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## Motivations for an arbitrary precision interval arithmetic and the MPFI library (2002)

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### Other Repositories/Bibliography

Venue: | Reliable Computing |

Citations: | 38 - 6 self |

### Citations

1275 |
Interval Analysis
- Moore
- 1966
(Show Context)
Citation Context ...nd also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic =-=[21, 2, 18]-=-. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but i... |

650 |
Interval Methods for Systems of Equations
- Neumaier
- 1990
(Show Context)
Citation Context ...complete and really reliable arbitrary precision interval arithmetic and this led us to implement our own library. 2 Theoretical Background The theoretical result underlying this idea can be found in =-=[23]-=-: let us denote by X an interval, by f a function and by F an interval extension of f, where F is given by a Lipschitz expression, let ε correspond to the current computing precision p: ε = 2 −p , the... |

590 | Global Optimization Using Interval Analysis, - Hansen - 1992 |

387 | Applied Interval Analysis. - Jaulin, Kieffer, et al. - 2001 |

350 |
Rigorous global search: continuous problems.
- Kearfott
- 1996
(Show Context)
Citation Context ..., without requiring restarting the program from the very beginning. This work will be pursued with a careful study of the solution of linear systems and of global optimization of continuous functions =-=[17, 22]-=-. Applications such as parameter estimation in control theory [20] will offer the opportunity to gain further insight in the development of new algorithms. Acknowledgements We would like to thank Geor... |

159 |
algorithms: an Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics
- Ideals
- 2001
(Show Context)
Citation Context ...vations for the development of MPFI was to propose effective methods for solving algebraic systems with a finite number of solutions. We focus on strategies relying on Gröbner bases calculations (see =-=[12]-=- for an introduction), which can handle realistic problems in numerous application fields. A Gröbner basis characterizes the ideal associated to a system of equations by a basis of the ideal and a fun... |

137 |
Solving zero-dimensional systems through the rational univariate representation.
- Rouillier
- 1999
(Show Context)
Citation Context ...detailed in [4] Another approach continues with exact computations and computes the so-called Rational Univariate Representation (RUR) as suggested RevolRouillier-v4.tex; 20/01/2005; 14:53; p.9s10 in =-=[36]-=-. A RUR has the following shape: ⎧ ⎪⎨ ⎪⎩ h(T ) = 0, X1 = h1(T )/h0(T ), . Xn = hn(T )/h0(T ), where T denotes a new variable, h and hi, i = 0 . . . n, belong to Q | [T ], and h and h0 are coprime. In ... |

114 |
Efficient isolation of polynomial real roots.
- Rouillier, Zimmermann
- 2003
(Show Context)
Citation Context ...itions and multiplications) and automatic differentiation (forward differentiation by overloading operators and functions). 4 Applications The MPFI library is already in use. Rouillier and Zimmermann =-=[26]-=- have developed a hybrid algorithm (symbolic/interval) for isolating real roots of polynomials, Revol [24] has implemented interval Newton algorithm [15] adapted to multiple precision computations. A ... |

96 |
An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations
- Auzinger, Stetter
- 1988
(Show Context)
Citation Context .... . , αd are the complex roots of S. As a consequence, the simultaneous computation of all eigenvalues of MX1 , . . . , MXn provides an approximation of all the complex solutions of S, as detailed in =-=[4]-=- Another approach continues with exact computations and computes the so-called Rational Univariate Representation (RUR) as suggested RevolRouillier-v4.tex; 20/01/2005; 14:53; p.9s10 in [36]. A RUR has... |

89 | Emden, Interval arithmetic: From principles to implementations,
- Hickey, Ju, et al.
- 2001
(Show Context)
Citation Context ...or this choice in IntLab does not hold for MPFI, since in MPFI, every multiple precision operation is a software one. The specifications used for the implementation are based on the IEEE 754 standard =-=[19]-=-: − an interval is a connected closed subset of IR; − if op is an n-ary operation and x1, . . . , xn are intervals, the result of op(x1, . . . , xn), the operation op performed with interval arguments... |

82 | Polynomial real root isolation using Descartes’ rule of signs.
- Collins, Akritas
- 1976
(Show Context)
Citation Context ...ives 1 (resp. 0) when applied to the polynomial (x + 1) dPk,c(1/(1 + x)), then the polynomial P has exactly one root (resp. no root) in the interval ( c 2k , c+1 2k ). Theorems due to Collins/Akritas =-=[10]-=- and Vincent [42] state that for sufficiently large values of k, Descartes rule of signs when applied to the polynomial (x + 1) dPk,c(1/(1 + x)) always yields 0 or 1. We are only interested in the sig... |

69 |
C-XSC: A C++ Class Library for Extended Scientific Computing
- Klatte, Kulisch, et al.
- 1993
(Show Context)
Citation Context ...nd also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic =-=[21, 2, 18]-=-. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but i... |

65 |
Algorithm 737: INTLIB: A Portable FORTRAN 77
- Kearfott, Dawande, et al.
(Show Context)
Citation Context ...g a molecular conformation entails the minimization of an energy function and requires accurate evaluations of this energy function. Most usual interval arithmetic libraries (PROFIL/BIAS [27], Intlib =-=[21]-=-. . . ) or compilers (Sun Forte, e.g. C++ [41]) are based on double floating-point numbers and do not propose arbitrary precision interval arithmetic. The XSC languages [25] include a long accumulator... |

64 |
Bounding solutions of systems of equations using interval analysis.
- Hansen, Sengupta
- 1981
(Show Context)
Citation Context ...of small boxes can be as accurate as desired and circumvents the overestimation induced by the inclusion into one single box. A second example comes from linear algebra: Hansen and Sengupta algorithm =-=[16]-=- “solves” a linear system Ax = b with A an interval matrix and b an interval vector, i.e. it computes an enclosure of the convex hull of {x : ∃A ∈ A, ∃b ∈ b, Ax = b}. This algorithm firstly computes C... |

62 | A fortran multiple-precision arithmetic package
- Brent
- 1978
(Show Context)
Citation Context ...ry precision computations but implements few related functionalities. Such an arithmetic was also mentioned as an easy-to-implement extension to Brent’s multiple precision package MP as early as 1981 =-=[7]-=-. Anyway, none of the aforementioned packages implements a complete and really reliable arbitrary precision interval arithmetic and this led us to implement our own library. 2 Theoretical Background T... |

41 |
An interval Newton method
- Hansen, Greenberg
- 1983
(Show Context)
Citation Context ...ry is already in use. Rouillier and Zimmermann [26] have developed a hybrid algorithm (symbolic/interval) for isolating real roots of polynomials, Revol [24] has implemented interval Newton algorithm =-=[15]-=- adapted to multiple precision computations. A main advantage of using MPFI is that one is no more limited by the computing precision: for instance one can impose arbitrary accuracy on both the root a... |

37 | Fast and Parallel Interval Arithmetic,
- Rump
- 1999
(Show Context)
Citation Context ...point numbers: this is not visible for the user but ensures that the swelling of intervals’ widths is less important than with the midpoint-radius representation such as implemented by Rump in IntLab =-=[27, 28]-=-. Indeed, switching the rounding modes incurs no penalty with multiple precision arithmetic and the motivation for this choice in IntLab does not hold for MPFI: every multiple precision operation is a... |

32 | The iRRAM: Exact arithmetic in C++.
- Muller
- 2001
(Show Context)
Citation Context ...other words, in most cases an increase in the computing precision yields an improved accuracy on the results. This is also the starting point of Müller’s work on an effective simulation of a Real RAM =-=[22]-=-, following the theoretical results by Brattka and Hertling [6] on the feasibility of a Real RAM. In Müller’s work, a computation is performed and, if the final accuracy is not sufficient, then the wh... |

23 |
Precise computation using range arithmetic via
- Aberth, Schaefer
- 1992
(Show Context)
Citation Context ... roundings of elementary functions (0.6 ulp for intpak in Maple, 1 ulp for Mathematica), incorrect roundings 1 and several other undue assumptions can be found. Other works include “range arithmetic” =-=[1, 2]-=- and IntLab [40]. The range library is a (now quite old) arbitrary precision library which primarily aims at indicating the number of correct digits rather than at performing interval arithmetic. Howe... |

19 | Polynomial root finding using iterated eigenvalue computation
- Fortune
- 2001
(Show Context)
Citation Context ...tte’s polynomials: our algorithm does not use the fact that those polynomials are sparse. (extract from [38], reference [4] corresponds to reference [6] in this paper and [9, 8] have been replaced by =-=[14]-=- here) Compared to the same algorithm where all operations are performed using exact arithmetic, the results are as follows. For Laguerre polynomials of high degree (900 or 1000), the times are very c... |

16 |
Introduction to interval analysis .
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...nd also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic =-=[21, 2, 18]-=-. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but i... |

16 |
Feasible real random access machines
- Brattka, Hertling
- 1998
(Show Context)
Citation Context ...n yields an improved accuracy on the results. This is also the starting point of Müller’s work on an effective simulation of a Real RAM [22], following the theoretical results by Brattka and Hertling =-=[6]-=- on the feasibility of a Real RAM. In Müller’s work, a computation is performed and, if the final accuracy is not sufficient, then the whole computation is restarted with an increased precision; this ... |

16 |
H.:“Sur la resolution des équations numériques
- Vincent
(Show Context)
Citation Context ...when applied to the polynomial (x + 1) dPk,c(1/(1 + x)), then the polynomial P has exactly one root (resp. no root) in the interval ( c 2k , c+1 2k ). Theorems due to Collins/Akritas [10] and Vincent =-=[42]-=- state that for sufficiently large values of k, Descartes rule of signs when applied to the polynomial (x + 1) dPk,c(1/(1 + x)) always yields 0 or 1. We are only interested in the signs of coefficient... |

13 | Solving nonlinear systems by constraint inversion and interval arithmetic
- Ceberio, Granvilliers
- 2000
(Show Context)
Citation Context ...terval arithmetic library), a library implementing arbitrary precision interval arithmetic which is described in this paper. This quest for extra accuracy can be found in other works such as those by =-=[9]-=- where polynomial expressions are symbolically rewritten before being evaluated, so as to reduce the overestimation due to dependency, or by [5] where high-order Taylor expansions are used. In this la... |

13 | An experimental interval arithmetic package in Maple
- Connell, Corless
- 1993
(Show Context)
Citation Context ... entails the minimization of an energy function and requires accurate evaluations of this energy function. Several multiple precision interval packages are available. Let us quote for instance intpak =-=[11]-=- and intpakX [13] for Maple or a similar package for Mathematica [17]. Due to unverified assumptions on the roundings of elementary functions (0.6 ulp for intpak in Maple, 1 ulp for Mathematica), to b... |

12 |
PROFIL/BIAS - A Fast Interval Library.
- Knueppel
- 1994
(Show Context)
Citation Context ...cluded as well (trigonometric and hyperbolic trigonometric functions and their reciprocals). The planned functionalities, that will be added in a near future, include a C++ interface à la Profil/BIAS =-=[19]-=- for ease of use, basic tools for linear algebra (vector and matrix data types, additions and multiplications) and automatic differentiation (forward differentiation by overloading operators and funct... |

12 |
Precise Numerical Methods Using C
- Aberth
- 1998
(Show Context)
Citation Context ... roundings of elementary functions (0.6 ulp for intpak in Maple, 1 ulp for Mathematica), incorrect roundings 1 and several other undue assumptions can be found. Other works include “range arithmetic” =-=[1, 2]-=- and IntLab [40]. The range library is a (now quite old) arbitrary precision library which primarily aims at indicating the number of correct digits rather than at performing interval arithmetic. Howe... |

11 | Automatic linear correction of rounding errors
- Langlois
- 2001
(Show Context)
Citation Context ...studies performed by the CEA (French Nuclear Agency) on the simulation of nuclear plant accidents and also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena =-=[20]-=-. Another approach consists in computing with certified enclosures, namely interval arithmetic [21, 2, 18]. The fundamental principle of this arithmetic consists in replacing every number by an interv... |

11 |
Interval arithmetic in Mathematica
- Keiper
- 1993
(Show Context)
Citation Context ...hose range of exponents is {−1022, · · · , 1023}. Several arbitrary precision interval packages are available, for instance intpak [11] and intpakX [15] for Maple or a similar package for Mathematica =-=[24]-=-. These packages cannot be considered as reliable: unverified assumptions on the roundings of elementary functions (0.6 ulp for intpak in Maple, 1 ulp for Mathematica), incorrect roundings 1 and sever... |

9 |
Interval newton iteration in multiple precision for the univariate case. Numerical Algorithms
- Revol
- 2003
(Show Context)
Citation Context .... Degree Computing precision (MPFI algorithm) Exact IS 53 107 215 431 863 algorithm 200 2 0 0 2 0 0 46400 400 2 0 0 0 2 0 185536 600 2 0 0 0 2 0 ? 5.3. Approximation of the zeroes of a function Revol =-=[34]-=- has implemented the interval Newton algorithm [18] and has adapted it to arbitrary precision computations. In this example, the algorithm is primarily intended for interval computations. It has been ... |

9 |
C++ Interval Arithmetic Programming Reference
- Microsystems
(Show Context)
Citation Context ...ation of an energy function and requires accurate evaluations of this energy function. Most usual interval arithmetic libraries (PROFIL/BIAS [27], Intlib [21]. . . ) or compilers (Sun Forte, e.g. C++ =-=[41]-=-) are based on double floating-point numbers and do not propose arbitrary precision interval arithmetic. The XSC languages [25] include a long accumulator type, initially for accurate dot product. It ... |

8 | Kearfott, Rigorous Global Search Continuous Problems - Baker - 1996 |

8 | On stopping criteria in verified nonlinear systems or optimization algorithms
- Kearfott, Walster
- 2000
(Show Context)
Citation Context ...tions. A main advantage of using MPFI is that one is no more limited by the computing precision: for instance one can impose arbitrary accuracy on both the root and the residual in Newton’s algorithm =-=[4]-=-. Furthermore, the aforementioned implementations manage to adapt dynamically the precision to the computing needs without restarting the whole program. This desirable feature will be sought after for... |

8 |
Newton’s algorithm using multiple precision interval arithmetic
- Revol
(Show Context)
Citation Context ...s and functions). 4 Applications The MPFI library is already in use. Rouillier and Zimmermann [26] have developed a hybrid algorithm (symbolic/interval) for isolating real roots of polynomials, Revol =-=[24]-=- has implemented interval Newton algorithm [15] adapted to multiple precision computations. A main advantage of using MPFI is that one is no more limited by the computing precision: for instance one c... |

7 |
Verified high-order inversion of functional dependencies and interval Newton methods
- Berz, Hoefkens
- 2001
(Show Context)
Citation Context ...a accuracy can be found in other works such as those by [9] where polynomial expressions are symbolically rewritten before being evaluated, so as to reduce the overestimation due to dependency, or by =-=[5]-=- where high-order Taylor expansions are used. In this latter work, the time overhead is about 1500 for a single evaluation, however it is compensated by the reduction in the number of steps performed ... |

6 |
W.: Intervallrechnung in Maple - Die Erweiterung intpakX zum Paket intpak der Share-Library, Preprint 99/2 des Instituts für Wissenschaftliches Rechnen und Mathematische Modellbildung (IWRMM), Universität
- Geulig, Krämer
- 1999
(Show Context)
Citation Context ...mization of an energy function and requires accurate evaluations of this energy function. Several multiple precision interval packages are available. Let us quote for instance intpak [11] and intpakX =-=[13]-=- for Maple or a similar package for Mathematica [17]. Due to unverified assumptions on the roundings of elementary functions (0.6 ulp for intpak in Maple, 1 ulp for Mathematica), to bugged roundings (... |

5 |
Numerical toolbox for verified computing II – Advanced Numerical Problems
- Krämer, Kulisch, et al.
- 1998
(Show Context)
Citation Context ...c. The XSC languages [25] include a long accumulator type, initially for accurate dot product. It has also been used for Horner evaluation of a polynomial in the interval Newton algorithm for example =-=[26]-=-. The maximal precision is thus limited, since the long accumulator type relies on the double type, whose range of exponents is {−1022, · · · , 1023}. Several arbitrary precision interval packages are... |

5 |
Portable software for interval arithmetic
- Yohe
(Show Context)
Citation Context ...t are available. Arbitrary precision interval arithmetic was also mentioned as an easy-to-implement extension to Brent’s multiple precision package MP as early as 1978 [8]. It was implemented in 1980 =-=[43]-=-, using the Augment preprocessor (which replaced arithmetic operators by calls to the appropriate functions, as overloading was not available). However, the precision was not really dynamic, since the... |

4 |
La bibliothèque CADNA : présentation et utilisation. Manual, Laboratoire d’Informatique de Paris 6
- Chesneaux, Guilain, et al.
- 1996
(Show Context)
Citation Context ... Let us cite studies performed by the CEA (French Nuclear Agency) on the simulation of nuclear plant accidents and also softwares controlling and possibly correcting numerical programs, such as Cadna =-=[10]-=- or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic [21, 2, 18]. The fundamental principle of this arithmetic consists in replacing every number... |

4 |
The MPFI library http://www.ens-lyon.fr/~nrevol
- Revol, Rouillier
- 2001
(Show Context)
Citation Context ...nd efficiency reasons (MPFR is based on GMP and efficiency is a motto for its developers) and also because of the availability of the source code, we chose MPFR. The corresponding library, named MPFI =-=[25]-=-, is a portable library written in C for arbitrary precision interval arithmetic. It is based on the GNU MP library and on the MPFR library and is part of the latter. The largest achievable computing ... |

3 |
Precise computation using range arithmetic
- Aberth, Schaefer
- 1992
(Show Context)
Citation Context ...g towards −∞ of 1−9.10 −5 gives 1 instead of 0.999 in Maple v6 and v7), and to several undue assumptions, these packages cannot be considered as reliable. Earlier works include the “range arithmetic” =-=[1]-=-, a multiple precision library which aims at indicating the number of correct digits rather than at performing interval arithmetic, and IntLab [27] which primarily implements efficiently interval algo... |

3 |
The interval library filib++ 2.0. http://www.math.uni-wuppertal.fr/org/WRST/ software/filib.html, Preprint 2001/4, Universität
- Lerch, Tischler, et al.
- 2001
(Show Context)
Citation Context ...se to 2 compared to MPFR, whatever the precision. To give a precise idea of the additional cost caused by the use of an arbitrary precision arithmetic, we compare MPFI with fi_lib. The library fi_lib =-=[29]-=- is based on the native double type for floating-point RevolRouillier-v4.tex; 20/01/2005; 14:53; p.6sArbitrary precision interval arithmetic and the MPFI library 7 numbers. A first important feature o... |

2 |
Arithmos: a reliable integrated computational environment
- Group
- 2001
(Show Context)
Citation Context ...functions. To sum up, compliance with the IEEE 754 standard for floating-point arithmetic, extended to elementary functions, is welcome. 3sThe Arithmos project of the CANT team, U. Antwerpen, Belgium =-=[8]-=-, or the MPFR library (Multiple Precision Floating-point Reliable library), developed by the Spaces team, INRIA Lorraine, France [12], are such libraries. For portability and efficiency reasons (MPFR ... |

2 |
The MPFR library
- Daney, Hanrot, et al.
- 2001
(Show Context)
Citation Context ...ome. 3sThe Arithmos project of the CANT team, U. Antwerpen, Belgium [8], or the MPFR library (Multiple Precision Floating-point Reliable library), developed by the Spaces team, INRIA Lorraine, France =-=[12]-=-, are such libraries. For portability and efficiency reasons (MPFR is based on GMP and efficiency is a motto for its developers) and also because of the availability of the source code, we chose MPFR.... |

2 |
The MPFR library. http://www.mpfr.org
- Daney, Hanrot, et al.
- 2001
(Show Context)
Citation Context ...thmos project of the CANT team, U. Antwerpen, Belgium [9], or in the MPFR library (Multiple Precision Floating-point Reliable arithmetic library), developed by the Spaces team, INRIA Lorraine, France =-=[13]-=-. For portability and efficiency reasons (MPFR is based on GMP and efficiency is a motto for its developers) and also because of the availability of the source code, we chose MPFR. The corresponding i... |

2 |
Validating polynomial numerical computations with complementary automatic methods
- Langlois, Revol
- 2001
(Show Context)
Citation Context ...been performed on Chebyshev polynomials of degree up to 40, on the Wilkinson polynomial of degree 20 [34], and on a polynomial with a double root to illustrate the correct behaviour of this algorithm =-=[28]-=-. To conclude, the two aforementioned implementations managed to adapt dynamically the precision to the computing needs without restarting the whole program. This desirable feature will be sought afte... |

1 |
Interval arithmetic
- Keiper
- 1993
(Show Context)
Citation Context ... evaluations of this energy function. Several multiple precision interval packages are available. Let us quote for instance intpak [11] and intpakX [13] for Maple or a similar package for Mathematica =-=[17]-=-. Due to unverified assumptions on the roundings of elementary functions (0.6 ulp for intpak in Maple, 1 ulp for Mathematica), to bugged roundings (for instance, with 3 decimal digits, the rounding to... |

1 |
Numerical computation of polynomial tools: MPSolve - version 2.0. http://www.dmi.unict.it/~marotta/Articoli/mpsolve. ps
- Bini, Fiorentino
- 1998
(Show Context)
Citation Context .... However, eigensolve is 100 times faster for Mignotte’s polynomials: our algorithm does not use the fact that those polynomials are sparse. (extract from [38], reference [4] corresponds to reference =-=[6]-=- in this paper and [9, 8] have been replaced by [14] here) Compared to the same algorithm where all operations are performed using exact arithmetic, the results are as follows. For Laguerre polynomial... |

1 |
Les Robots parallèles. Hermès Paris
- Merlet
- 1990
(Show Context)
Citation Context ...ithmetics, and namely the accuracy of the result or simply for overcoming a numerical instability. An illustration of this comes from the solution of the direct kinematics problem for parallel robots =-=[30]-=-. In our problem, the studied parallel manipulators were hexapod manipulators: they consist in a mobile platform and a base, linked by 6 rigid kinematic chains, cf. figure 1. The joints between the ki... |

1 |
The MPFI library. http://perso.ens-lyon.fr/ nathalie.revol/software.html
- Revol, Rouillier
- 2001
(Show Context)
Citation Context ...ts developers) and also because of the availability of the source code, we chose MPFR. The corresponding interval library, named MPFI for Multiple Precision Floating-point Interval arithmetic library =-=[35]-=-, is a portable library written in C for arbitrary precision interval arithmetic (source code and documentation can be freely downloaded at the url http://perso.ens-lyon.fr/nathalie.revol/software.htm... |