Data.Complex

haskell2010-1.1.1.0: Compatibility with Haskell 2010

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

Instances

Typeable1 Complex |
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) |

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

Extracts the real part of a complex number.

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

Extracts the imaginary part of a complex number.

# Polar form

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

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

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

``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 -> aSource

The nonnegative magnitude of a complex number.

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

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 aSource

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))