Copyright	(c) Scott N. Walck 2012-2017
License	GPL-3 (see LICENSE)
Maintainer	Scott N. Walck <walck@lvc.edu>
Stability	experimental
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Complex.Cyclotomic

Description

The cyclotomic numbers are a subset of the complex numbers with the following properties:

The cyclotomic numbers are represented exactly, enabling exact computations and equality comparisons.
The cyclotomic numbers contain the Gaussian rationals (complex numbers of the form p + q i with p and q rational). As a consequence, the cyclotomic numbers are a dense subset of the complex numbers.
The cyclotomic numbers contain the square roots of all rational numbers.
The cyclotomic numbers form a field: they are closed under addition, subtraction, multiplication, and division.
The cyclotomic numbers contain the sine and cosine of all rational multiples of pi.
The cyclotomic numbers can be thought of as the rational field extended with nth roots of unity for arbitrarily large integers n.

Floating point numbers do not do well with equality comparison:

(sqrt 2 + sqrt 3)^2 == 5 + 2 * sqrt 6
 -> False

Data.Complex.Cyclotomic represents these numbers exactly, allowing equality comparison:

(sqrtRat 2 + sqrtRat 3)^2 == 5 + 2 * sqrtRat 6
 -> True

Cyclotomics can be exported as inexact complex numbers using the toComplex function:

e 6
 -> -e(3)^2
real $ e 6
 -> 1/2
imag $ e 6
 -> -1/2*e(12)^7 + 1/2*e(12)^11
imag (e 6) == sqrtRat 3 / 2
 -> True
toComplex $ e 6
 -> 0.5000000000000003 :+ 0.8660254037844384

The algorithms for cyclotomic numbers are adapted from code by Martin Schoenert and Thomas Breuer in the GAP project http://www.gap-system.org/ (in particular source files gap4r4/src/cyclotom.c and gap4r4/lib/cyclotom.gi).

Synopsis

data Cyclotomic
i :: Cyclotomic
e :: Integer -> Cyclotomic
sqrtInteger :: Integer -> Cyclotomic
sqrtRat :: Rational -> Cyclotomic
sinDeg :: Rational -> Cyclotomic
cosDeg :: Rational -> Cyclotomic
sinRev :: Rational -> Cyclotomic
cosRev :: Rational -> Cyclotomic
gaussianRat :: Rational -> Rational -> Cyclotomic
polarRat :: Rational -> Rational -> Cyclotomic
polarRatDeg :: Rational -> Rational -> Cyclotomic
polarRatRev :: Rational -> Rational -> Cyclotomic
conj :: Cyclotomic -> Cyclotomic
real :: Cyclotomic -> Cyclotomic
imag :: Cyclotomic -> Cyclotomic
isReal :: Cyclotomic -> Bool
isRat :: Cyclotomic -> Bool
isGaussianRat :: Cyclotomic -> Bool
toComplex :: RealFloat a => Cyclotomic -> Complex a
toReal :: RealFloat a => Cyclotomic -> Maybe a
toRat :: Cyclotomic -> Maybe Rational
goldenRatio :: Cyclotomic
dft :: [Cyclotomic] -> [Cyclotomic]
dftInv :: [Cyclotomic] -> [Cyclotomic]
rootsQuadEq :: Rational -> Rational -> Rational -> Maybe (Cyclotomic, Cyclotomic)
heron :: Rational -> Rational -> Rational -> Cyclotomic

Documentation

data Cyclotomic #

A cyclotomic number.

Instances

Eq Cyclotomic #
Instance details Defined in Data.Complex.Cyclotomic Methods (==) :: Cyclotomic -> Cyclotomic -> Bool # (/=) :: Cyclotomic -> Cyclotomic -> Bool #
Fractional Cyclotomic #
Instance details Defined in Data.Complex.Cyclotomic Methods (/) :: Cyclotomic -> Cyclotomic -> Cyclotomic # recip :: Cyclotomic -> Cyclotomic # fromRational :: Rational -> Cyclotomic #
Num Cyclotomic #	`abs` and `signum` are partial functions. A cyclotomic number is not guaranteed to have a cyclotomic absolute value. When defined, `signum c` is the complex number with magnitude 1 that has the same argument as c; `signum c = c / abs c`.
Instance details Defined in Data.Complex.Cyclotomic Methods (+) :: Cyclotomic -> Cyclotomic -> Cyclotomic # (-) :: Cyclotomic -> Cyclotomic -> Cyclotomic # (*) :: Cyclotomic -> Cyclotomic -> Cyclotomic # negate :: Cyclotomic -> Cyclotomic # abs :: Cyclotomic -> Cyclotomic # signum :: Cyclotomic -> Cyclotomic # fromInteger :: Integer -> Cyclotomic #
Show Cyclotomic #
Instance details Defined in Data.Complex.Cyclotomic Methods showsPrec :: Int -> Cyclotomic -> ShowS # show :: Cyclotomic -> String # showList :: [Cyclotomic] -> ShowS #

i :: Cyclotomic #

The square root of -1.

e :: Integer -> Cyclotomic #

The primitive nth root of unity. For example, e(4) = i is the primitive 4th root of unity, and e(5) = exp(2*pi*i/5) is the primitive 5th root of unity. In general, e n = exp(2*pi*i/n).

sqrtInteger :: Integer -> Cyclotomic #

The square root of an Integer.

sqrtRat :: Rational -> Cyclotomic #

The square root of a Rational number.

sinDeg :: Rational -> Cyclotomic #

Sine function with argument in degrees.

cosDeg :: Rational -> Cyclotomic #

Cosine function with argument in degrees.

sinRev :: Rational -> Cyclotomic #

Sine function with argument in revolutions.

cosRev :: Rational -> Cyclotomic #

Cosine function with argument in revolutions.

gaussianRat :: Rational -> Rational -> Cyclotomic #

Make a Gaussian rational; gaussianRat p q is the same as p + q * i.

polarRat #

Arguments

:: Rational	magnitude
-> Rational	angle, in revolutions
-> Cyclotomic	cyclotomic number

A complex number in polar form, with rational magnitude r and rational angle s of the form r * exp(2*pi*i*s); polarRat r s is the same as r * e q ^ p, where s = p/q. This function is the same as polarRatRev.

polarRatDeg #

Arguments

:: Rational	magnitude
-> Rational	angle, in degrees
-> Cyclotomic	cyclotomic number

A complex number in polar form, with rational magnitude and rational angle in degrees.

polarRatRev #

Arguments

:: Rational	magnitude
-> Rational	angle, in revolutions
-> Cyclotomic	cyclotomic number

A complex number in polar form, with rational magnitude and rational angle in revolutions.

conj :: Cyclotomic -> Cyclotomic #

Complex conjugate.

real :: Cyclotomic -> Cyclotomic #

Real part of the cyclotomic number.

imag :: Cyclotomic -> Cyclotomic #

Imaginary part of the cyclotomic number.

isReal :: Cyclotomic -> Bool #

Is the cyclotomic a real number?

isRat :: Cyclotomic -> Bool #

Is the cyclotomic a rational?

isGaussianRat :: Cyclotomic -> Bool #

Is the cyclotomic a Gaussian rational?

toComplex :: RealFloat a => Cyclotomic -> Complex a #

Export as an inexact complex number.

toReal :: RealFloat a => Cyclotomic -> Maybe a #

Export as an inexact real number if possible.

toRat :: Cyclotomic -> Maybe Rational #

Return an exact rational number if possible.

goldenRatio :: Cyclotomic #

The golden ratio, (1 + √5)/2.

dft :: [Cyclotomic] -> [Cyclotomic] #

Discrete Fourier transform, X_k = sum_{n=0}^{N-1} x_n cdot e^{-i 2 pi frac{k}{N} n}.

dftInv :: [Cyclotomic] -> [Cyclotomic] #

Inverse discrete Fourier transform, x_n = frac{1}{N} sum_{k=0}^{N-1} X_k cdot e^{i 2 pi frac{k}{N} n}.

rootsQuadEq #

Arguments

:: Rational	a
-> Rational	b
-> Rational	c
-> Maybe (Cyclotomic, Cyclotomic)	roots

Solutions to the quadratic equation a x^2 + b x + c = 0. Returns Nothing if a == 0.

heron #

Arguments

:: Rational	a
-> Rational	b
-> Rational	c
-> Cyclotomic	area of triangle

Heron's formula for the area of a triangle with side lengths a, b, c.