Safe Haskell	Safe
Language	Haskell2010

Num.Complex

Contents

Complex
- Optics

Synopsis

data Complex a
realPart :: Complex a -> a
imagPart :: Complex a -> a
mkPolar :: Floating a => a -> a -> Complex a
cis :: Floating a => a -> Complex a
polar :: RealFloat a => Complex a -> (a, a)
magnitude :: RealFloat a => Complex a -> a
phase :: RealFloat a => Complex a -> a
conjugate :: Num a => Complex a -> Complex a
_realPart :: Functor f => (a -> f a) -> Complex a -> f (Complex a)
_imagPart :: Functor f => (a -> f a) -> Complex a -> f (Complex a)
_polar :: RealFloat a => Iso' (Complex a) (a, a)
_magnitude :: RealFloat a => Lens' (Complex a) a
_phase :: RealFloat a => Lens' (Complex a) a
_conjugate :: RealFloat a => Iso' (Complex a) (Complex a)

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
Instance details 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
Instance details 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
Instance details 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
Instance details 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
Instance details 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
Instance details 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
Instance details 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
Instance details Defined in Data.Hashable.Class Methods liftHashWithSalt :: (Int -> a -> Int) -> Int -> Complex a -> Int #
Apply Complex
Instance details 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
Instance details 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
Instance details 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
Instance details 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)
Instance details 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)
Instance details 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)
Instance details Defined in Data.Complex Methods (==) :: Complex a -> Complex a -> Bool # (/=) :: Complex a -> Complex a -> Bool #
RealFloat a => Floating (Complex a)	Since: base-2.1
Instance details 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
Instance details 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)
Instance details 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
Instance details 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)
Instance details 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)
Instance details Defined in Data.Complex Methods showsPrec :: Int -> Complex a -> ShowS # show :: Complex a -> String # showList :: [Complex a] -> ShowS #
Generic (Complex a)
Instance details 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)
Instance details 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
Instance details 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)
Instance details Defined in Control.DeepSeq Methods rnf :: Complex a -> () #
Unbox a => Unbox (Complex a)
Instance details Defined in Data.Vector.Unboxed.Base
(RealFloat a, Precise a) => Precise (Complex a)
Instance details 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
Instance details 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
Instance details 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))
Instance details Defined in Data.Constraint Methods ins :: Eq a :- Eq (Complex a) #
(Read a) :=> (Read (Complex a))
Instance details Defined in Data.Constraint Methods ins :: Read a :- Read (Complex a) #
(RealFloat a) :=> (Num (Complex a))
Instance details Defined in Data.Constraint Methods ins :: RealFloat a :- Num (Complex a) #
(RealFloat a) :=> (Fractional (Complex a))
Instance details Defined in Data.Constraint Methods ins :: RealFloat a :- Fractional (Complex a) #
(RealFloat a) :=> (Floating (Complex a))
Instance details Defined in Data.Constraint Methods ins :: RealFloat a :- Floating (Complex a) #
(Show a) :=> (Show (Complex a))
Instance details 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
Instance details Defined in Control.Lens.Each Methods each :: Traversal (Complex a) (Complex b) a b #
type Rep Complex
Instance details Defined in Data.Functor.Rep type Rep Complex = Bool
data MVector s (Complex a)
Instance details Defined in Data.Vector.Unboxed.Base data MVector s (Complex a) = MV_Complex (MVector s (a, a))
type Rep (Complex a)
Instance details 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)
Instance details Defined in Data.Vector.Unboxed.Base data Vector (Complex a) = V_Complex (Vector (a, a))
type Index (Complex a)
Instance details Defined in Control.Lens.At type Index (Complex a) = Int
type Rep1 Complex
Instance details 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