## Time- and space-efficient evaluation of some hypergeometric constants (2007)

Citations: | 8 - 1 self |

### BibTeX

@MISC{Cheng07time-and,

author = {Howard Cheng and Guillaume Hanrot and Emmanuel Thomé and Eugene Zima Paul and Thème Sym and Howard Cheng and Guillaume Hanrot and Emmanuel Thomé and Eugene Zima and Paul Zimmermann and Projet Cacao},

title = {Time- and space-efficient evaluation of some hypergeometric constants},

year = {2007}

}

### OpenURL

### Abstract

apport de recherche ISSN 0249-6399 ISRN INRIA/RR--6105--FR+ENG

### Citations

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Citation Context ...n denominator �N−2 i=0 q(i) has a relatively small size of O(N log N). An approach commonly known as “binary splitting” has been independently discovered and used by many authors in such computations =-=[1, 2, 3, 4, 9, 10, 12, 13]-=-. In binary splitting, the use of fast integer multiplication yields a total time complexity of O(M(d log d) log d) = O(M(d) log 2 d), where M(t) = O(t log t log log t) is the complexity of multiplica... |

73 | MPFR: A multiple-precision binary floating-point library with correct rounding
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Citation Context ...actored form to a flat integer. For this purpose, we use the same kind of algorithm as mentioned in [14] for the computation of n!. We implemented our algorithm in C++ with the GMP and MPFR libraries =-=[11, 7]-=-. We modified the GMP library with an improved FFT multiplication code [8]. We compare our results with the two programs mentioned in Section 2, which compute digits of π using Formula (3). For the pu... |

41 |
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Citation Context ...use of fast integer multiplication yields a total time complexity of O(M(d log d) log d) = O(M(d) log 2 d), where M(t) = O(t log t log log t) is the complexity of multiplication of two t-bit integers =-=[15]-=-. The O(d logd) space complexity of the algorithm is the same as the size of the computed numerator and denominator. The numerator and denominator computed by the binary splitting approach typically h... |

35 |
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Citation Context ... the PartialAdd operation mentioned in Section 4, and also the final expansion of T and Q from the factored form to a flat integer. For this purpose, we use the same kind of algorithm as mentioned in =-=[14]-=- for the computation of n!. We implemented our algorithm in C++ with the GMP and MPFR libraries [11, 7]. We modified the GMP library with an improved FFT multiplication code [8]. We compare our result... |

22 | Fast multiprecision evaluation of series of rational numbers
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Citation Context ...tions and other constants —including the exponential function, logarithms, trigonometric functions, and constants such as π or Apéry’s constant ζ(3)— is commonly carried out by evaluating such series =-=[10, 12]-=-. For example, we have or 1 π = 12 2ζ(3) = i=0 p(i) q(i) ∞� (−1) n545140134n + 13591409 6403203n+3/2 (6n)! (3n)!n! 3 n=0 ∞� n=0 (−1) n (205n 2 + 250n + 77) (n + 1)!5 n! 5 (2) (3) . (4) (2n + 2)! 5 Ass... |

20 | Fast multiplication and its applications
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Citation Context ...n denominator �N−2 i=0 q(i) has a relatively small size of O(N log N). An approach commonly known as “binary splitting” has been independently discovered and used by many authors in such computations =-=[1, 2, 3, 4, 9, 10, 12, 13]-=-. In binary splitting, the use of fast integer multiplication yields a total time complexity of O(M(d log d) log d) = O(M(d) log 2 d), where M(t) = O(t log t log log t) is the complexity of multiplica... |

12 | A GMP-based implementation of Schonhage-Strassen‘s large integer multiplication algorithm
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Citation Context ...rithm as mentioned in [14] for the computation of n!. We implemented our algorithm in C++ with the GMP and MPFR libraries [11, 7]. We modified the GMP library with an improved FFT multiplication code =-=[8]-=-. We compare our results with the two programs mentioned in Section 2, which compute digits of π using Formula (3). For the purpose of comparison, we focus on the time for the evaluation of the fracti... |

11 | Acceleration of Euclidean Algorithm and Rational Number Reconstruction
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Citation Context ...epresentation by factoring. Additionally practicality of the algorithm depends on the availability of the implementation of the asymptotically fast rational reconstruction algorithm (for example, see =-=[16]-=-). We also mention the gmp-chudnovsky program [17], which uses the binary splitting method to compute digits of π using Formula (3). Two modifications are made to the classical method described above.... |

8 |
Fast Multiplication and its Applications,” http://cr.yp.to/papers.html#multapps
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Citation Context ...n denominator �N−2 i=0 q(i) has a relatively small size of O(N log N). An approach commonly known as “binary splitting” has been independently discovered and used by many authors in such computations =-=[1, 2, 3, 4, 9, 10, 12, 13]-=-. In binary splitting, the use of fast integer multiplication yields a total time complexity of O(M(d log d) log d) = O(M(d) log 2 d), where M(t) = O(t log t log log t) is the complexity of multiplica... |

7 |
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4 | Space-efficient evaluation of hypergeometric series
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Citation Context ...y removing some of these factors from the computation. For this purpose, a partially factored representation was introduced in the binary splitting process. Subsequently, Cheng, Gergel, Kim, and Zima =-=[5]-=- applied modular computation and rational number reconstruction to obtain the reduced fraction. If the reduced numerator and denominator have size O(d), the resulting algorithm has a space complexity ... |

4 |
On accelerated methods to evaluate sums of produts of rational numbers
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Citation Context ...ethod, by avoiding the unneeded computation of the common divisor between the numerator and denominator. Several approaches have already been taken in that direction.s4 H. Cheng et al. In particular, =-=[6]-=- suggests to use a partially factored form for the computed quantities, in order to efficiently identify and remove common factors, and [17] goes further by explicitly constructing the common divisor ... |

4 |
Binary splitting method. http://numbers.computation.free.fr/Constants/Algorithms/splitting.html
- Gourdon, Sebah
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Citation Context ...tions and other constants —including the exponential function, logarithms, trigonometric functions, and constants such as π or Apéry’s constant ζ(3)— is commonly carried out by evaluating such series =-=[10, 12]-=-. For example, we have or 1 π = 12 2ζ(3) = i=0 p(i) q(i) ∞� (−1) n545140134n + 13591409 6403203n+3/2 (6n)! (3n)!n! 3 n=0 ∞� n=0 (−1) n (205n 2 + 250n + 77) (n + 1)!5 n! 5 (2) (3) . (4) (2n + 2)! 5 Ass... |

4 |
gmp-chudnovsky.c code for computing digits of π using the Gnu MP library. Available at http://www.swox.com/gmp/pi-with-gmp.html, 2002. de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 1
- Xue
(Show Context)
Citation Context ...en taken in that direction.s4 H. Cheng et al. In particular, [6] suggests to use a partially factored form for the computed quantities, in order to efficiently identify and remove common factors, and =-=[17]-=- goes further by explicitly constructing the common divisor and dividing out the numerator and denominator. The present work builds on top of this strategy and uses a fully factored form in the binary... |

2 |
Fast evaluation of hypergeometric functions by FEE
- Karatsuba
- 1997
(Show Context)
Citation Context |

1 |
Binary splitting method. http://numbers. computation.free.fr/Constants/Algorithms/splitting.html
- Gourdon, Sebah
- 2001
(Show Context)
Citation Context ...tions and other constants —including the exponential function, logarithms, trigonometric functions, and constants such as π or Apéry’s constant ζ(3)— is commonly carried out by evaluating such series =-=[10, 12]-=-. For example, we have or 1 π = 12 2ζ(3) = i=0 p(i) q(i) ∞� (−1) n545140134n + 13591409 6403203n+3/2 (6n)! (3n)!n! 3 n=0 ∞� n=0 (−1) n (205n 2 + 250n + 77) (n + 1)!5 n! 5 (2) (3) . (4) (2n + 2)! 5 Ass... |