Certified Roundoff Error Bounds Using Bernstein Expansions and Sparse Krivine-Stengle Representations

Rocca, Alexandre; Magron, Victor; Dang, Thao

doi:10.1109/arith.2017.36

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…which together with (18) implies that l ′ − l ′ k ∈ H kn+1 (K). For this, let us choose u α,j := |b α | ≥ 0, which shows that w α ≥ 0.…”

Section: Convergence Rate Of Fpbern and Fpkristenmentioning

confidence: 79%

“…Here is a summary of our key contributions: This work is the follow-up of our previous contribution [18]. The main novelties, both theoretical and practical, are the following: in [18], we could only handle polynomial programs with box input constraints. For FPBern, the extension to rational functions relies on [13].…”

Section: Key Contributionsmentioning

confidence: 99%

“…We brought major updates to the C++ code of FPBern (b). For FPKriSten, an extension to semialgebraic input sets was already theoretically possible in [18] with the hierarchy of LP relaxations based on sparse Krivine-Stengle representations, and in this current version, we have updated Section 3.2 accordingly to handle this more general case. We have carefully implemented this extension in our software package FPKriSten in order to not compromise efficiency.…”

Section: Key Contributionsmentioning

confidence: 99%

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Certified Roundoff Error Bounds Using Bernstein Expansions and Sparse Krivine-Stengle Representations

Rocca

Magron

Dang

2019

IEEE Trans. Comput.

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Floating point error is a drawback of embedded systems implementation that is difficult to avoid. Computing rigorous upper bounds of roundoff errors is absolutely necessary for the validation of critical software. This problem of computing rigorous upper bounds is even more challenging when addressing non-linear programs. In this paper, we propose and compare two new algorithms based on Bernstein expansions and sparse Krivine-Stengle representations, adapted from the field of the global optimization, to compute upper bounds of roundoff errors for programs implementing polynomial and rational functions. We also provide the convergence rate of these two algorithms. We release two related software package FPBern and FPKriSten, and compare them with the state-of-the-art tools. We show that these two methods achieve competitive performance, while providing accurate upper bounds by comparison with the other tools.

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“…which together with (18) implies that l ′ − l ′ k ∈ H kn+1 (K). For this, let us choose u α,j := |b α | ≥ 0, which shows that w α ≥ 0.…”

Section: Convergence Rate Of Fpbern and Fpkristenmentioning

confidence: 79%

Section: Key Contributionsmentioning

confidence: 99%

Section: Key Contributionsmentioning

confidence: 99%

See 1 more Smart Citation

Certified Roundoff Error Bounds Using Bernstein Expansions and Sparse Krivine-Stengle Representations

Rocca

Magron

Dang

2019

IEEE Trans. Comput.

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“…Once the coefficients have been chosen, the approximation and the evaluation error can a posteriori be certified by several existing algorithms and tools, like Sollya [11], Gappa [14], Rosa [13] or Real2Float [25].…”

Section: Introductionmentioning

confidence: 99%

Exchange Algorithm for Evaluation and Approximation Error-Optimized Polynomials

Arzelier

Bréhard

Joldeş

2019

2019 IEEE 26th Symposium on Computer Arithmetic (ARITH)

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Machine implementation of mathematical functions often relies on polynomial approximations. The particularity is that rounding errors occur both when representing the polynomial coefficients on a finite number of bits, and when evaluating it in finite precision. Hence, for finding the best polynomial (for a given fixed degree, norm and interval), one has to consider both types of errors: approximation and evaluation. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad-hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is the best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez algorithm. A discussion and comparison of the obtained results on different examples are finally presented.

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“…These tools are mainly based on interval arithmetic (e.g. Gappa [Daumas and Melquiond 2010], Fluctuat [Delmas et al 2009], Rosa [Darulova and Kuncak 2014]) or methods coming from global optimization such as Taylor approximation in FPTaylor by [Solovyev et al 2015], Bernstein expansion in FPBern by [Rocca et al 2016]. The recent framework by ], related to the Real2Float software package, employs semidefinite programming (SDP) to obtain a hierarchy of upper bounds converging to the absolute roundoff error.…”

Section: Introductionmentioning

confidence: 99%

Interval Enclosures of Upper Bounds of Roundoff Errors Using Semidefinite Programming

Magron

2018

ACM Trans. Math. Softw.

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A longstanding problem related to floating-point implementation of numerical programs is to provide efficient yet precise analysis of output errors.We present a framework to compute lower bounds on largest absolute roundoff errors, for a particular rounding model. This method applies to numerical programs implementing polynomial functions with box constrained input variables. Our study is based on three different hierarchies, relying respectively on generalized eigenvalue problems, elementary computations and semidefinite programming (SDP) relaxations. This is complementary of over-approximation frameworks, consisting of obtaining upper bounds on the largest absolute roundoff error. Combining the results of both frameworks allows to get enclosures for upper bounds on roundoff errors.The under-approximation framework provided by the third hierarchy is based on a new sequence of convergent robust SDP approximations for certain classes of polynomial optimization problems. Each problem in this hierarchy can be solved exactly via SDP. By using this hierarchy, one can provide a monotone nondecreasing sequence of lower bounds converging to the absolute roundoff error of a program implementing a polynomial function, applying for a particular rounding model.We investigate the efficiency and precision of our method on non-trivial polynomial programs coming from space control, optimization and computational biology.

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Certified Roundoff Error Bounds Using Bernstein Expansions and Sparse Krivine-Stengle Representations

Cited by 4 publications

References 14 publications

Certified Roundoff Error Bounds Using Bernstein Expansions and Sparse Krivine-Stengle Representations

Certified Roundoff Error Bounds Using Bernstein Expansions and Sparse Krivine-Stengle Representations

Exchange Algorithm for Evaluation and Approximation Error-Optimized Polynomials

Interval Enclosures of Upper Bounds of Roundoff Errors Using Semidefinite Programming

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