Copyright	(c) The University of Glasgow 2001
License	BSD-style (see the file libraries/base/LICENSE)
Maintainer	libraries@haskell.org
Stability	provisional
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Complex

Contents

Rectangular form
Polar form
Conjugate

Description

Complex numbers.

Synopsis

data Complex a = !a :+ !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

Rectangular form

data Complex a Source #

A data type representing complex numbers.

You can read about complex numbers on wikipedia.

In haskell, complex numbers are represented as a :+ b which can be thought of as representing $a + bi$. 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. Apart from the loss of precision due to IEEE754 floating point numbers, it holds that z == abs z * signum z.

Note that Complex's instances inherit the deficiencies from the type parameter's. For example, Complex Float's Ord instance has similar problems to Float's.

As can be seen in the examples, the Foldable and Traversable instances traverse the real part first.

Examples

Expand

>>> (5.0 :+ 2.5) + 6.5
11.5 :+ 2.5

>>> abs (1.0 :+ 1.0) - sqrt 2.0
0.0 :+ 0.0

>>> abs (signum (4.0 :+ 3.0))
1.0 :+ 0.0

>>> foldr (:) [] (1 :+ 2)
[1,2]

>>> mapM print (1 :+ 2)
1
2

Constructors

!a :+ !a infix 6

forms a complex number from its real and imaginary rectangular components.

Instances

Instances details

MonadFix Complex Source #	Since: base-4.15.0.0
Instance details Defined in Data.Complex Methods mfix :: (a -> Complex a) -> Complex a Source #
MonadZip Complex Source #	Since: base-4.15.0.0
Instance details Defined in Data.Complex Methods mzip :: Complex a -> Complex b -> Complex (a, b) Source # mzipWith :: (a -> b -> c) -> Complex a -> Complex b -> Complex c Source # munzip :: Complex (a, b) -> (Complex a, Complex b) Source #
Foldable Complex Source #	Since: base-4.9.0.0
Instance details Defined in Data.Complex Methods fold :: Monoid m => Complex m -> m Source # foldMap :: Monoid m => (a -> m) -> Complex a -> m Source # foldMap' :: Monoid m => (a -> m) -> Complex a -> m Source # foldr :: (a -> b -> b) -> b -> Complex a -> b Source # foldr' :: (a -> b -> b) -> b -> Complex a -> b Source # foldl :: (b -> a -> b) -> b -> Complex a -> b Source # foldl' :: (b -> a -> b) -> b -> Complex a -> b Source # foldr1 :: (a -> a -> a) -> Complex a -> a Source # foldl1 :: (a -> a -> a) -> Complex a -> a Source # toList :: Complex a -> [a] Source # null :: Complex a -> Bool Source # length :: Complex a -> Int Source # elem :: Eq a => a -> Complex a -> Bool Source # maximum :: Ord a => Complex a -> a Source # minimum :: Ord a => Complex a -> a Source # sum :: Num a => Complex a -> a Source # product :: Num a => Complex a -> a Source #
Foldable1 Complex Source #	Since: base-4.18.0.0
Instance details Defined in Data.Foldable1 Methods fold1 :: Semigroup m => Complex m -> m Source # foldMap1 :: Semigroup m => (a -> m) -> Complex a -> m Source # foldMap1' :: Semigroup m => (a -> m) -> Complex a -> m Source # toNonEmpty :: Complex a -> NonEmpty a Source # maximum :: Ord a => Complex a -> a Source # minimum :: Ord a => Complex a -> a Source # head :: Complex a -> a Source # last :: Complex a -> a Source # foldrMap1 :: (a -> b) -> (a -> b -> b) -> Complex a -> b Source # foldlMap1' :: (a -> b) -> (b -> a -> b) -> Complex a -> b Source # foldlMap1 :: (a -> b) -> (b -> a -> b) -> Complex a -> b Source # foldrMap1' :: (a -> b) -> (a -> b -> b) -> Complex a -> b Source #
Eq1 Complex Source #	`>>>` `eq1 (1 :+ 2) (1 :+ 2)`True `>>>` `eq1 (1 :+ 2) (1 :+ 3)`False Since: base-4.16.0.0
Instance details Defined in Data.Functor.Classes Methods liftEq :: (a -> b -> Bool) -> Complex a -> Complex b -> Bool Source #
Read1 Complex Source #	`>>>` `readPrec_to_S readPrec1 0 "(2 % 3) :+ (3 % 4)" :: [(Complex Rational, String)]`[(2 % 3 :+ 3 % 4,"")] Since: base-4.16.0.0
Instance details Defined in Data.Functor.Classes Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Complex a) Source # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Complex a] Source # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Complex a) Source # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Complex a] Source #
Show1 Complex Source #	`>>>` `showsPrec1 0 (2 :+ 3) ""`"2 :+ 3" Since: base-4.16.0.0
Instance details Defined in Data.Functor.Classes Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Complex a -> ShowS Source # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Complex a] -> ShowS Source #
Traversable Complex Source #	Since: base-4.9.0.0
Instance details Defined in Data.Complex Methods traverse :: Applicative f => (a -> f b) -> Complex a -> f (Complex b) Source # sequenceA :: Applicative f => Complex (f a) -> f (Complex a) Source # mapM :: Monad m => (a -> m b) -> Complex a -> m (Complex b) Source # sequence :: Monad m => Complex (m a) -> m (Complex a) Source #
Applicative Complex Source #	Since: base-4.9.0.0
Instance details Defined in Data.Complex Methods pure :: a -> Complex a Source # (<>) :: Complex (a -> b) -> Complex a -> Complex b Source # liftA2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c Source # (>) :: Complex a -> Complex b -> Complex b Source # (<*) :: Complex a -> Complex b -> Complex a Source #
Functor Complex Source #	Since: base-4.9.0.0
Instance details Defined in Data.Complex Methods fmap :: (a -> b) -> Complex a -> Complex b Source # (<$) :: a -> Complex b -> Complex a Source #
Monad Complex Source #	Since: base-4.9.0.0
Instance details Defined in Data.Complex Methods (>>=) :: Complex a -> (a -> Complex b) -> Complex b Source # (>>) :: Complex a -> Complex b -> Complex b Source # return :: a -> Complex a Source #
Generic1 Complex Source #
Instance details Defined in Data.Complex Associated Types type Rep1 Complex :: k -> Type Source # Methods from1 :: forall (a :: k). Complex a -> Rep1 Complex a Source # to1 :: forall (a :: k). Rep1 Complex a -> Complex a Source #
Data a => Data (Complex a) Source #	Since: base-2.1
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) Source # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Complex a) Source # toConstr :: Complex a -> Constr Source # dataTypeOf :: Complex a -> DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Complex a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Complex a)) Source # gmapT :: (forall b. Data b => b -> b) -> Complex a -> Complex a Source # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r Source # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r Source # gmapQ :: (forall d. Data d => d -> u) -> Complex a -> [u] Source # gmapQi :: Int -> (forall d. Data d => d -> u) -> Complex a -> u Source # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a) Source #
Storable a => Storable (Complex a) Source #	Since: base-4.8.0.0
Instance details Defined in Data.Complex Methods sizeOf :: Complex a -> Int Source # alignment :: Complex a -> Int Source # peekElemOff :: Ptr (Complex a) -> Int -> IO (Complex a) Source # pokeElemOff :: Ptr (Complex a) -> Int -> Complex a -> IO () Source # peekByteOff :: Ptr b -> Int -> IO (Complex a) Source # pokeByteOff :: Ptr b -> Int -> Complex a -> IO () Source # peek :: Ptr (Complex a) -> IO (Complex a) Source # poke :: Ptr (Complex a) -> Complex a -> IO () Source #
RealFloat a => Floating (Complex a) Source #	Since: base-2.1
Instance details Defined in Data.Complex Methods pi :: Complex a Source # exp :: Complex a -> Complex a Source # log :: Complex a -> Complex a Source # sqrt :: Complex a -> Complex a Source # (**) :: Complex a -> Complex a -> Complex a Source # logBase :: Complex a -> Complex a -> Complex a Source # sin :: Complex a -> Complex a Source # cos :: Complex a -> Complex a Source # tan :: Complex a -> Complex a Source # asin :: Complex a -> Complex a Source # acos :: Complex a -> Complex a Source # atan :: Complex a -> Complex a Source # sinh :: Complex a -> Complex a Source # cosh :: Complex a -> Complex a Source # tanh :: Complex a -> Complex a Source # asinh :: Complex a -> Complex a Source # acosh :: Complex a -> Complex a Source # atanh :: Complex a -> Complex a Source # log1p :: Complex a -> Complex a Source # expm1 :: Complex a -> Complex a Source # log1pexp :: Complex a -> Complex a Source # log1mexp :: Complex a -> Complex a Source #
Generic (Complex a) Source #
Instance details Defined in Data.Complex Associated Types type Rep (Complex a) :: Type -> Type Source # Methods from :: Complex a -> Rep (Complex a) x Source # to :: Rep (Complex a) x -> Complex a Source #
RealFloat a => Num (Complex a) Source #	Since: base-2.1
Instance details Defined in Data.Complex Methods (+) :: Complex a -> Complex a -> Complex a Source # (-) :: Complex a -> Complex a -> Complex a Source # (*) :: Complex a -> Complex a -> Complex a Source # negate :: Complex a -> Complex a Source # abs :: Complex a -> Complex a Source # signum :: Complex a -> Complex a Source # fromInteger :: Integer -> Complex a Source #
Read a => Read (Complex a) Source #	Since: base-2.1
Instance details Defined in Data.Complex Methods readsPrec :: Int -> ReadS (Complex a) Source # readList :: ReadS [Complex a] Source # readPrec :: ReadPrec (Complex a) Source # readListPrec :: ReadPrec [Complex a] Source #
RealFloat a => Fractional (Complex a) Source #	Since: base-2.1
Instance details Defined in Data.Complex Methods (/) :: Complex a -> Complex a -> Complex a Source # recip :: Complex a -> Complex a Source # fromRational :: Rational -> Complex a Source #
Show a => Show (Complex a) Source #	Since: base-2.1
Instance details Defined in Data.Complex Methods showsPrec :: Int -> Complex a -> ShowS Source # show :: Complex a -> String Source # showList :: [Complex a] -> ShowS Source #
Eq a => Eq (Complex a) Source #	Since: base-2.1
Instance details Defined in Data.Complex Methods (==) :: Complex a -> Complex a -> Bool Source # (/=) :: Complex a -> Complex a -> Bool Source #
type Rep1 Complex Source #	Since: base-4.9.0.0
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))
type Rep (Complex a) Source #	Since: base-4.9.0.0
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)))

realPart :: Complex a -> a Source #

Extracts the real part of a complex number.

Examples

Expand

>>> realPart (5.0 :+ 3.0)
5.0

>>> realPart ((5.0 :+ 3.0) * (2.0 :+ 3.0))
1.0

imagPart :: Complex a -> a Source #

Extracts the imaginary part of a complex number.

Examples

Expand

>>> imagPart (5.0 :+ 3.0)
3.0

>>> imagPart ((5.0 :+ 3.0) * (2.0 :+ 3.0))
21.0

Polar form

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

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

Examples

Expand

>>> mkPolar 1 (pi / 4)
0.7071067811865476 :+ 0.7071067811865475

>>> mkPolar 1 0
1.0 :+ 0.0

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

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

cis = mkPolar 1

Examples

Expand

>>> cis 0
1.0 :+ 0.0

The following examples are not perfectly zero due to IEEE 754

>>> cis pi
(-1.0) :+ 1.2246467991473532e-16

>>> cis (4 * pi) - cis (2 * pi)
0.0 :+ (-2.4492935982947064e-16)

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

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

polar z = (magnitude z, phase z)

Examples

Expand

>>> polar (1.0 :+ 1.0)
(1.4142135623730951,0.7853981633974483)

>>> polar ((-1.0) :+ 0.0)
(1.0,3.141592653589793)

>>> polar (0.0 :+ 0.0)
(0.0,0.0)

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

The non-negative magnitude of a complex number.

Examples

Expand

>>> magnitude (1.0 :+ 1.0)
1.4142135623730951

>>> magnitude (1.0 + 0.0)
1.0

>>> magnitude (0.0 :+ (-5.0))
5.0

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

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

Examples

Expand

>>> phase (0.5 :+ 0.5) / pi
0.25

>>> phase (0 :+ 4) / pi
0.5

Conjugate

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

The conjugate of a complex number.

conjugate (conjugate x) = x

Examples

Expand

>>> conjugate (3.0 :+ 3.0)
3.0 :+ (-3.0)

>>> conjugate ((3.0 :+ 3.0) * (2.0 :+ 2.0))
0.0 :+ (-12.0)