Difference between revisions of "Exact real arithmetic"
(inserted content from Hawiki) 
(adapted Hawiki syntax) 

Line 39:  Line 39:  
This HaWiki article provides links to many implementations. 
This HaWiki article provides links to many implementations. 

−  
+  == Implementations == 

−  = Exact Real Arithmetic = 

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

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

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

Line 53:  Line 52:  
This sometimes means that [http://medialab.freaknet.org/bignum/dudeney.html no more data is output]. 
This sometimes means that [http://medialab.freaknet.org/bignum/dudeney.html no more data is output]. 

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

−  == Era == 
+  === 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. 
[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 20051026 12:16:05.835361616 +0200 

−  +++ Era.hs 20051026 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 == 
+  === 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. 
[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. 

−  == ICReals == 
+  === ICReals === 
[http://www.doc.ic.ac.uk/~ae/exactcomputation/#bm:implementations ICReals] is an implementation by Abbas Edalat, Marko Krznarć and Peter J. Potts. This implementation uses linear fractional transformations. 
[http://www.doc.ic.ac.uk/~ae/exactcomputation/#bm:implementations ICReals] is an implementation by Abbas Edalat, Marko Krznarć and Peter J. Potts. This implementation uses linear fractional transformations. 

−  == NumericPrelude/Positional == 
+  === NumericPrelude/Positional === 
−  Represents a real number as pair 
+  Represents a real number as pair <hask>(exponent,[digit])</hask>, where the digits are <hask>Int</hask>s in the open range <hask>(basis,basis)</hask>. 
There is no need for an extra sign item in the number data structure. 
There is no need for an extra sign item in the number data structure. 

−  The 
+  The <hask>basis</hask> can range from <hask>10</hask> to <hask>1000</hask>. 
(Binary representations can be derived from the hexadecimal representation.) 
(Binary representations can be derived from the hexadecimal representation.) 

Showing the numbers in traditional format (nonnegative digits) 
Showing the numbers in traditional format (nonnegative digits) 

Line 112:  Line 84:  
It features 
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.haskell.org/numericprelude/src/Number/Positional.hs NumericPrelude: positional numbers] 

⚫  
+  [[Category:Packages]] 

[[Category:theoretical foundations]] 
[[Category:theoretical foundations]] 
Revision as of 12:01, 22 October 2006
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 numbersasprograms 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 on exact real arithmetic: using some Mandelbrotplotting 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.
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 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.
 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 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.
Portallike homepages
Exact Computation
There are functional programming materials too, even with downloadable Haskell source.
ExactRealArithmetic
This HaWiki article provides links to many implementations.
Implementations
Exact real arithmetic refers to an implementation of the computable real numbers. There are several implementations of exact real arithmetic in Haskell.
BigFloat
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 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
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 mirror: http://darcs.augustsson.net/Darcs/CReal/
Few Digits
Few Digits is an implementation by Russell O'Connor. This is a prototype of the implementation he intendeds to write in Coq. Once the Coq implementation is complete, the Haskell code could be extracted producing an implementation that would be proved correct.
ICReals
ICReals 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 (nonnegative 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