# Difference between revisions of "Exact real arithmetic"

(inserted content from Hawiki) |
(→Theory: about the paper - it's very good! read it!) |
||

(7 intermediate revisions by 3 users not shown) | |||

Line 2: | Line 2: | ||

== Introduction == |
== Introduction == |
||

− | Exact real arithmetic is an interesting area: it is a deep connection between |
+ | Exact real arithmetic is an interesting area: it is based on a deep connection between |

* numeric methods |
* numeric methods |
||

− | * and deep theoretic |
+ | * and deep theoretic foundations of algorithms (and mathematics). |

Its topic: computable real numbers raise a lot of interesting questions rooted in mathematical analysis, arithmetic, but also [[Computer science#Computability theory|Computability theory]] (see numbers-as-programs approaches). |
Its topic: computable real numbers raise a lot of interesting questions rooted in mathematical analysis, arithmetic, but also [[Computer science#Computability theory|Computability theory]] (see numbers-as-programs approaches). |
||

Line 14: | Line 14: | ||

Exact reals must allow us to run a huge series of computations, prescribing only the precision of the end result. Intermediate computations, and determining their necessary precision must be achieved automatically, dynamically. |
Exact reals must allow us to run a huge series of computations, prescribing only the precision of the end result. Intermediate computations, and determining their necessary precision must be achieved automatically, dynamically. |
||

− | Maybe another problem, but it was that lead me to think |
+ | Maybe another problem, but it was that lead me to think about exact real arithmetic: |

− | using some Mandelbrot-plotting programs, the number of iterations must be prescribed by the user at the beginning. And when we zoom too deep into these Mandelbrot worlds, it will become ragged or smooth. Maybe solving this particular problem does not |
+ | using some Mandelbrot-plotting programs, the number of iterations must be prescribed by the user at the beginning. And when we zoom too deep into these Mandelbrot worlds, it will become ragged or smooth. Maybe solving this particular problem does not necessarily need the concept of exact real arithmetic, but it was the first time I began to think about such problems. |

See other numeric algorithms at [[Libraries and tools/Mathematics]]. |
See other numeric algorithms at [[Libraries and tools/Mathematics]]. |
||

− | === Why |
+ | === Why are there reals at all which are defined exactly, but are not computable? === |

See e.g. [[Chaitin's construction]]. |
See e.g. [[Chaitin's construction]]. |
||

Line 25: | Line 25: | ||

== Theory == |
== Theory == |
||

− | * Jean Vuillemin's [http://www.inria.fr/rrrt/rr-0760.html Exact real computer arithmetic with continued fractions] is very good article on the topic itself. It can serve |
+ | * Jean Vuillemin's [http://www.inria.fr/rrrt/rr-0760.html Exact real computer arithmetic with continued fractions] is very good article on the topic itself. It can also serve as a good introductory article, because it presents the connections to both mathematical analysis and [[Computer science#Computability theory|Computability theory]]. It discusses several methods, and it describes some of them in more details. |

* [http://www.cs.bham.ac.uk/~mhe/ Martín Escardó]'s project [http://www.dcs.ed.ac.uk/home/mhe/plume/ A Calculator for Exact Real Number Computation] -- its chosen functional language is Haskell, mainly because of its purity, lazyness, presence of lazy lists, pattern matching. Martín Escardó has many exact real arithetic materials also among his many [http://www.cs.bham.ac.uk/~mhe/papers/index.html papers]. |
* [http://www.cs.bham.ac.uk/~mhe/ Martín Escardó]'s project [http://www.dcs.ed.ac.uk/home/mhe/plume/ A Calculator for Exact Real Number Computation] -- its chosen functional language is Haskell, mainly because of its purity, lazyness, presence of lazy lists, pattern matching. Martín Escardó has many exact real arithetic materials also among his many [http://www.cs.bham.ac.uk/~mhe/papers/index.html papers]. |
||

− | * [http://users.info.unicaen.fr/~karczma/arpap/ Jerzy Karczmarczuk]'s paper with the funny title [http://users.info.unicaen.fr/~karczma/arpap/lazypi.ps.gz The Most Unreliable Technique in the World to compute pi] describes how to compute Pi as a lazy list of digits. |
+ | * [http://users.info.unicaen.fr/~karczma/arpap/ Jerzy Karczmarczuk]'s ''(wonderful, wonderful! gem of a)'' paper with the funny title [http://users.info.unicaen.fr/~karczma/arpap/lazypi.ps.gz The Most Unreliable Technique in the World to compute pi] describes how to compute Pi as a lazy list of digits. |

⚫ | |||

+ | == Implementations == |
||

⚫ | |||

+ | See [[Libraries and tools/Mathematics]] |
||

− | There are functional programming materials too, even with downloadable Haskell source. |
||

− | === [http://www.haskell.org/hawiki/ExactRealArithmetic ExactRealArithmetic] === |
||

− | This HaWiki article provides links to many implementations. |
||

⚫ | |||

− | = Exact Real Arithmetic = |
||

⚫ | |||

− | |||

− | Exact real arithmetic refers to an implementation of the computable real numbers. |
||

− | There are several implementations of exact real arithmetic in Haskell. |
||

− | |||

− | == BigFloat == |
||

− | |||

− | [http://medialab.freaknet.org/bignum/ BigFloat] is an implementation by Martin Guy. |
||

− | It works with streams of decimal digits (strictly in the range from 0 to 9) and a separate sign. |
||

− | The produced digits are always correct. |
||

− | Output is postponed until the code is certain what the next digit is. |
||

− | This sometimes means that [http://medialab.freaknet.org/bignum/dudeney.html no more data is output]. |
||

− | |||

− | == COMP == |
||

− | |||

− | COMP is an implementation by Yann Kieffer. The work is in beta, and the library isn't available yet. |
||

− | |||

− | == Era == |
||

− | |||

− | [http://www.cs.man.ac.uk/arch/dlester/exact.html Era] is an implementation (in Haskell 1.2) by David Lester. It is quite fast, possibly the fastest Haskell implementation. At 220 lines it is also the shortest. Probably the shortest implementation of exact real arithmetic in any language. |
||

− | |||

− | Here is a patch to get Era 1.0 to compile in Haskell 98. |
||

− | |||

− | {{{ |
||

− | --- Era.hs 2005-10-26 12:16:05.835361616 +0200 |
||

− | +++ Era.hs 2005-10-26 12:15:28.396053256 +0200 |
||

− | @@ -8,6 +8,10 @@ |
||

− | |||

− | module Era where |
||

− | |||

− | +import Ratio |
||

− | +import Char |
||

− | +import Numeric (readDec, readSigned) |
||

− | + |
||

− | data CR = CR_ (Int -> Integer) |
||

− | |||

− | instance Eq CR where |
||

− | @@ -179,8 +183,10 @@ |
||

− | digits :: Int |
||

− | digits = 40 |
||

− | |||

− | -instance Text CR where |
||

− | +instance Read CR where |
||

− | readsPrec p = readSigned readFloat |
||

− | + |
||

− | +instance Show CR where |
||

− | showsPrec p x = let xs = get_str digits x in |
||

− | if head xs == '-' then showParen (p > 6) (showString xs) |
||

− | else showString xs |
||

− | }}} |
||

− | |||

− | == Few Digits == |
||

− | |||

− | [http://r6.ca/ Few Digits] is an implementation by Russell O'Connor. This is a prototype of the implementation he intendeds to write in [http://coq.inria.fr/ Coq]. Once the Coq implementation is complete, the Haskell code could be extracted producing an implementation that would be proved correct. |
||

− | |||

− | == IC-Reals == |
||

− | |||

− | [http://www.doc.ic.ac.uk/~ae/exact-computation/#bm:implementations IC-Reals] is an implementation by Abbas Edalat, Marko Krznarć and Peter J. Potts. This implementation uses linear fractional transformations. |
||

− | |||

− | == NumericPrelude/Positional == |
||

− | |||

− | Represents a real number as pair {{{(exponent,[digit])}}}, where the digits are {{{Int}}}s in the open range {{{(-basis,basis)}}}. |
||

− | There is no need for an extra sign item in the number data structure. |
||

− | The {{{basis}}} can range from {{{10}}} to {{{1000}}}. |
||

− | (Binary representations can be derived from the hexadecimal representation.) |
||

− | Showing the numbers in traditional format (non-negative digits) |
||

− | fails for fractions ending with a run of zeros. |
||

− | However the internal representation with negative digits can always be shown |
||

− | and is probably more useful for further processing. |
||

− | An interface for the numeric type hierarchy of the NumericPrelude project is provided. |
||

− | |||

− | It features |
||

− | * basis conversion |
||

− | * basic arithmetic: addition, subtraction, multiplication, division |
||

− | * algebraic arithmetic: square root, other roots (no general polynomial roots) |
||

− | * transcendental arithmetic: pi, exponential, logarithm, trigonometric and inverse trigonometric functions |
||

− | |||

− | [http://darcs.haskell.org/numericprelude/src/Number/Positional.hs NumericPrelude: positional numbers] |
||

− | |||

− | http://darcs.augustsson.net/Darcs/CReal/ |
||

+ | [[Category:Mathematics]] |
||

[[Category:theoretical foundations]] |
[[Category:theoretical foundations]] |

## Latest revision as of 19:45, 26 December 2012

## Contents

## Introduction

Exact real arithmetic is an interesting area: it is based on a deep connection between

- numeric methods
- and deep theoretic foundations of algorithms (and mathematics).

Its topic: computable real numbers raise a lot of interesting questions rooted in mathematical analysis, arithmetic, but also Computability theory (see numbers-as-programs approaches).

Computable reals can be achieved by many approaches -- it is not one single theory.

### What it is *not*

Exact real arithmetic is not the same as *fixed* arbitrary precision reals (see `Precision(n)`

of Yacas).

Exact reals must allow us to run a huge series of computations, prescribing only the precision of the end result. Intermediate computations, and determining their necessary precision must be achieved automatically, dynamically.

Maybe another problem, but it was that lead me to think about exact real arithmetic: using some Mandelbrot-plotting programs, the number of iterations must be prescribed by the user at the beginning. And when we zoom too deep into these Mandelbrot worlds, it will become ragged or smooth. Maybe solving this particular problem does not necessarily need the concept of exact real arithmetic, but it was the first time I began to think about such problems.

See other numeric algorithms at Libraries and tools/Mathematics.

### Why are there reals at all which are defined exactly, but are not computable?

See e.g. Chaitin's construction.

## Theory

- Jean Vuillemin's Exact real computer arithmetic with continued fractions is very good article on the topic itself. It can also serve as a good introductory article, because it presents the connections to both mathematical analysis and Computability theory. It discusses several methods, and it describes some of them in more details.

- Martín Escardó's project A Calculator for Exact Real Number Computation -- its chosen functional language is Haskell, mainly because of its purity, lazyness, presence of lazy lists, pattern matching. Martín Escardó has many exact real arithetic materials also among his many papers.

- Jerzy Karczmarczuk's
*(wonderful, wonderful! gem of a)*paper with the funny title The Most Unreliable Technique in the World to compute pi describes how to compute Pi as a lazy list of digits.

## Implementations

See Libraries and tools/Mathematics

## Portal-like homepages

- Exact Computation: There are functional programming materials too, even with downloadable Haskell source.