Num.Complex

Complex

data Complex a

Complex numbers are an algebraic type.

For a complex number z, abs z is a number with the magnitude of z, but oriented in the positive real direction, whereas signum z has the phase of z, but unit magnitude.

The Foldable and Traversable instances traverse the real part first.

Instances

Monad Complex

Since: base-4.9.0.0

Defined in Data.Complex

Methods

(>>=) :: Complex a -> (a -> Complex b) -> Complex b

(>>) :: Complex a -> Complex b -> Complex b

return :: a -> Complex a

fail :: String -> Complex a

Functor Complex

Defined in Data.Complex

Methods

fmap :: (a -> b) -> Complex a -> Complex b

(<$) :: a -> Complex b -> Complex a

Applicative Complex

Since: base-4.9.0.0

Defined in Data.Complex

Methods

pure :: a -> Complex a

(<*>) :: Complex (a -> b) -> Complex a -> Complex b

liftA2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c

(*>) :: Complex a -> Complex b -> Complex b

(<*) :: Complex a -> Complex b -> Complex a

Foldable Complex

Defined in Data.Complex

Methods

fold :: Monoid m => Complex m -> m

foldMap :: Monoid m => (a -> m) -> Complex a -> m

foldr :: (a -> b -> b) -> b -> Complex a -> b

foldr' :: (a -> b -> b) -> b -> Complex a -> b

foldl :: (b -> a -> b) -> b -> Complex a -> b

foldl' :: (b -> a -> b) -> b -> Complex a -> b

foldr1 :: (a -> a -> a) -> Complex a -> a

foldl1 :: (a -> a -> a) -> Complex a -> a

toList :: Complex a -> [a]

null :: Complex a -> Bool

length :: Complex a -> Int

elem :: Eq a => a -> Complex a -> Bool

maximum :: Ord a => Complex a -> a

minimum :: Ord a => Complex a -> a

sum :: Num a => Complex a -> a

product :: Num a => Complex a -> a

Traversable Complex

Defined in Data.Complex

Methods

traverse :: Applicative f => (a -> f b) -> Complex a -> f (Complex b)

sequenceA :: Applicative f => Complex (f a) -> f (Complex a)

mapM :: Monad m => (a -> m b) -> Complex a -> m (Complex b)

sequence :: Monad m => Complex (m a) -> m (Complex a)

Distributive Complex

Defined in Data.Distributive

Methods

distribute :: Functor f => f (Complex a) -> Complex (f a)

collect :: Functor f => (a -> Complex b) -> f a -> Complex (f b)

distributeM :: Monad m => m (Complex a) -> Complex (m a)

collectM :: Monad m => (a -> Complex b) -> m a -> Complex (m b)

Representable Complex

Defined in Data.Functor.Rep

Associated Types

type Rep Complex :: *

Methods

tabulate :: (Rep Complex -> a) -> Complex a

index :: Complex a -> Rep Complex -> a

Hashable1 Complex

Defined in Data.Hashable.Class

Methods

liftHashWithSalt :: (Int -> a -> Int) -> Int -> Complex a -> Int

Apply Complex

Defined in Data.Functor.Bind.Class

Methods

(<.>) :: Complex (a -> b) -> Complex a -> Complex b

(.>) :: Complex a -> Complex b -> Complex b

(<.) :: Complex a -> Complex b -> Complex a

liftF2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c

Traversable1 Complex

Defined in Data.Semigroup.Traversable.Class

Methods

traverse1 :: Apply f => (a -> f b) -> Complex a -> f (Complex b)

sequence1 :: Apply f => Complex (f b) -> f (Complex b)

Foldable1 Complex

Defined in Data.Semigroup.Foldable.Class

Methods

fold1 :: Semigroup m => Complex m -> m

foldMap1 :: Semigroup m => (a -> m) -> Complex a -> m

toNonEmpty :: Complex a -> NonEmpty a

Bind Complex

Defined in Data.Functor.Bind.Class

Methods

(>>-) :: Complex a -> (a -> Complex b) -> Complex b

join :: Complex (Complex a) -> Complex a

Unbox a => Vector Vector (Complex a)

Defined in Data.Vector.Unboxed.Base

Methods

basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) (Complex a) -> m (Vector (Complex a))

basicUnsafeThaw :: PrimMonad m => Vector (Complex a) -> m (Mutable Vector (PrimState m) (Complex a))

basicLength :: Vector (Complex a) -> Int

basicUnsafeSlice :: Int -> Int -> Vector (Complex a) -> Vector (Complex a)

basicUnsafeIndexM :: Monad m => Vector (Complex a) -> Int -> m (Complex a)

basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) (Complex a) -> Vector (Complex a) -> m ()

elemseq :: Vector (Complex a) -> Complex a -> b -> b

Unbox a => MVector MVector (Complex a)

Defined in Data.Vector.Unboxed.Base

Methods

basicLength :: MVector s (Complex a) -> Int

basicUnsafeSlice :: Int -> Int -> MVector s (Complex a) -> MVector s (Complex a)

basicOverlaps :: MVector s (Complex a) -> MVector s (Complex a) -> Bool

basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) (Complex a))

basicInitialize :: PrimMonad m => MVector (PrimState m) (Complex a) -> m ()

basicUnsafeReplicate :: PrimMonad m => Int -> Complex a -> m (MVector (PrimState m) (Complex a))

basicUnsafeRead :: PrimMonad m => MVector (PrimState m) (Complex a) -> Int -> m (Complex a)

basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) (Complex a) -> Int -> Complex a -> m ()

basicClear :: PrimMonad m => MVector (PrimState m) (Complex a) -> m ()

basicSet :: PrimMonad m => MVector (PrimState m) (Complex a) -> Complex a -> m ()

basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) (Complex a) -> MVector (PrimState m) (Complex a) -> m ()

basicUnsafeMove :: PrimMonad m => MVector (PrimState m) (Complex a) -> MVector (PrimState m) (Complex a) -> m ()

basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) (Complex a) -> Int -> m (MVector (PrimState m) (Complex a))

Eq a => Eq (Complex a)

Defined in Data.Complex

Methods

(==) :: Complex a -> Complex a -> Bool

(/=) :: Complex a -> Complex a -> Bool

RealFloat a => Floating (Complex a)

Since: base-2.1

Defined in Data.Complex

Methods

pi :: Complex a

exp :: Complex a -> Complex a

log :: Complex a -> Complex a

sqrt :: Complex a -> Complex a

(**) :: Complex a -> Complex a -> Complex a

logBase :: Complex a -> Complex a -> Complex a

sin :: Complex a -> Complex a

cos :: Complex a -> Complex a

tan :: Complex a -> Complex a

asin :: Complex a -> Complex a

acos :: Complex a -> Complex a

atan :: Complex a -> Complex a

sinh :: Complex a -> Complex a

cosh :: Complex a -> Complex a

tanh :: Complex a -> Complex a

asinh :: Complex a -> Complex a

acosh :: Complex a -> Complex a

atanh :: Complex a -> Complex a

log1p :: Complex a -> Complex a

expm1 :: Complex a -> Complex a

log1pexp :: Complex a -> Complex a

log1mexp :: Complex a -> Complex a

RealFloat a => Fractional (Complex a)

Since: base-2.1

Defined in Data.Complex

Methods

(/) :: Complex a -> Complex a -> Complex a

recip :: Complex a -> Complex a

fromRational :: Rational -> Complex a

Data a => Data (Complex a)

Defined in Data.Complex

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Complex a -> c (Complex a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Complex a)

toConstr :: Complex a -> Constr

dataTypeOf :: Complex a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Complex a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Complex a))

gmapT :: (forall b. Data b => b -> b) -> Complex a -> Complex a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQ :: (forall d. Data d => d -> u) -> Complex a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Complex a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

RealFloat a => Num (Complex a)

Since: base-2.1

Defined in Data.Complex

Methods

(+) :: Complex a -> Complex a -> Complex a

(-) :: Complex a -> Complex a -> Complex a

(*) :: Complex a -> Complex a -> Complex a

negate :: Complex a -> Complex a

abs :: Complex a -> Complex a

signum :: Complex a -> Complex a

fromInteger :: Integer -> Complex a

Read a => Read (Complex a)

Defined in Data.Complex

Methods

readsPrec :: Int -> ReadS (Complex a)

readList :: ReadS [Complex a]

readPrec :: ReadPrec (Complex a)

readListPrec :: ReadPrec [Complex a]

Show a => Show (Complex a)

Defined in Data.Complex

Methods

showsPrec :: Int -> Complex a -> ShowS

show :: Complex a -> String

showList :: [Complex a] -> ShowS

Generic (Complex a)

Defined in Data.Complex

Associated Types

type Rep (Complex a) :: * -> *

Methods

from :: Complex a -> Rep (Complex a) x

to :: Rep (Complex a) x -> Complex a

Hashable a => Hashable (Complex a)

Defined in Data.Hashable.Class

Methods

hashWithSalt :: Int -> Complex a -> Int

hash :: Complex a -> Int

Storable a => Storable (Complex a)

Since: base-4.8.0.0

Defined in Data.Complex

Methods

sizeOf :: Complex a -> Int

alignment :: Complex a -> Int

peekElemOff :: Ptr (Complex a) -> Int -> IO (Complex a)

pokeElemOff :: Ptr (Complex a) -> Int -> Complex a -> IO ()

peekByteOff :: Ptr b -> Int -> IO (Complex a)

pokeByteOff :: Ptr b -> Int -> Complex a -> IO ()

peek :: Ptr (Complex a) -> IO (Complex a)

poke :: Ptr (Complex a) -> Complex a -> IO ()

NFData a => NFData (Complex a)

Defined in Control.DeepSeq

Methods

rnf :: Complex a -> ()

Unbox a => Unbox (Complex a)

Defined in Data.Vector.Unboxed.Base

(RealFloat a, Precise a) => Precise (Complex a)

Defined in Numeric.Log

Methods

log1p :: Complex a -> Complex a

expm1 :: Complex a -> Complex a

log1pexp :: Complex a -> Complex a

log1mexp :: Complex a -> Complex a

Serialise a => Serialise (Complex a)

Since: serialise-0.2.0.0

Defined in Codec.Serialise.Class

Methods

encode :: Complex a -> Encoding

decode :: Decoder s (Complex a)

encodeList :: [Complex a] -> Encoding

decodeList :: Decoder s [Complex a]

Generic1 Complex

Defined in Data.Complex

Associated Types

type Rep1 Complex :: k -> *

Methods

from1 :: Complex a -> Rep1 Complex a

to1 :: Rep1 Complex a -> Complex a

(Eq a) :=> (Eq (Complex a))

Defined in Data.Constraint

Methods

ins :: Eq a :- Eq (Complex a)

(Read a) :=> (Read (Complex a))

Defined in Data.Constraint

Methods

ins :: Read a :- Read (Complex a)

(RealFloat a) :=> (Num (Complex a))

Defined in Data.Constraint

Methods

ins :: RealFloat a :- Num (Complex a)

(RealFloat a) :=> (Fractional (Complex a))

Defined in Data.Constraint

Methods

ins :: RealFloat a :- Fractional (Complex a)

(RealFloat a) :=> (Floating (Complex a))

Defined in Data.Constraint

Methods

ins :: RealFloat a :- Floating (Complex a)

(Show a) :=> (Show (Complex a))

Defined in Data.Constraint

Methods

ins :: Show a :- Show (Complex a)

Each (Complex a) (Complex b) a b

each :: (RealFloat a, RealFloat b) => Traversal (Complex a) (Complex b) a b

Defined in Control.Lens.Each

Methods

each :: Traversal (Complex a) (Complex b) a b

type Rep Complex

Defined in Data.Functor.Rep

type Rep Complex = Bool

data MVector s (Complex a)

Defined in Data.Vector.Unboxed.Base

data MVector s (Complex a) = MV_Complex (MVector s (a, a))

type Rep (Complex a)

Defined in Data.Complex

type Rep (Complex a) = D1 (MetaData "Complex" "Data.Complex" "base" False) (C1 (MetaCons ":+" (InfixI NotAssociative 6) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)))

data Vector (Complex a)

Defined in Data.Vector.Unboxed.Base

data Vector (Complex a) = V_Complex (Vector (a, a))

type Index (Complex a)

Defined in Control.Lens.At

type Index (Complex a) = Int

type Rep1 Complex

Defined in Data.Complex

type Rep1 Complex = D1 (MetaData "Complex" "Data.Complex" "base" False) (C1 (MetaCons ":+" (InfixI NotAssociative 6) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1 :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1))

realPart :: Complex a -> a

Extracts the real part of a complex number.

imagPart :: Complex a -> a

Extracts the imaginary part of a complex number.

mkPolar :: Floating a => a -> a -> Complex a

Form a complex number from polar components of magnitude and phase.

cis :: Floating a => a -> Complex a

cis t is a complex value with magnitude 1 and phase t (modulo 2*pi).

polar :: RealFloat a => Complex a -> (a, a)

The function polar takes a complex number and returns a (magnitude, phase) pair in canonical form: the magnitude is nonnegative, and the phase in the range (-pi, pi]; if the magnitude is zero, then so is the phase.

magnitude :: RealFloat a => Complex a -> a

The nonnegative magnitude of a complex number.

phase :: RealFloat a => Complex a -> a

The phase of a complex number, in the range (-pi, pi]. If the magnitude is zero, then so is the phase.

conjugate :: Num a => Complex a -> Complex a

The conjugate of a complex number.

Optics

_realPart :: Functor f => (a -> f a) -> Complex a -> f (Complex a)

Access the realPart of a Complex number.

>>> (a :+ b)^._realPart
a

>>> a :+ b & _realPart *~ 2
a * 2 :+ b

_realPart :: Functor f => (a -> f a) -> Complex a -> f (Complex a)

_imagPart :: Functor f => (a -> f a) -> Complex a -> f (Complex a)

Access the imagPart of a Complex number.

>>> (a :+ b)^._imagPart
b

>>> a :+ b & _imagPart *~ 2
a :+ b * 2

_imagPart :: Functor f => (a -> f a) -> Complex a -> f (Complex a)

_polar :: RealFloat a => Iso' (Complex a) (a, a)

This isn't quite a legal Lens. Notably the

view l (set l b a) = b

law is violated when you set a polar value with 0 magnitude and non-zero phase as the phase information is lost, or with a negative magnitude which flips the phase and retains a positive magnitude. So don't do that!

Otherwise, this is a perfectly cromulent Lens.

_magnitude :: RealFloat a => Lens' (Complex a) a

Access the magnitude of a Complex number.

>>> (10.0 :+ 20.0) & _magnitude *~ 2
20.0 :+ 40.0

This isn't quite a legal Lens. Notably the

view l (set l b a) = b

law is violated when you set a negative magnitude. This flips the phase and retains a positive magnitude. So don't do that!

Otherwise, this is a perfectly cromulent Lens.

Setting the magnitude of a zero Complex number assumes the phase is 0.

_phase :: RealFloat a => Lens' (Complex a) a

Access the phase of a Complex number.

>>> (mkPolar 10 (2-pi) & _phase +~ pi & view _phase) ≈ 2
True

This isn't quite a legal Lens. Notably the

view l (set l b a) = b

law is violated when you set a phase outside the range (-pi, pi]. The phase is always in that range when queried. So don't do that!

Otherwise, this is a perfectly cromulent Lens.

_conjugate :: RealFloat a => Iso' (Complex a) (Complex a)

Access the conjugate of a Complex number.

>>> (2.0 :+ 3.0) & _conjugate . _imagPart -~ 1
2.0 :+ 4.0

>>> (mkPolar 10.0 2.0 ^. _conjugate . _phase) ≈ (-2.0)
True