Data.Complex

haskell2010-1.1.2.0: Compatibility with Haskell 2010

Safe Haskell | Safe
Language | Haskell2010

Data.Complex

Contents

- Rectangular form
- Polar form
- Conjugate
- Specification

Synopsis

- data Complex a :: * -> * = !a :+ !a
- realPart :: RealFloat a => Complex a -> a
- imagPart :: RealFloat a => Complex a -> a
- mkPolar :: RealFloat a => a -> a -> Complex a
- cis :: RealFloat a => a -> Complex a
- polar :: RealFloat a => Complex a -> (a, a)
- magnitude :: RealFloat a => Complex a -> a
- phase :: RealFloat a => Complex a -> a
- conjugate :: RealFloat a => Complex a -> Complex a

# Rectangular form

data Complex a :: * -> * Source

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.

Constructors

!a :+ !a infix 6 | forms a complex number from its real and imaginary rectangular components.

Instances

Eq a => Eq (Complex a) |
RealFloat a => Floating (Complex a) |
RealFloat a => Fractional (Complex a) |
Data a => Data (Complex a) |
RealFloat a => Num (Complex a) |
Read a => Read (Complex a) |
Show a => Show (Complex a) |
Typeable (* -> *) Complex |

realPart :: RealFloat a => Complex a -> a Source

Extracts the real part of a complex number.

imagPart :: RealFloat a => Complex a -> a Source

Extracts the imaginary part of a complex number.

# Polar form

mkPolar :: RealFloat a => a -> a -> Complex a Source

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

cis :: RealFloat a => a -> Complex a Source

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

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 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 Source

The nonnegative magnitude of a complex number.

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.

# Conjugate

conjugate :: RealFloat a => Complex a -> Complex a Source

The conjugate of a complex number.

# Specification

    module Data.Complex(Complex((:+)), realPart, imagPart, conjugate, mkPolar,
                        cis, polar, magnitude, phase)  where

    infix  6  :+

    data  (RealFloat a)     => Complex a = !a :+ !a  deriving (Eq,Read,Show)

    realPart, imagPart :: (RealFloat a) => Complex a -> a
    realPart (x:+y)        =  x
    imagPart (x:+y)        =  y

    conjugate      :: (RealFloat a) => Complex a -> Complex a
    conjugate (x:+y) =  x :+ (-y)

    mkPolar                :: (RealFloat a) => a -> a -> Complex a
    mkPolar r theta        =  r * cos theta :+ r * sin theta

    cis            :: (RealFloat a) => a -> Complex a
    cis theta      =  cos theta :+ sin theta

    polar          :: (RealFloat a) => Complex a -> (a,a)
    polar z                =  (magnitude z, phase z)

    magnitude :: (RealFloat a) => Complex a -> a
    magnitude (x:+y) =  scaleFloat k
                       (sqrt ((scaleFloat mk x)^2 + (scaleFloat mk y)^2))
                      where k  = max (exponent x) (exponent y)
                            mk = - k

    phase :: (RealFloat a) => Complex a -> a
    phase (0 :+ 0) = 0
    phase (x :+ y) = atan2 y x

    instance  (RealFloat a) => Num (Complex a)  where
        (x:+y) + (x':+y') =  (x+x') :+ (y+y')
        (x:+y) - (x':+y') =  (x-x') :+ (y-y')
        (x:+y) * (x':+y') =  (x*x'-y*y') :+ (x*y'+y*x')
        negate (x:+y)     =  negate x :+ negate y
        abs z             =  magnitude z :+ 0
        signum 0          =  0
        signum z@(x:+y)   =  x/r :+ y/r  where r = magnitude z
        fromInteger n     =  fromInteger n :+ 0

    instance  (RealFloat a) => Fractional (Complex a)  where
        (x:+y) / (x':+y') =  (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
                             where x'' = scaleFloat k x'
                                   y'' = scaleFloat k y'
                                   k   = - max (exponent x') (exponent y')
                                   d   = x'*x'' + y'*y''

        fromRational a    =  fromRational a :+ 0

    instance  (RealFloat a) => Floating (Complex a)       where
        pi             =  pi :+ 0
        exp (x:+y)     =  expx * cos y :+ expx * sin y
                          where expx = exp x
        log z          =  log (magnitude z) :+ phase z

        sqrt 0         =  0
        sqrt z@(x:+y)  =  u :+ (if y < 0 then -v else v)
                          where (u,v) = if x < 0 then (v',u') else (u',v')
                                v'    = abs y / (u'*2)
                                u'    = sqrt ((magnitude z + abs x) / 2)

        sin (x:+y)     =  sin x * cosh y :+ cos x * sinh y
        cos (x:+y)     =  cos x * cosh y :+ (- sin x * sinh y)
        tan (x:+y)     =  (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
                          where sinx  = sin x
                                cosx  = cos x
                                sinhy = sinh y
                                coshy = cosh y

        sinh (x:+y)    =  cos y * sinh x :+ sin  y * cosh x
        cosh (x:+y)    =  cos y * cosh x :+ sin y * sinh x
        tanh (x:+y)    =  (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
                          where siny  = sin y
                                cosy  = cos y
                                sinhx = sinh x
                                coshx = cosh x

        asin z@(x:+y)  =  y':+(-x')
                          where  (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
        acos z@(x:+y)  =  y'':+(-x'')
                          where (x'':+y'') = log (z + ((-y'):+x'))
                                (x':+y')   = sqrt (1 - z*z)
        atan z@(x:+y)  =  y':+(-x')
                          where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))

        asinh z        =  log (z + sqrt (1+z*z))
        acosh z        =  log (z + (z+1) * sqrt ((z-1)/(z+1)))
        atanh z        =  log ((1+z) / sqrt (1-z*z))