## Exact Arithmetic on the Stern-Brocot Tree (2003)

Venue: | NIJMEEGS INSTITUUT VOOR INFORMATICA EN INFORMATEIKUNDE, 2003. HTTP://WWW.CS.RU.NL/RESEARCH/REPORTS/FULL/NIII-R0325.PDF |

Citations: | 8 - 2 self |

### BibTeX

@TECHREPORT{Niqui03exactarithmetic,

author = {Milad Niqui},

title = {Exact Arithmetic on the Stern-Brocot Tree},

institution = {NIJMEEGS INSTITUUT VOOR INFORMATICA EN INFORMATEIKUNDE, 2003. HTTP://WWW.CS.RU.NL/RESEARCH/REPORTS/FULL/NIII-R0325.PDF},

year = {2003}

}

### OpenURL

### Abstract

In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simpli ed version of the homographic and the quadratic algorithms [19, 12]. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant representation for real numbers.

### Citations

533 | Concrete mathematics
- Graham, Knuth, et al.
- 1989
(Show Context)
Citation Context ...d: on-line, call-by-need, corecursive, coalgebraic, etc.. 1 is a symmetric mathematical structure with remarkable algebraic and combinatorial properties. It was recently reintroduced by Graham et al. =-=[6]-=-. Bates [1] studies the tree thoroughly and compares it with other similar combinatorial structures such as Farey sequence, hyperbinary tree, Gray-code sequence and paper folding sequence. Both in [6]... |

237 |
Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ...ound this is extend our alphabet in order to get what is known as an admissible representation. Admissible representations are those which arise out of the Cantor space topology on the set of streams =-=[20]-=-. There are various ways of forming a ternary representation based on Stern{Brocot representation. We add one new symbol to our alphabet and consider ternary streams instead of binary streams. Thus we... |

73 |
Exact Real Computer Arithmetic with Continued Fractions
- Vuillemin
- 1990
(Show Context)
Citation Context ...e present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simplied version of the homographic and the quadratic algorithms [=-=19, 12]-=-. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant repr... |

42 | A new representation for exact real numbers
- Edalat, Potts
- 1997
(Show Context)
Citation Context ...f real numbers using an innite composition of contracting functions on a compact interval. These include the representations by means of Mobius transformations that is developed by Potts and Edalat [1=-=5, 4]-=-. Potts and Edalat's work generalises earlier works by Gosper [5], Vuillemin [19] and MenissierMorain [12]. Gosper, in his famous unpublished work [5], showed how to add and multiply two continued fra... |

35 |
Über eine zahlentheoretische Funktion
- Stern
(Show Context)
Citation Context ...a redundant representation for real numbers. Our basis is the Stern{Brocot tree which wassrst discovered by 19th century German mathematician Moritz Abraham Stern and French clockmaker Achille Brocot =-=[3, 17, 7]-=-. The tree itself 1 Also called: on-line, call-by-need, corecursive, coalgebraic, etc.. 1 is a symmetric mathematical structure with remarkable algebraic and combinatorial properties. It was recently ... |

18 |
Calcul des rouages par approximation, nouvelle méthode
- Brocot
(Show Context)
Citation Context ...a redundant representation for real numbers. Our basis is the Stern{Brocot tree which wassrst discovered by 19th century German mathematician Moritz Abraham Stern and French clockmaker Achille Brocot =-=[3, 17, 7]-=-. The tree itself 1 Also called: on-line, call-by-need, corecursive, coalgebraic, etc.. 1 is a symmetric mathematical structure with remarkable algebraic and combinatorial properties. It was recently ... |

14 |
Exact Real Arithmetic Using Möbius Transformations
- Potts
- 1998
(Show Context)
Citation Context ...f real numbers using an innite composition of contracting functions on a compact interval. These include the representations by means of Mobius transformations that is developed by Potts and Edalat [1=-=5, 4]-=-. Potts and Edalat's work generalises earlier works by Gosper [5], Vuillemin [19] and MenissierMorain [12]. Gosper, in his famous unpublished work [5], showed how to add and multiply two continued fra... |

11 |
exacte, conception, algorithmique et performances d’une implémentation informatique en précision arbitraire. Thèse, Université Paris 7
- Ménissier-Morain, Arithmétique
- 1994
(Show Context)
Citation Context ...e present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simplied version of the homographic and the quadratic algorithms [=-=19, 12]-=-. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant repr... |

10 | An On-line Arithmetic Unit for Bit-Pipelined Rational Arithmetic - Kornerup, Matula - 1988 |

9 | QArith: Coq formalisation of lazy rational arithmetic
- Niqui, Bertot
- 2003
(Show Context)
Citation Context ...the coecients to satisfy the precondition. Thus assuming 5 For this to work we need to dene a well-ordering on the set of tensors. This is always possible (See the discussion before Theorem 4.21 and [14]) 16 that Sn (Tn ; 1 ; ; n ) = (s; Tns ; 1s ; ; ns ), and that Tns = h as 2 n as 2 n 1 ::: as 0 bs 2 n bs 2 n 1 ::: bs 0 i we use the notations T ns ,T ns 1 and T ns 2 to denote the ... |

7 | LCF: A lexicographic binary representation of the rationals
- Kornerup, Matula
- 1995
(Show Context)
Citation Context ...'s algorithm on a bit-serial arithmetic unit. Their binary encoding of continued fraction expansion was a lexicographic one and they could use it to obtain a redundant representation for real numbers =-=[10]-=-. Our approach in the present paper is similar to the one by Kornerup and Matula. We introduce a binary representation of rational numbers based on continued fractions and we expand it to a redundant ... |

6 |
Simple canonical representation of rational numbers
- Bertot
- 2003
(Show Context)
Citation Context ... called transducers. Liardet and Stambul [11] present the quadratic algorithm using Raney's transducers and generalise it to compute rational functions involving continued fraction expansions. Bertot =-=[2]-=- discusses the fact that by taking this binary encoding for rational numbers one can simplify the complexity of mathematical proofs. In Section 2 we introduce the Stern{Brocot tree and some of its bas... |

5 | On the Teeth of Wheels
- Hayes
- 2000
(Show Context)
Citation Context ...a redundant representation for real numbers. Our basis is the Stern{Brocot tree which wassrst discovered by 19th century German mathematician Moritz Abraham Stern and French clockmaker Achille Brocot =-=[3, 17, 7]-=-. The tree itself 1 Also called: on-line, call-by-need, corecursive, coalgebraic, etc.. 1 is a symmetric mathematical structure with remarkable algebraic and combinatorial properties. It was recently ... |

5 |
Many-Valued Real Functions Computable by Finite Transducers using IFSRepresentations
- Konečn´y
- 2000
(Show Context)
Citation Context ... which case we absorb the next element of the input stream. There have been many theoretical and practical instances of applying this idea. A rather general theoretical approach is taken by Konecny [8=-=-=-], where the limitation theorems for IFSrepresentations of real numbers is given. IFS-representations are representations of real numbers using an innite composition of contracting functions on a comp... |

5 |
Algebraic computations with continued fractions
- Liardet, Stambul
- 1998
(Show Context)
Citation Context ...esented in Raney [16]. Raney uses this binary representation to devise the homographic algorithm on continued fractions and he makes use ofsnite-state automata called transducers. Liardet and Stambul =-=[11]-=- present the quadratic algorithm using Raney's transducers and generalise it to compute rational functions involving continued fraction expansions. Bertot [2] discusses the fact that by taking this bi... |

4 |
Self-Matching and Interleaving in Some Integer Sequences and the Gauss Map
- Bates
- 2001
(Show Context)
Citation Context ... call-by-need, corecursive, coalgebraic, etc.. 1 is a symmetric mathematical structure with remarkable algebraic and combinatorial properties. It was recently reintroduced by Graham et al. [6]. Bates =-=[1]-=- studies the tree thoroughly and compares it with other similar combinatorial structures such as Farey sequence, hyperbinary tree, Gray-code sequence and paper folding sequence. Both in [6] and in [1]... |

2 |
On continued fractions and automata. Mathematische Annalen
- Raney
- 1973
(Show Context)
Citation Context ...ional numbers based on the tree is introduced. This representation basically boils down to the unary encoding of the regular continued fraction expansion of rational numbers and is presented in Raney =-=[16]-=-. Raney uses this binary representation to devise the homographic algorithm on continued fractions and he makes use ofsnite-state automata called transducers. Liardet and Stambul [11] present the quad... |

1 |
http://coqcvs.inria.fr/cgi-bin/cvswebcoq.cgi/ contrib/Nijmegen/QArith/, may 2003. Files under cvs version of Coq 7.4
- Niqui, Bertot
(Show Context)
Citation Context ...which is a special case of the proof of theorem 4.21), is also formalised in Coq. The algorithms and their formalisation in Coq are available online as contribution to the Coq contribution library at =-=[13]-=-. This formalisation is joint work with Yves Bertot and is explained in [14]. 7 With the terminology of [8] the above representation is a slow IFS-representation. 23 0 1 1 0 1 1 1 2 2 1 3 3 L R M 1 3 ... |