| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
NumHask.Algebra
Contents
Description
The basic algebraic class structure of a number.
import NumHask.Algebra import Prelude hiding (Integral(..), (*), (**), (+), (-), (/), (^), (^^), abs, acos, acosh, asin, asinh, atan, atan2, atanh, ceiling, cos, cosh, exp, floor, fromInteger, fromIntegral, log, logBase, negate, pi, product, recip, round, sin, sinh, sqrt, sum, tan, tanh, toInteger, fromRational)
- module NumHask.Algebra.Additive
- module NumHask.Algebra.Basis
- module NumHask.Algebra.Distribution
- module NumHask.Algebra.Field
- module NumHask.Algebra.Integral
- module NumHask.Algebra.Magma
- module NumHask.Algebra.Metric
- module NumHask.Algebra.Module
- module NumHask.Algebra.Multiplicative
- module NumHask.Algebra.Rational
- module NumHask.Algebra.Ring
- data Complex a :: * -> * = !a :+ !a
Mapping from Num
Num is a very old part of haskell, and a lot of different numeric concepts are tossed in there. The closest analogue in numhask is the Ring class, which combines the classical +, - and *, together with the distribution laws.
No attempt is made, however, to reconstruct the particular combination of laws and classes that represent the old Num. A rough mapping of Num to numhask classes follows:
-- | Basic numeric class.
class Num a where
{-# MINIMAL (+), (*), abs, signum, fromInteger, (negate | (-)) #-}
(+), (-), (*) :: a -> a -> a
-- | Unary negation.
negate :: a -> a+ is a function of the Additive class,
- is a function of the AdditiveGroup class, and
* is a function of the Multiplicative class.
negate is specifically in the AdditiveInvertible class. There are many useful constructions between negate and (-), involving cancellative properties.
-- | Absolute value. abs :: a -> a -- | Sign of a number. -- The functions 'abs' and 'signum' should satisfy the law: -- -- > abs x * signum x == x -- -- For real numbers, the 'signum' is either @-1@ (negative), @0@ (zero) -- or @1@ (positive). signum :: a -> a
abs is a function in the Signed class. The concept of an absolute value of a number can include situations where the domain and codomain are different, and size as a function in the Normed class is supplied for these cases.
sign replaces signum, because signum is a heinous name.
-- | Conversion from an 'Integer'.
-- An integer literal represents the application of the function
-- 'fromInteger' to the appropriate value of type 'Integer',
-- so such literals have type @('Num' a) => a@.
fromInteger :: Integer -> afromInteger is given its own class FromInteger
module NumHask.Algebra.Additive
module NumHask.Algebra.Basis
module NumHask.Algebra.Distribution
module NumHask.Algebra.Field
module NumHask.Algebra.Integral
module NumHask.Algebra.Magma
module NumHask.Algebra.Metric
module NumHask.Algebra.Module
module NumHask.Algebra.Rational
module NumHask.Algebra.Ring
Complex numbers are an algebraic type.
For a complex number z, abs zz,
but oriented in the positive real direction, whereas signum zz, but unit magnitude.
The Foldable and Traversable instances traverse the real part first.
Constructors
| !a :+ !a infix 6 | forms a complex number from its real and imaginary rectangular components. |
Instances