# Exact real arithmetic

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EndreyMark (Talk | contribs) (or ,,computable reals'', a topic embracing analysis, arithmetic, computability. Introductory text, and links: 1) Vuillemin's strong theoretical foundation article 2) Exact Computation, a portal-like) |
EndreyMark (Talk | contribs) (What exact real arithmetic is NOT: _fixed_ arbitrary precision decimals.) |
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* numeric methods | * numeric methods | ||

* and deep theoretic fondations of algorithms (and mathematics). | * and deep theoretic fondations 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). | + | Its topic: computable real numbers raise a lot of interesting questions rooted in mathematical analysis, arithmetic, but also [http://en.wikipedia.org/wiki/Computability_theory computability theory] (see numbers-as-programs approaches). |

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

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+ | === What it is ''not'' === | ||

+ | Exact real arithmetic is not the same as ''fixed'' arbitrary precision reals (see <code>Precision(n)</code> of [http://yacas.sourceforge.net/ Yacas]). | ||

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+ | 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. | ||

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+ | Maybe another problem, but it was that lead me to think on 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 need necessarily the concept of exact real arithmetic, but it was the first time I began to think on such problems. | ||

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

## Revision as of 12:33, 22 April 2006

## Contents |

## 1 Introduction

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

- numeric methods
- and deep theoretic fondations 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.

### 1.1 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 on 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 need necessarily the concept of exact real arithmetic, but it was the first time I began to think on such problems.

## 2 Theory

Jean Vuillemin's Exact real computer arithmetic with continued fractions is very good article on the topic itself. It can serve also as a good introductory article, too, 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.

## 3 Portal-like homepages

### 3.1 Exact Computation

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