int -> Rational -> [Int] -> Rational
eigenvalue n a mu =
let mu' = fromPartition $ dual (toPartition mu) in
a * b mu' - b mu + toRational ((n-1) * sum mu)
ev = eigenvalue 4 alpha lambda
jp = jackPol' 4 lambda alpha 'J'
jp' = laplaceBeltrami alpha jp
assertEqual "" jp' (ev *^ jp)
, testCase "Jack polynomial is eigenpolynomial for Calogero-Sutherland" $ do
let
eigenval :: Int -> Rational -> [Int] -> Rational
eigenval n a mu = sum $ map
(\i -> let r = toRational (mu !! (i-1)) in
a/2 * r*r + ((toRational $ n + 1 - 2*i) / 2) * r)
[1 .. length mu]
alpha = 3 % 4
lambda = [3, 1]
ev = eigenval 4 alpha lambda
jp = jackPol' 4 lambda alpha 'J'
jp' = calogeroSutherland alpha jp
assertEqual "" jp' (ev *^ jp)
, testCase "Jack P-polynomial for alpha=1 is Schur polynomial" $ do
let
n = 5
lambda = [5, 4, 3, 2, 1]
jp = jackPol' n lambda 1 'P'
sp = schurPol' n lambda
assertEqual "" jp sp
, 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 (Groebner)" $ do
let sp = schurPol' 3 [3, 2, 1]
assertBool "" (Hspray.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 "skewSchur" $ do
let x = [2, 3, 4] :: [Int]
assertEqual "" (skewSchur x [3, 2, 1] [1, 1]) 1890
, testCase "skewSchurPol" $ do
let x = lone 1 :: Spray Rational
y = lone 2 :: Spray Rational
z = lone 3 :: Spray Rational
skp = skewSchurPol' 3 [2, 2, 1] [1, 1]
p = x^**^2 ^*^ y ^+^ x^**^2 ^*^ z ^+^ x ^*^ y^**^2
^+^ 3 *^ (x ^*^ y ^*^ z) ^+^ x ^*^ z^**^2
^+^ y^**^2 ^*^ z ^+^ y ^*^ z^**^2
assertEqual "" skp p
, testCase "skewSchurPol is symmetric (Groebner)" $ do
let skp = skewSchurPol' 3 [3, 2, 1] [1, 1]
assertBool "" (Hspray.isSymmetricSpray skp)
, testCase "zonalPol" $ do
let zp1 = zonalPol' 4 [3]
zp2 = zonalPol' 4 [2, 1]
zp3 = zonalPol' 4 [1, 1, 1]
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 :: Rational
h2 <- hypergeomat 10 2 a b x
assertEqual "" h1 h2
, testCase "Hall inner product" $ do
let
alpha = 2
poly1 = psPolynomial 4 [4]
poly2 = psPolynomial 4 [3, 1]
poly3 = psPolynomial 4 [2, 2]
poly4 = psPolynomial 4 [2, 1, 1]
poly5 = psPolynomial 4 [1, 1, 1, 1]
h1 = hallInnerProduct poly1 poly1 alpha
h2 = hallInnerProduct poly2 poly2 alpha
h3 = hallInnerProduct poly3 poly3 alpha
h4 = hallInnerProduct poly4 poly4 alpha
h5 = hallInnerProduct poly5 poly5 alpha
pow :: Rational -> Int -> Rational
pow = (^)
assertEqual ""
(h1, h2, h3, h4, h5)
(
4 * alpha
, 3 * pow alpha 2
, 8 * pow alpha 2
, 4 * pow alpha 3
, 24 * pow alpha 4
)
, testCase "Hall inner product of Schur polynomials" $ do
let
sp1 = schurPol 7 [4, 2, 1] :: Spray Int
sp2 = schurPol 7 [2, 2, 2, 1] :: Spray Int
h1 = hallInnerProduct'' sp1 sp1 1
h2 = hallInnerProduct'' sp2 sp2 1
h12 = hallInnerProduct'' sp1 sp2 1
assertEqual "" (h1, h2, h12) (1, 1, 0)
, testCase "Symbolic Hall inner product" $ do
let
poly1 = psPolynomial 4 [4] :: QSpray
poly2 = psPolynomial 4 [3, 1] :: QSpray
poly3 = psPolynomial 4 [2, 2] :: QSpray
poly4 = psPolynomial 4 [2, 1, 1] :: QSpray
poly5 = psPolynomial 4 [1, 1, 1, 1] :: QSpray
h1 = symbolicHallInnerProduct poly1 poly1
h2 = symbolicHallInnerProduct poly2 poly2
h3 = symbolicHallInnerProduct poly3 poly3
h4 = symbolicHallInnerProduct poly4 poly4
h5 = symbolicHallInnerProduct poly5 poly5
alpha = qlone 1
assertEqual ""
(h1, h2, h3, h4, h5)
(
4 *^ alpha
, 3 *^ alpha^**^2
, 8 *^ alpha^**^2
, 4 *^ alpha^**^3
, 24 *^ alpha^**^4
)
, testCase "Symbolic Hall inner product of Schur polynomials" $ do
let
sp1 = schurPol 7 [4, 2, 1] :: Spray Int
sp2 = schurPol 7 [2, 2, 2, 1] :: Spray Int
h1 = evaluateAt [1] (symbolicHallInnerProduct'' sp1 sp1)
h2 = evaluateAt [1] (symbolicHallInnerProduct'' sp2 sp2)
h12 = evaluateAt [1] (symbolicHallInnerProduct'' sp1 sp2)
assertEqual "" (h1, h2, h12) (1, 1, 0)
, testCase "Symbolic Hall inner product with parametric sprays" $ do
let
jp1 = jackSymbolicPol' 3 [1, 1, 1] 'P'
jp2 = jackSymbolicPol' 3 [2, 1] 'P'
jp3 = jackSymbolicPol' 3 [3] 'P'
t = qlone 1
t' = asRatioOfSprays t
h1 = hallInnerProduct jp1 jp1 t'
h2 = hallInnerProduct jp2 jp2 t'
h3 = hallInnerProduct jp3 jp3 t'
assertEqual ""
(h1, h2, h3)
(
asRatioOfSprays (t^**^3 /^ 6 ^+^ t^**^2 /^ 2 ^+^ t /^ 3)
, (2*^t^**^3 ^+^ t^**^2) %//% (t <+ 2)
, (3*^t^**^3) %//% ((t^**^2 ^+^ (3%2)*^t) <+ (1%2))
)
, testCase "Power sum polynomial and power sum combination" $ do
let
psPoly = 3*^psPolynomial 4 [2, 1, 1] ^-^ psPolynomial 4 [2, 1] :: QSpray
psCombo = psCombination psPoly
assertEqual ""
psCombo
(
DM.fromList [([2, 1, 1], 3), ([2, 1], -1)]
)
]