#### DMCA

## ∗∗∗LIP (CNRS/ÉNS Lyon/INRIA/Universite ́ de Lyon) (2010)

### Citations

588 |
Global optimization using interval analysis
- Hansen
- 1992
(Show Context)
Citation Context ...o remark that obtaining a tight upper-bound for ‖ε‖∞ is equivalent to rigorously solving a univariate global optimization problem. This question has already been extensively studied in the literature =-=[7, 8, 9]-=-. However, these algorithms are completely inefficient in our case: this is due to the particular nature of the function ε. Indeed, this function ε is the difference between two functions very close t... |

520 |
Evaluating Derivatives. Principle and Techniques of Algorithmic Dierentiation
- Griewank
- 2000
(Show Context)
Citation Context ...ty is to bound the term f (n+1)(ξ) for ξ ∈ I. This can be achieved using a technique often called automatic differentiation [26, 27, 15]∗, differentiation arithmetic [27], algorithmic differentiation =-=[28]-=- or Taylor arithmetic [26]. Automatic differentiation allows for evaluating the first n derivatives of f in a point x0 without doing any symbolic differentiation. For bounding the f (n+1)(ξ) when ξ ∈ ... |

440 |
Introduction to Approximation Theory
- Cheney
- 1966
(Show Context)
Citation Context ...ese extrema are usually found by searching for the zeros of the derivative of the error function. Wellknown numerical algorithms like bisection, Newton’s algorithm or the secant algorithm can be used =-=[4]-=-. These techniques offer a good and fast estimation of the needed bound, and implementations are available in most numerical software tools, like Maple or Matlab. ∗http://www.cl.cam.ac.uk/~jrh13/hol-l... |

359 | Semidefinite programming relaxations for semialgebraic problems
- Parrilo
(Show Context)
Citation Context ...a, b], slightly more elaborate ‘Positivstellensatz’ certificates involving sums of squares and multiplication by b− x or x− a work well. In a more general context of multivariate polynomials, Parrilo =-=[49]-=- pioneered the approach of generating such certificates using semidefinite programming (SDP). However, the main high-performance SDP solvers involve complicated nonlinear algorithms implemented in flo... |

350 |
Rigorous global search: continuous problems.
- Kearfott
- 1996
(Show Context)
Citation Context ...o remark that obtaining a tight upper-bound for ‖ε‖∞ is equivalent to rigorously solving a univariate global optimization problem. This question has already been extensively studied in the literature =-=[7, 8, 9]-=-. However, these algorithms are completely inefficient in our case: this is due to the particular nature of the function ε. Indeed, this function ε is the difference between two functions very close t... |

129 | Fast algorithms for manipulating formal power series.
- Brent, Kung
- 1978
(Show Context)
Citation Context ...approximation errors 19 • etc. Indeed, manipulating these arrays is nothing but manipulating truncated formal series. There exist fast algorithms for multiplying, composing or inverting formal series =-=[29, 30]-=-. For instance, computing the first n terms of the product of two series can be performed in O(n log(n)) operations only (instead of the O(n2) operations required when using Leibniz formula). These so... |

58 |
FADBAD, a flexible C++ package for automatic differentiation — using the forward and backward methods
- Bendtsen, Stauning
- 1996
(Show Context)
Citation Context ...eview of existing techniques). For bounding the remainder, the only difficulty is to bound the term f (n+1)(ξ) for ξ ∈ I. This can be achieved using a technique often called automatic differentiation =-=[26, 27, 15]-=-∗, differentiation arithmetic [27], algorithmic differentiation [28] or Taylor arithmetic [26]. Automatic differentiation allows for evaluating the first n derivatives of f in a point x0 without doing... |

43 | Formal verification of floating point trigonometric functions.
- Harrison
- 2000
(Show Context)
Citation Context ...mal proof that can be verified by a highly reliable proof-checking program. This is doubly attractive because such proof checkers are now often used for verifying floating-point hardware and software =-=[11, 37, 38, 39, 40, 41, 42, 43]-=-, In such cases bounds on approximation errors often arise as key lemmas in a larger formal proof, so an integrated way of handling them is desirable. There is a substantial literature on using proof ... |

42 | Floating point verification in HOL Light: the exponential function.
- Harrison
- 1997
(Show Context)
Citation Context ...ill explain this phenomenon in Section 2. 1.0.2 Specific approaches Since generic algorithms are not able to solve this problem, more specific approaches have been developed in the past fifteen years =-=[10, 11, 12, 13, 14]-=-. In the following we analyze them based on the above-mentioned key features for our algorithm. Krämer needed to bound approximation errors while he was developing the FI LIB library [10]. His method... |

39 |
Formally verifying IEEE compliance of floating-point hardware.
- O’Leary, Zhao, et al.
- 1999
(Show Context)
Citation Context ...mal proof that can be verified by a highly reliable proof-checking program. This is doubly attractive because such proof checkers are now often used for verifying floating-point hardware and software =-=[11, 37, 38, 39, 40, 41, 42, 43]-=-, In such cases bounds on approximation errors often arise as key lemmas in a larger formal proof, so an integrated way of handling them is desirable. There is a substantial literature on using proof ... |

38 | A Mechanically Checked Proof of the Correctness of the Kernel of the AMD5k86 Floating-Point Division Program.
- Moore, Lynch, et al.
- 1998
(Show Context)
Citation Context ...mal proof that can be verified by a highly reliable proof-checking program. This is doubly attractive because such proof checkers are now often used for verifying floating-point hardware and software =-=[11, 37, 38, 39, 40, 41, 42, 43]-=-, In such cases bounds on approximation errors often arise as key lemmas in a larger formal proof, so an integrated way of handling them is desirable. There is a substantial literature on using proof ... |

36 |
A Mechanically Checked Proof of IEEE Compliance of a Register-Transfer-Level Specification of the AMD
- Russinoff
- 1998
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35 |
Preuves formelles en arithmétiques à virgule flottante
- Boldo
- 2004
(Show Context)
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34 |
Rigorous Analysis of Nonlinear Motion in Particle Accelerators
- Makino
- 1998
(Show Context)
Citation Context ... J. Harrison, M. Joldeş, Ch. Lauter Another possibility consists of computing both the polynomial and a bound on the remainder simultaneously. Such a technique has been introduced by Berz and Makino =-=[34]-=- under the name of Taylor model. A good introduction to this subject is [21]. As the name indicates, Taylor Models have been designed to compute Taylor polynomials. Their advantage is that the remaind... |

34 | Checking geometric programs or verification of geometric structures. Computational Geometry: Theory and Applications
- Mehlhorn, Näher, et al.
- 1999
(Show Context)
Citation Context ...fact, the basic observation that ‘result checking’ can be more productive than ‘proving correctness’ has been emphasized by Blum [45] and appears in many other contexts such as computational geometry =-=[46]-=-.) The two principal phases of our approach illustrate this dichotomy quite well: • In order to bound the difference |f − T | between the function f and its Taylor series T , there seems to be no shor... |

31 | Verifying Nonlinear Real Formulas Via Sums of Squares. In - Harrison - 2007 |

27 | Efficiency in a fully-expansive theorem prover
- Boulton
- 1993
(Show Context)
Citation Context ...rmal proof, so an integrated way of handling them is desirable. There is a substantial literature on using proof checkers to verify the results of various logical and mathematical decision procedures =-=[44]-=-. In some cases, a direct approach seems necessary, where the algorithm is expressed logically inside the theorem prover, formally proved correct and ‘executed’ in a mathematically precise way via log... |

24 |
Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Co
- V
- 1978
(Show Context)
Citation Context ... in the complex plane. It is not necessary to know complex analysis to understand the general ideas of what follows. The reader interested by this subject is advised to consult a reference book (e.g. =-=[31]-=-) for a general background on complex analysis. The functions of interest for us are generally analytic on the whole complex plane, except maybe in a given list of points (called the singularities of ... |

21 |
Models and Other Validated Functional Inclusion Methods
- Makino, Berz, et al.
- 2003
(Show Context)
Citation Context ...ypass the high dependency problem present in ε = p− f , we have to write p− f differently, so that we can avoid the decorrelation between f and p. For this, one widespread technique in the literature =-=[11, 20, 21, 14]-=-, firstly informally proposed by David Wheeler, consists in replacing the function f by another polynomial T that approximates it and for which a bound on the remainder f − T is not too difficult to o... |

21 |
Interval Methods for Bounding the Range of Polynomials and Solving Systems of Nonlinear Equations
- Stahl
- 1995
(Show Context)
Citation Context ... from it a polynomial and how to bound the remainder. Suppose that a recursive call gives us a polynomial Tu ≃ u together with a bound on the remainder∆u. Since Tu is a polynomial, many methods exist =-=[35]-=- for efficiently computing an accurate enclosure J of Tu(I). e n sl -0 04 45 34 3,sv er sio ns1s- 8sJ ans2 01 0 Efficient and accurate computation of upper bounds of approximation errors 23 Reasoning ... |

20 | Assisted verification of elementary functions using Gappa
- Dinechin, Lauter, et al.
- 2006
(Show Context)
Citation Context ...lopment of libms, IEEE 754-2008 standard [2] recommends that the functions be correctly rounded. Correct rounding of transcendental function requires strict discipline on accuracy bounding and proofs =-=[3]-=-. Currently most libms offer strong guarantees: they are made with care and pass many tests before being published. However, in the core of libms, the error between polynomial approximations and funct... |

18 |
ªProgram Result Checking: A New Approach to Making Programs More Reliable,º
- Blum
- 1993
(Show Context)
Citation Context ...often be both simpler and more efficient than the direct approach. (In fact, the basic observation that ‘result checking’ can be more productive than ‘proving correctness’ has been emphasized by Blum =-=[45]-=- and appears in many other contexts such as computational geometry [46].) The two principal phases of our approach illustrate this dichotomy quite well: • In order to bound the difference |f − T | bet... |

14 |
Divider Circuit Verification With Model Checking and Theorem Proving.
- Kaivola, Aagaard
- 2000
(Show Context)
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13 |
Méthodes d’optimisation globale basées sur l’analyse d’intervalle pour la résolution des problèmes avec contraintes
- Messine
- 1997
(Show Context)
Citation Context ...o remark that obtaining a tight upper-bound for ‖ε‖∞ is equivalent to rigorously solving a univariate global optimization problem. This question has already been extensively studied in the literature =-=[7, 8, 9]-=-. However, these algorithms are completely inefficient in our case: this is due to the particular nature of the function ε. Indeed, this function ε is the difference between two functions very close t... |

13 |
The Arithmetic of Differentiation
- Rall
- 1986
(Show Context)
Citation Context ...eview of existing techniques). For bounding the remainder, the only difficulty is to bound the term f (n+1)(ξ) for ξ ∈ I. This can be achieved using a technique often called automatic differentiation =-=[26, 27, 15]-=-∗, differentiation arithmetic [27], algorithmic differentiation [28] or Taylor arithmetic [26]. Automatic differentiation allows for evaluating the first n derivatives of f in a point x0 without doing... |

12 | Effective bounds for P-recursive sequences
- Mezzarobba, Salvy
- 2010
(Show Context)
Citation Context ...)(x0)/i! satisfies a finite recurrence: there exists an integer r such that, for all i, ci+r is completely determined by the values ci, . . . , ci+r−1. Such a function is called differentially finite =-=[33]-=- (or also D-finite or holonomic). As a direct consequence, the whole sequence (ci)i∈N depends only on the first r terms c0, . . . , cr−1. Hence, in principle, the behavior of the whole sequence can be... |

11 |
O((n logn)3/2) Algorithms for composition and reversion of power series, Analytic Computational Complexity
- Brent, Kung
- 1975
(Show Context)
Citation Context ...approximation errors 19 • etc. Indeed, manipulating these arrays is nothing but manipulating truncated formal series. There exist fast algorithms for multiplying, composing or inverting formal series =-=[29, 30]-=-. For instance, computing the first n terms of the product of two series can be performed in O(n log(n)) operations only (instead of the O(n2) operations required when using Leibniz formula). These so... |

10 |
A certified infinite norm for the implementation of elementary functions
- Chevillard, Lauter
(Show Context)
Citation Context ..., in the hope of obtaining a more tightly bounded result. This can be a challenging issue because the exact relationship between an algorithm’s input parameters and result tightness are often unknown =-=[12, 14]-=-. Let us stress again one fact about the previous techniques: the computation of the very digits of the supremum norm is intermingled with the validation of the fact that u is an upper bound. However,... |

10 | Globsol: History, composition, and advice on use
- Kearfott
- 2003
(Show Context)
Citation Context ...the results obtained, one well established technique is Interval Arithmetic [15]. The applications of Interval Arithmetic to rigorous global optimization have been broadly developed in the literature =-=[17, 7, 9]-=-. These methods are based on a general interval branchand-bound algorithm. It involves an exhaustive search over the initial interval. This interval is subdivided recursively (“branching”), and those ... |

10 |
Formal Verification of a fully IEEE compliant floating point unit
- JACOBI
- 2002
(Show Context)
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9 |
Certified and fast computation of supremum norms of approximation errors
- Chevillard, Joldes, et al.
- 2009
(Show Context)
Citation Context ...ill explain this phenomenon in Section 2. 1.0.2 Specific approaches Since generic algorithms are not able to solve this problem, more specific approaches have been developed in the past fifteen years =-=[10, 11, 12, 13, 14]-=-. In the following we analyze them based on the above-mentioned key features for our algorithm. Krämer needed to bound approximation errors while he was developing the FI LIB library [10]. His method... |

9 |
ACETAF: A software package for computing validated bounds for Taylor coefficients of analytic functions
- Eble, Neher
- 2003
(Show Context)
Citation Context ...es is not limited to geometric series: bi can take other forms than M/di, provided that the series ∑+∞ i=n+1 bi |x − x0| i can easily be bounded. Neher and Eble proposed a software tool called ACETAF =-=[32]-=- for automatically computing suitable majorizing series. ACETAF uses different techniques, all based on a theorem of complex analysis called Cauchy’s estimate. We refer to [32] for the details on this... |

8 |
Newton’s algorithm using multiple precision interval arithmetic
- Revol
(Show Context)
Citation Context ...ests are based on using Interval Arithmetic for bounding the image of a function. Many variants for accelerating the rejection process have been implemented. One example is the Interval Newton method =-=[18]-=- used for safely enclosing all the zeros of a univariate function. However, when using interval calculations, the image of the function is overestimated. As discussed in [18], for a large class of fun... |

7 |
Long-term stability of the Tevatron by verified global optimization
- Berz, Makino, et al.
(Show Context)
Citation Context ...e available software implementations are scarce. The best known is COSY [22], written in FORTRAN by Berz and his group. Although highly optimized and used for rigorous global optimization problems in =-=[23, 24]-=- and articles referenced therein, currently, for our specific problem, COSY has two major drawbacks. First, it does not provide multiple precision arithmetic, and thus fails to solve the cancellation ... |

6 |
Rigorous global search using taylor models.
- Berz, Makino
- 2009
(Show Context)
Citation Context ...e available software implementations are scarce. The best known is COSY [22], written in FORTRAN by Berz and his group. Although highly optimized and used for rigorous global optimization problems in =-=[23, 24]-=- and articles referenced therein, currently, for our specific problem, COSY has two major drawbacks. First, it does not provide multiple precision arithmetic, and thus fails to solve the cancellation ... |

6 |
Formal Global Optimization with Taylor Models
- Zumkeller
- 2006
(Show Context)
Citation Context ...e have described in this paper, which can handle a wider range of functions, expanded about arbitrary points and nested in complex ways. This is certainly feasible, and related work has been reported =-=[47, 48]-=-, but much remains to be done, and performing the whole operation inside a formal checker appears to be very time-consuming. 5.2 Formalizing polynomial nonnegativity Several approaches to formally pro... |

6 | A library of taylor models for pvs automatic proof checker
- Cháves, Daumas
- 2006
(Show Context)
Citation Context ...e have described in this paper, which can handle a wider range of functions, expanded about arbitrary points and nested in complex ways. This is certainly feasible, and related work has been reported =-=[47, 48]-=-, but much remains to be done, and performing the whole operation inside a formal checker appears to be very time-consuming. 5.2 Formalizing polynomial nonnegativity Several approaches to formally pro... |

5 |
Taylor forms–use and limits. Reliable Computing 9
- Neumaier
- 2003
(Show Context)
Citation Context ...dency Problem for Approximation Errors While the above-mentioned methods can be successfully used in general, when trying to solve our problem, one is faced with the so-called “dependency phenomenon” =-=[20]-=-. Roughly speaking, it is due to the fact that multiple occurrences of the same variable are not exploited by Interval Arithmetic. The computations are performed “blindly”, taking into account only th... |

5 |
Efficient Isolation of Polynomial Real Roots, in "Journal of
- ROUILLIER, ZIMMERMANN
(Show Context)
Citation Context ... in Section 5. The second technique, establishing the absence of zeros, is easy to implement using traditional polynomial real roots counting techniques, such as Sturm sequences or the Descartes test =-=[25]-=-. Our current implementation computes a Sturm sequence. 3.3 The complete supremum norm algorithm for absolute error problems ε = p− f When combining all results presented in Sections 3.1 and 3.2, we o... |

4 |
Sichere und genaue Abschätzung des Approximationsfehlers bei rationalen Approximationen, Tech. rep., Institut für angewandte Mathematik, Universität Karlsruhe
- Krämer
- 1996
(Show Context)
Citation Context ...ill explain this phenomenon in Section 2. 1.0.2 Specific approaches Since generic algorithms are not able to solve this problem, more specific approaches have been developed in the past fifteen years =-=[10, 11, 12, 13, 14]-=-. In the following we analyze them based on the above-mentioned key features for our algorithm. Krämer needed to bound approximation errors while he was developing the FI LIB library [10]. His method... |

2 |
The MPFI library, http://gforge.inria.fr/projects/mpfi
- Revol, Rouillier
(Show Context)
Citation Context ... of ϕ(x) can be made arbitrarily tight by increasing the precision used for evaluation [18]. This can be achieved using multiprecision interval arithmetic libraries, like for example the MPFI Library =-=[19]-=-. 2.3 Dependency Problem for Approximation Errors While the above-mentioned methods can be successfully used in general, when trying to solve our problem, one is faced with the so-called “dependency p... |

2 |
Global Optimization in Type Theory. PhD thesis, École Polytechnique Paris. Bibliography 139 Glossary of Symbols ∀x.ϕ(x
- Zumkeller
- 2008
(Show Context)
Citation Context ...ious section we just gave a rough idea of Taylor models. In practice, several optimizations are possible and implementations vary in speed and quality depending on choices made for the implementation =-=[21, 36]-=-. It is also possible to use the technique of Taylor models for computing polynomials that are not Taylor polynomials [36]. However even a straightforward out-of-the box implementation of Taylor model... |

1 |
Users manual for the Sollya tool, Release 1.1, http://gforge.inria.fr/frs/download.php/7055/sollya.pdf
- Chevillard, Lauter, et al.
- 2008
(Show Context)
Citation Context ...de using basic functions such as exp or cos. The precise list of basic functions is not important for our purpose: we can consider the list of functions defined in software tools like Maple or Sollya =-=[6]-=- for instance.The only requirement for basic functions is that they be differentiable up to a sufficiently high order. iii. The algorithm should be able to automatically handle a particular difficulty... |

1 |
Lectures in Real Geometry, Vol. 23 of Expositions in Mathematics, de Gruyter
- Broglia
- 1996
(Show Context)
Citation Context ...putation step before formal verification, the positivity of difference polynomials s1 and s2 (see Section 3) is shown using an Interval Arithmetic based implementation of the Sturm Sequence algorithm =-=[50]-=-.The implementation has a fall-back to rational arithmetic if Interval Arithmetic fails to give an unambiguous answer because the enclosure is not sufficiently tight [12] . In the examples presented, ... |