-- |-- Module: Math.NumberTheory.ArithmeticFunctions.Class-- Copyright: (c) 2016 Andrew Lelechenko-- Licence: MIT-- Maintainer: Andrew Lelechenko <andrew.lelechenko@gmail.com>---- Generic type for arithmetic functions over arbitrary unique-- factorisation domains.--{-# LANGUAGE GADTs #-}moduleMath.NumberTheory.ArithmeticFunctions.Class(ArithmeticFunction (..),runFunction ,runFunctionOnFactors )whereimportControl.ApplicativeimportPreludehiding(Applicative(..))importMath.NumberTheory.Primes -- | A typical arithmetic function operates on the canonical factorisation of-- a number into prime's powers and consists of two rules. The first one-- determines the values of the function on the powers of primes. The second-- one determines how to combine these values into final result.---- In the following definition the first argument is the function on prime's-- powers, the monoid instance determines a rule of combination (typically-- 'Data.Semigroup.Product' or 'Data.Semigroup.Sum'), and the second argument is convenient for unwrapping-- (typically, 'Data.Semigroup.getProduct' or 'Data.Semigroup.getSum').dataArithmeticFunction n a whereArithmeticFunction ::Monoidm =>(Prime n ->Word->m )->(m ->a )->ArithmeticFunction n a -- | Convert to a function. The value on 0 is undefined.runFunction ::UniqueFactorisation n =>ArithmeticFunction n a ->n ->a runFunction :: forall n a.
UniqueFactorisation n =>
ArithmeticFunction n a -> n -> a
runFunction ArithmeticFunction n a
f =ArithmeticFunction n a -> [(Prime n, Word)] -> a
forall n a. ArithmeticFunction n a -> [(Prime n, Word)] -> a
runFunctionOnFactors ArithmeticFunction n a
f ([(Prime n, Word)] -> a) -> (n -> [(Prime n, Word)]) -> n -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
.n -> [(Prime n, Word)]
forall a. UniqueFactorisation a => a -> [(Prime a, Word)]
factorise -- | Convert to a function on prime factorisation.runFunctionOnFactors ::ArithmeticFunction n a ->[(Prime n ,Word)]->a runFunctionOnFactors :: forall n a. ArithmeticFunction n a -> [(Prime n, Word)] -> a
runFunctionOnFactors (ArithmeticFunction Prime n -> Word -> m
f m -> a
g )=m -> a
g (m -> a) -> ([(Prime n, Word)] -> m) -> [(Prime n, Word)] -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
.[m] -> m
forall a. Monoid a => [a] -> a
mconcat([m] -> m) -> ([(Prime n, Word)] -> [m]) -> [(Prime n, Word)] -> m
forall b c a. (b -> c) -> (a -> b) -> a -> c
.((Prime n, Word) -> m) -> [(Prime n, Word)] -> [m]
forall a b. (a -> b) -> [a] -> [b]
map((Prime n -> Word -> m) -> (Prime n, Word) -> m
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurryPrime n -> Word -> m
f )instanceFunctor(ArithmeticFunction n )wherefmap :: forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
fmapa -> b
f (ArithmeticFunction Prime n -> Word -> m
g m -> a
h )=(Prime n -> Word -> m) -> (m -> b) -> ArithmeticFunction n b
forall m n a.
Monoid m =>
(Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a
ArithmeticFunction Prime n -> Word -> m
g (a -> b
f (a -> b) -> (m -> a) -> m -> b
forall b c a. (b -> c) -> (a -> b) -> a -> c
.m -> a
h )instanceApplicative(ArithmeticFunction n )wherepure :: forall a. a -> ArithmeticFunction n a
purea
x =(Prime n -> Word -> ()) -> (() -> a) -> ArithmeticFunction n a
forall m n a.
Monoid m =>
(Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a
ArithmeticFunction (\Prime n
_Word
_->())(a -> () -> a
forall a b. a -> b -> a
consta
x )(ArithmeticFunction Prime n -> Word -> m
f1 m -> a -> b
g1 )<*> :: forall a b.
ArithmeticFunction n (a -> b)
-> ArithmeticFunction n a -> ArithmeticFunction n b
<*>(ArithmeticFunction Prime n -> Word -> m
f2 m -> a
g2 )=(Prime n -> Word -> (m, m))
-> ((m, m) -> b) -> ArithmeticFunction n b
forall m n a.
Monoid m =>
(Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a
ArithmeticFunction (\Prime n
p Word
k ->(Prime n -> Word -> m
f1 Prime n
p Word
k ,Prime n -> Word -> m
f2 Prime n
p Word
k ))(\(m
a1 ,m
a2 )->m -> a -> b
g1 m
a1 (m -> a
g2 m
a2 ))instanceSemigroupa =>Semigroup(ArithmeticFunction n a )where<> :: ArithmeticFunction n a
-> ArithmeticFunction n a -> ArithmeticFunction n a
(<>)=(a -> a -> a)
-> ArithmeticFunction n a
-> ArithmeticFunction n a
-> ArithmeticFunction n a
forall a b c.
(a -> b -> c)
-> ArithmeticFunction n a
-> ArithmeticFunction n b
-> ArithmeticFunction n c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2a -> a -> a
forall a. Semigroup a => a -> a -> a
(<>)instanceMonoida =>Monoid(ArithmeticFunction n a )wheremempty :: ArithmeticFunction n a
mempty=a -> ArithmeticFunction n a
forall a. a -> ArithmeticFunction n a
forall (f :: * -> *) a. Applicative f => a -> f a
purea
forall a. Monoid a => a
mempty-- | Factorisation is expensive, so it is better to avoid doing it twice.-- Write 'runFunction (f + g) n' instead of 'runFunction f n + runFunction g n'.instanceNuma =>Num(ArithmeticFunction n a )wherefromInteger :: Integer -> ArithmeticFunction n a
fromInteger=a -> ArithmeticFunction n a
forall a. a -> ArithmeticFunction n a
forall (f :: * -> *) a. Applicative f => a -> f a
pure(a -> ArithmeticFunction n a)
-> (Integer -> a) -> Integer -> ArithmeticFunction n a
forall b c a. (b -> c) -> (a -> b) -> a -> c
.Integer -> a
forall a. Num a => Integer -> a
fromIntegernegate :: ArithmeticFunction n a -> ArithmeticFunction n a
negate=(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Num a => a -> a
negatesignum :: ArithmeticFunction n a -> ArithmeticFunction n a
signum =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Num a => a -> a
signumabs :: ArithmeticFunction n a -> ArithmeticFunction n a
abs =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Num a => a -> a
abs+ :: ArithmeticFunction n a
-> ArithmeticFunction n a -> ArithmeticFunction n a
(+) =(a -> a -> a)
-> ArithmeticFunction n a
-> ArithmeticFunction n a
-> ArithmeticFunction n a
forall a b c.
(a -> b -> c)
-> ArithmeticFunction n a
-> ArithmeticFunction n b
-> ArithmeticFunction n c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2a -> a -> a
forall a. Num a => a -> a -> a
(+)(-)=(a -> a -> a)
-> ArithmeticFunction n a
-> ArithmeticFunction n a
-> ArithmeticFunction n a
forall a b c.
(a -> b -> c)
-> ArithmeticFunction n a
-> ArithmeticFunction n b
-> ArithmeticFunction n c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2(-)* :: ArithmeticFunction n a
-> ArithmeticFunction n a -> ArithmeticFunction n a
(*) =(a -> a -> a)
-> ArithmeticFunction n a
-> ArithmeticFunction n a
-> ArithmeticFunction n a
forall a b c.
(a -> b -> c)
-> ArithmeticFunction n a
-> ArithmeticFunction n b
-> ArithmeticFunction n c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2a -> a -> a
forall a. Num a => a -> a -> a
(*)instanceFractionala =>Fractional(ArithmeticFunction n a )wherefromRational :: Rational -> ArithmeticFunction n a
fromRational=a -> ArithmeticFunction n a
forall a. a -> ArithmeticFunction n a
forall (f :: * -> *) a. Applicative f => a -> f a
pure(a -> ArithmeticFunction n a)
-> (Rational -> a) -> Rational -> ArithmeticFunction n a
forall b c a. (b -> c) -> (a -> b) -> a -> c
.Rational -> a
forall a. Fractional a => Rational -> a
fromRationalrecip :: ArithmeticFunction n a -> ArithmeticFunction n a
recip =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Fractional a => a -> a
recip/ :: ArithmeticFunction n a
-> ArithmeticFunction n a -> ArithmeticFunction n a
(/) =(a -> a -> a)
-> ArithmeticFunction n a
-> ArithmeticFunction n a
-> ArithmeticFunction n a
forall a b c.
(a -> b -> c)
-> ArithmeticFunction n a
-> ArithmeticFunction n b
-> ArithmeticFunction n c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2a -> a -> a
forall a. Fractional a => a -> a -> a
(/)instanceFloatinga =>Floating(ArithmeticFunction n a )wherepi :: ArithmeticFunction n a
pi =a -> ArithmeticFunction n a
forall a. a -> ArithmeticFunction n a
forall (f :: * -> *) a. Applicative f => a -> f a
purea
forall a. Floating a => a
piexp :: ArithmeticFunction n a -> ArithmeticFunction n a
exp =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
explog :: ArithmeticFunction n a -> ArithmeticFunction n a
log =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
logsin :: ArithmeticFunction n a -> ArithmeticFunction n a
sin =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
sincos :: ArithmeticFunction n a -> ArithmeticFunction n a
cos =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
cosasin :: ArithmeticFunction n a -> ArithmeticFunction n a
asin =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
asinacos :: ArithmeticFunction n a -> ArithmeticFunction n a
acos =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
acosatan :: ArithmeticFunction n a -> ArithmeticFunction n a
atan =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
atansinh :: ArithmeticFunction n a -> ArithmeticFunction n a
sinh =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
sinhcosh :: ArithmeticFunction n a -> ArithmeticFunction n a
cosh =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
coshasinh :: ArithmeticFunction n a -> ArithmeticFunction n a
asinh =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
asinhacosh :: ArithmeticFunction n a -> ArithmeticFunction n a
acosh =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
acoshatanh :: ArithmeticFunction n a -> ArithmeticFunction n a
atanh =(a -> a) -> ArithmeticFunction n a -> ArithmeticFunction n a
forall a b.
(a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmapa -> a
forall a. Floating a => a -> a
atanh