Data.Number

A library for real number arithmetics

Classes

data Continued a

Continued fraction type

Represents a simple continued fraction of the form:

Supports Haskell arithmetic operators though the use of the operator function is to be preferred since provide shortcuts to calculations.

It can be an infinite sequence. Be careful with functions like show and precision: they may not terminate.

Cons operator

n :| x equivalent to n + 1/x

Negate operator

M x equivalent to -x

Constructors

M (Continued a)

Negative number

a :| (Continued a) infixr 5

Positive number

E

Zero

Instances

Eq a => Eq (Continued a)

Ord a => Ord (Continued a)

Read a => Read (Continued a)

Show a => Show (Continued a)

type Number = Continued Nat

Real numbers datatype (a continued fraction of naturals)

data Nat

Lazy Peano numbers. Allow calculation with infinite values.

Constructors

Z

Zero

S Nat

Successor

Instances

Bounded Nat

The lower bound is zero, the upper bound is infinity.

Enum Nat

The pred function is bounded at Zero.

Eq Nat

== and /= work as long as at least one operand is finite.

Integral Nat

Since negative numbers are not allowed, quot = div and rem = mod. The methods quot, rem, div, mod, quotRem and divMod will terminate as long as their first argument is finite. Infinities in their second arguments are permitted and are handled as follows:

  n quot infinity   =   n div infinity   =   0
  n rem  infinity   =   n mod infinity   =   n

Num Nat

Addition, multiplication, and subtraction are lazy in both arguments, meaning that, in the case of infinite values, they can produce an infinite stream of S-constructors. As long as the callers of these functions only consume a finite amount of these, the program will not hang.

fromInteger is not injective in case of Nat, since negative integers are all converted to zero (Z).

Ord Nat

All methods work as long as at least one operand is finite.

Real Nat

Since toRational returns a Ratio Integer, it WILL NOT terminate on infinities.

Show Nat

Peano Nat

Peano class instance

data Whole

Whole numbers (Z).

Constructors

Whole Nat Sign

Construct a whole number out of a magnitue and a sign.

Instances

Bounded Whole

The bounds are negative and positive infinity.

Enum Whole

succ and pred work according to the total order on the whole numbers, i.e. succ n = n+1 and pred n = n-1.

Eq Whole

Positive and negative zero are considered equal.

Integral Whole

Integer conversions and division

Num Whole

Implements arithmetics for Whole numbers

Ord Whole

The ordering is the standard total order on Z. Positive and negative zero are equal.

Real Whole

Since toRational returns a Ratio Integer, it WILL NOT terminate on infinities.

Peano Whole

Peano class instance defines infinity (positive) and other functions handling the sign

Functions

fromList :: [Nat] -> Number

Create a Number from a list of naturals

toList :: Number -> [Nat]

Convert a Number to a list of naturals (losing the sign)

fromNumber :: RealFrac a => Number -> a

Convert a Number into a RealFrac number

toNumber :: RealFrac a => a -> Number

Convert a RealFrac number into a Number

Constants

σ :: Number

The infinite continued fraction whose terms are naturals numbers

φ :: Number

The golden ratio

π :: Number

Pi: the ratio of a circle's circumference to its diameter

e :: Number

Euler's number: the base of the natural logarithm

Internals

hom :: Hom -> Number -> Number

Homographic function

Given the Hom matrix

and a Number x calculates

See http://perl.plover.com/yak/cftalk/INFO/gosper.txt for a complete explanation.

biHom :: BiHom -> Number -> Number -> Number

Bihomographic function

Given a BiHom matrix

and two Number x and y calculates

cut :: Nat -> Number -> Number

Truncate a Number to a given length n