-- | -- Module: Math.NumberTheory.Recurrences.BilinearTests -- Copyright: (c) 2016 Andrew Lelechenko -- Licence: MIT -- Maintainer: Andrew Lelechenko -- -- Tests for Math.NumberTheory.Recurrences.Bilinear -- {-# OPTIONS_GHC -fno-warn-type-defaults #-} module Math.NumberTheory.Recurrences.BilinearTests ( testSuite ) where import Test.Tasty import Test.Tasty.HUnit import Data.Ratio import Math.NumberTheory.Recurrences.Bilinear (bernoulli, binomial, euler, eulerian1, eulerian2, eulerPolyAt1, lah, stirling1, stirling2) import Math.NumberTheory.TestUtils binomialProperty1 :: NonNegative Int -> Bool binomialProperty1 (NonNegative i) = length (binomial !! i) == i + 1 binomialProperty2 :: NonNegative Int -> Bool binomialProperty2 (NonNegative i) = binomial !! i !! 0 == 1 binomialProperty3 :: NonNegative Int -> Bool binomialProperty3 (NonNegative i) = binomial !! i !! i == 1 binomialProperty4 :: Positive Int -> Positive Int -> Bool binomialProperty4 (Positive i) (Positive j) = j >= i || binomial !! i !! j == binomial !! (i - 1) !! (j - 1) + binomial !! (i - 1) !! j stirling1Property1 :: NonNegative Int -> Bool stirling1Property1 (NonNegative i) = length (stirling1 !! i) == i + 1 stirling1Property2 :: NonNegative Int -> Bool stirling1Property2 (NonNegative i) = stirling1 !! i !! 0 == if i == 0 then 1 else 0 stirling1Property3 :: NonNegative Int -> Bool stirling1Property3 (NonNegative i) = stirling1 !! i !! i == 1 stirling1Property4 :: Positive Int -> Positive Int -> Bool stirling1Property4 (Positive i) (Positive j) = j >= i || stirling1 !! i !! j == stirling1 !! (i - 1) !! (j - 1) + (toInteger i - 1) * stirling1 !! (i - 1) !! j stirling2Property1 :: NonNegative Int -> Bool stirling2Property1 (NonNegative i) = length (stirling2 !! i) == i + 1 stirling2Property2 :: NonNegative Int -> Bool stirling2Property2 (NonNegative i) = stirling2 !! i !! 0 == if i == 0 then 1 else 0 stirling2Property3 :: NonNegative Int -> Bool stirling2Property3 (NonNegative i) = stirling2 !! i !! i == 1 stirling2Property4 :: Positive Int -> Positive Int -> Bool stirling2Property4 (Positive i) (Positive j) = j >= i || stirling2 !! i !! j == stirling2 !! (i - 1) !! (j - 1) + toInteger j * stirling2 !! (i - 1) !! j lahProperty1 :: NonNegative Int -> Bool lahProperty1 (NonNegative i) = length (lah !! i) == i + 1 lahProperty2 :: NonNegative Int -> Bool lahProperty2 (NonNegative i) = lah !! i !! 0 == product [1 .. i+1] lahProperty3 :: NonNegative Int -> Bool lahProperty3 (NonNegative i) = lah !! i !! i == 1 lahProperty4 :: Positive Int -> Positive Int -> Bool lahProperty4 (Positive i) (Positive j) = j >= i || lah !! i !! j == sum [ stirling1 !! (i + 1) !! k * stirling2 !! k !! (j + 1) | k <- [j + 1 .. i + 1] ] eulerian1Property1 :: NonNegative Int -> Bool eulerian1Property1 (NonNegative i) = length (eulerian1 !! i) == i eulerian1Property2 :: Positive Int -> Bool eulerian1Property2 (Positive i) = eulerian1 !! i !! 0 == 1 eulerian1Property3 :: Positive Int -> Bool eulerian1Property3 (Positive i) = eulerian1 !! i !! (i - 1) == 1 eulerian1Property4 :: Positive Int -> Positive Int -> Bool eulerian1Property4 (Positive i) (Positive j) = j >= i - 1 || eulerian1 !! i !! j == (toInteger $ i - j) * eulerian1 !! (i - 1) !! (j - 1) + (toInteger j + 1) * eulerian1 !! (i - 1) !! j eulerian2Property1 :: NonNegative Int -> Bool eulerian2Property1 (NonNegative i) = length (eulerian2 !! i) == i eulerian2Property2 :: Positive Int -> Bool eulerian2Property2 (Positive i) = eulerian2 !! i !! 0 == 1 eulerian2Property3 :: Positive Int -> Bool eulerian2Property3 (Positive i) = eulerian2 !! i !! (i - 1) == product [1 .. toInteger i] eulerian2Property4 :: Positive Int -> Positive Int -> Bool eulerian2Property4 (Positive i) (Positive j) = j >= i - 1 || eulerian2 !! i !! j == (toInteger $ 2 * i - j - 1) * eulerian2 !! (i - 1) !! (j - 1) + (toInteger j + 1) * eulerian2 !! (i - 1) !! j bernoulliSpecialCase1 :: Assertion bernoulliSpecialCase1 = assertEqual "B_0 = 1" (bernoulli !! 0) 1 bernoulliSpecialCase2 :: Assertion bernoulliSpecialCase2 = assertEqual "B_1 = -1/2" (bernoulli !! 1) (- 1 % 2) bernoulliProperty1 :: NonNegative Int -> Bool bernoulliProperty1 (NonNegative m) = case signum (bernoulli !! m) of 1 -> m == 0 || m `mod` 4 == 2 0 -> m /= 1 && odd m -1 -> m == 1 || (m /= 0 && m `rem` 4 == 0) _ -> False bernoulliProperty2 :: NonNegative Int -> Bool bernoulliProperty2 (NonNegative m) = bernoulli !! m == (if m == 0 then 1 else 0) - sum [ bernoulli !! k * (binomial !! m !! k % (toInteger $ m - k + 1)) | k <- [0 .. m - 1] ] -- | For every odd positive integer @n@, @E_n@ is @0@. eulerProperty1 :: Positive Int -> Bool eulerProperty1 (Positive n) = euler !! (2 * n - 1) == 0 -- | Every positive even index produces a negative result. eulerProperty2 :: NonNegative Int -> Bool eulerProperty2 (NonNegative n) = euler !! (2 + 4 * n) < 0 -- | The Euler number sequence is https://oeis.org/A122045 eulerSpecialCase1 :: Assertion eulerSpecialCase1 = assertEqual "euler" (take 20 euler) [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0, -199360981, 0, 19391512145, 0, -2404879675441, 0] -- | For any even positive integer @n@, @E_n(1)@ is @0@. eulerPAt1Property1 :: Positive Int -> Bool eulerPAt1Property1 (Positive n) = (eulerPolyAt1 !! (2 * n)) == 0 -- | The numerators in this sequence are from https://oeis.org/A198631 while the -- denominators are from https://oeis.org/A006519. eulerPAt1SpecialCase1 :: Assertion eulerPAt1SpecialCase1 = assertEqual "eulerPolyAt1" (take 20 eulerPolyAt1) (zipWith (%) [1, 1, 0, -1, 0, 1, 0, -17, 0, 31, 0, -691, 0, 5461, 0, -929569, 0, 3202291, 0, -221930581] [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4]) testSuite :: TestTree testSuite = testGroup "Bilinear" [ testGroup "binomial" [ testSmallAndQuick "shape" binomialProperty1 , testSmallAndQuick "left side" binomialProperty2 , testSmallAndQuick "right side" binomialProperty3 , testSmallAndQuick "recurrency" binomialProperty4 ] , testGroup "stirling1" [ testSmallAndQuick "shape" stirling1Property1 , testSmallAndQuick "left side" stirling1Property2 , testSmallAndQuick "right side" stirling1Property3 , testSmallAndQuick "recurrency" stirling1Property4 ] , testGroup "stirling2" [ testSmallAndQuick "shape" stirling2Property1 , testSmallAndQuick "left side" stirling2Property2 , testSmallAndQuick "right side" stirling2Property3 , testSmallAndQuick "recurrency" stirling2Property4 ] , testGroup "lah" [ testSmallAndQuick "shape" lahProperty1 , testSmallAndQuick "left side" lahProperty2 , testSmallAndQuick "right side" lahProperty3 , testSmallAndQuick "zip stirlings" lahProperty4 ] , testGroup "eulerian1" [ testSmallAndQuick "shape" eulerian1Property1 , testSmallAndQuick "left side" eulerian1Property2 , testSmallAndQuick "right side" eulerian1Property3 , testSmallAndQuick "recurrency" eulerian1Property4 ] , testGroup "eulerian2" [ testSmallAndQuick "shape" eulerian2Property1 , testSmallAndQuick "left side" eulerian2Property2 , testSmallAndQuick "right side" eulerian2Property3 , testSmallAndQuick "recurrency" eulerian2Property4 ] , testGroup "bernoulli" [ testCase "B_0" bernoulliSpecialCase1 , testCase "B_1" bernoulliSpecialCase2 , testSmallAndQuick "sign" bernoulliProperty1 , testSmallAndQuick "recursive definition" bernoulliProperty2 ] , testGroup "Euler numbers" [ testCase "First 20 elements of E_n are correct" eulerSpecialCase1 , testSmallAndQuick "E_n with n odd is 0" eulerProperty1 , testSmallAndQuick "E_n for n in [2,6,8,12..] is negative" eulerProperty2 ] , testGroup "Euler Polynomial of order N evaluated at 1" [ testCase "First 20 elements of E_n(1) are correct" eulerPAt1SpecialCase1 , testSmallAndQuick "E_n(1) with n in [2,4,6..] is 0" eulerPAt1Property1 ] ]