arithmoi-0.8.0.0: test-suite/Math/NumberTheory/Recurrencies/BilinearTests.hs
-- |
-- Module: Math.NumberTheory.Recurrencies.BilinearTests
-- Copyright: (c) 2016 Andrew Lelechenko
-- Licence: MIT
-- Maintainer: Andrew Lelechenko <andrew.lelechenko@gmail.com>
-- Stability: Provisional
--
-- Tests for Math.NumberTheory.Recurrencies.Bilinear
--
{-# OPTIONS_GHC -fno-warn-type-defaults #-}
module Math.NumberTheory.Recurrencies.BilinearTests
( testSuite
) where
import Test.Tasty
import Test.Tasty.HUnit
import Data.Ratio
import Math.NumberTheory.Recurrencies.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 `mod` 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
]
]