Data.Number.Flint.NF.Fmpzi

Gaussian integers

data Fmpzi

Constructors

Fmpzi !(ForeignPtr CFmpzi)

Instances

Num Fmpzi

Defined in Data.Number.Flint.NF.Fmpzi.Instances

Methods

(+) :: Fmpzi -> Fmpzi -> Fmpzi

(-) :: Fmpzi -> Fmpzi -> Fmpzi

(*) :: Fmpzi -> Fmpzi -> Fmpzi

negate :: Fmpzi -> Fmpzi

abs :: Fmpzi -> Fmpzi

signum :: Fmpzi -> Fmpzi

fromInteger :: Integer -> Fmpzi

Show Fmpzi

Defined in Data.Number.Flint.NF.Fmpzi.Instances

Methods

showsPrec :: Int -> Fmpzi -> ShowS

show :: Fmpzi -> String

showList :: [Fmpzi] -> ShowS

Eq Fmpzi

Defined in Data.Number.Flint.NF.Fmpzi.Instances

Methods

(==) :: Fmpzi -> Fmpzi -> Bool

(/=) :: Fmpzi -> Fmpzi -> Bool

data CFmpzi

Constructors

CFmpzi CFmpz CFmpz

Instances

Storable CFmpzi

Defined in Data.Number.Flint.NF.Fmpzi.FFI

Methods

sizeOf :: CFmpzi -> Int

alignment :: CFmpzi -> Int

peekElemOff :: Ptr CFmpzi -> Int -> IO CFmpzi

pokeElemOff :: Ptr CFmpzi -> Int -> CFmpzi -> IO ()

peekByteOff :: Ptr b -> Int -> IO CFmpzi

pokeByteOff :: Ptr b -> Int -> CFmpzi -> IO ()

peek :: Ptr CFmpzi -> IO CFmpzi

poke :: Ptr CFmpzi -> CFmpzi -> IO ()

newFmpzi :: IO Fmpzi

Create Fmpzi.

newFmpzi_ :: CLong -> CLong -> IO Fmpzi

Create Fmpzi.

withFmpzi :: Fmpzi -> (Ptr CFmpzi -> IO a) -> IO (Fmpzi, a)

Use Fmpzi

withNewFmpzi :: (Ptr CFmpzi -> IO a) -> IO (Fmpzi, a)

Use new Fmpzi

withFmpziReal :: Fmpzi -> (Ptr b -> IO a) -> IO (Fmpzi, a)

Use real part of Fmpzi

withFmpziImag :: Storable a => Fmpzi -> (Ptr a -> IO a) -> IO (Fmpzi, a)

Use imaginary part of Fmpzi

Types

Basic manipulation

fmpzi_init :: Ptr CFmpzi -> IO ()

fmpzi_clear :: Ptr CFmpzi -> IO ()

fmpzi_swap :: Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_zero :: Ptr CFmpzi -> IO ()

fmpzi_one :: Ptr CFmpzi -> IO ()

fmpzi_set :: Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_set_si_si :: Ptr CFmpzi -> CLong -> CLong -> IO ()

Input and output

fmpzi_get_str :: Ptr CFmpzi -> IO CString

fmpzi_fprint :: Ptr CFile -> Ptr CFmpzi -> IO ()

fmpzi_print :: Ptr CFmpzi -> IO ()

Random number generation

fmpzi_randtest :: Ptr CFmpzi -> Ptr CFRandState -> CMpBitCnt -> IO ()

Properties

fmpzi_equal :: Ptr CFmpzi -> Ptr CFmpzi -> IO CInt

fmpzi_is_zero :: Ptr CFmpzi -> IO CInt

fmpzi_is_one :: Ptr CFmpzi -> IO CInt

Units

fmpzi_is_unit :: Ptr CFmpzi -> IO CInt

fmpzi_canonical_unit_i_pow :: Ptr CFmpzi -> IO CLong

fmpzi_canonicalise_unit :: Ptr CFmpzi -> Ptr CFmpzi -> IO ()

Norms

fmpzi_bits :: Ptr CFmpzi -> IO CLong

fmpzi_norm :: Ptr CFmpz -> Ptr CFmpzi -> IO ()

Arithmetic

fmpzi_conj :: Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_neg :: Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_add :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_sub :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_sqr :: Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_mul :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_pow_ui :: Ptr CFmpzi -> Ptr CFmpzi -> CULong -> IO ()

Division

fmpzi_divexact :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_divexact q x y

Sets q to the quotient of x and y, assuming that the division is exact.

fmpzi_divrem :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_divrem q r x y

Computes a quotient and remainder satisfying \(x = q y + r\) with \(N(r) \le N(y)/2\), with a canonical choice of remainder when breaking ties.

fmpzi_divrem_approx :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_divrem_approx q r x y

Computes a quotient and remainder satisfying \(x = q y + r\) with \(N(r) < N(y)\), with an implementation-defined, non-canonical choice of remainder.

fmpzi_remove_one_plus_i :: Ptr CFmpzi -> Ptr CFmpzi -> IO CLong

fmpzi_remove_one_plus_i res x

Divide x exactly by the largest possible power \((1+i)^k\) and return the exponent k.

GCD

fmpzi_gcd_euclidean :: Ptr CFmpzi -> Ptr CFmpzi -> Ptr CFmpzi -> IO ()

fmpzi_gcd_euclidean res x y

Computes the GCD of x and y. The result is in canonical unit form.

The euclidean version is a straightforward implementation of Euclid's algorithm. The euclidean_improved version is optimized by performing approximate divisions. The binary version uses a (1+i)-ary analog of the binary GCD algorithm for integers [Wei2000]. The shortest version finds the GCD as the shortest vector in a lattice. The default version chooses an algorithm automatically.