yap-0.2: yet another prelude - a simplistic refactoring with algebraic classes

Data.YAP.Complex

Contents

Description

A version of Data.Complex, using the same type, but with less constrained operations. In particular this version permits Gaussian integers.

Synopsis

# Rectangular form

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.

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) RealFloat a => Field (Complex a) `RealFloat` is used to do scaling to reduce the incidence of overflow. Integral a => EuclideanDomain (Complex a) Gaussian integers: if `b` is non-zero, the norm (squared magnitude) of `mod a b` is at most half that of `b`. Standard associates lie in the positive quadrant. Ring a => Ring (Complex a) AbelianGroup a => AbelianGroup (Complex a) RealFloat a => Floating (Complex a) RealFloat a => Fractional (Complex a) RealFloat a => Num (Complex a)

realPart :: Complex a -> aSource

Extracts the real part of a complex number.

imagPart :: Complex a -> aSource

Extracts the imaginary part of a complex number.

# Polar form

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

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

cis :: Floating 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.

`RealFloat` is used to do scaling to reduce the incidence of overflow.

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.

`RealFloat` is need for `atan2`.

# Conjugate

conjugate :: AbelianGroup a => Complex a -> Complex aSource

The conjugate of a complex number.