Estimation of Round-off Errors in OpenMP Codes

Eberhart, Pacôme; Brajard, Julien; Fortin, Pierre; Jézéquel, Fabienne

doi:10.1007/978-3-319-45550-1_1

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…In this section, the CESTAC method is described and the algorithm of this method is presented. Also, a sample program of the CADNA library is demonstrated and finally advantages of the presented method based on the SA in discrete case are investigated in comparison with the traditional FPA [32,[34][35][36][37]40].…”

Section: Cestac Method-cadna Librarymentioning

confidence: 99%

“…Graillat et al [38,39] studied the SA in multi precision, dynamical control of Newton's method for multiple roots of polynomials and numerical validation of compensated summation algorithms with SA. Eberhart et al [40] presented a high performance numerical validation using the SA. In recent years, the CESTAC method was applied to validate the results of several numerical methods such as estimating the value of interpolation polynomials [41], improper and definite integrals [42,43], dynamical control on Gauss-Laguerre integration rule [44], evaluating the fuzzy definite integrals [45], numerical validation of the Sinccollocation method to solve linear integral equations [46], finding optimal iteration of the power and inverse iteration methods, solving linear systems, fuzzy Newton-Cotes integration rules [47], finding the optimal parameter of the homotopy analysis method for solving integral equations [48], solving fuzzy integral equations [49], validating Taylorcollocation method for solving Volterra integral equations with discontinuous kernels [50] and load leveling problem [51].…”

Section: Introductionmentioning

confidence: 99%

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A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

Noeiaghdam¹,

Araghi²

2020

MMEP

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Finding the optimal iteration of Gaussian quadrature rule is one of the important problems in the computational methods. In this study, we apply the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library to find the optimal iteration and optimal approximation of the Gauss-Legendre integration rule (G-LIR). A theorem is proved to show the validation of the presented method based on the concept of the common significant digits. Applying this method, an improper integral in the solution of the model of the osmosis system is evaluated and the optimal results are obtained. Moreover, the accuracy of method is demonstrated by evaluating other definite integrals. The results of examples illustrate the importance of using the stochastic arithmetic in discrete case in comparison with the common computer arithmetic.

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Section: Cestac Method-cadna Librarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

Noeiaghdam¹,

Araghi²

2020

MMEP

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“…Those advantages can be more pronounced on many-core processors. While CADNA supports OpenMP, the performance overhead can be larger than singlethread due to the existence of some private sections [8]. CADNA for CUDA is also available, but it is observed that the overhead (especially on compute-bound operations) becomes higher than that on CPUs [7].…”

Section: Matrix-vector Multiplication (Memory-bound)mentioning

confidence: 99%

Can We Avoid Rounding-Error Estimation in HPC Codes and Still Get Trustworthy Results?

Jézéquel

Graillat

Mukunoki

et al. 2020

Lecture Notes in Computer Science

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Numerical validation enables one to ensure the reliability of numerical computations that rely on floating-point operations. Discrete Stochastic Arithmetic (DSA) makes it possible to validate the accuracy of floating-point computations using random rounding. However, it may bring a large performance overhead compared with the standard floatingpoint operations. In this article, we show that with perturbed data it is possible to use standard floating-point arithmetic instead of DSA for the purpose of numerical validation. For instance, for codes including matrix multiplications, we can directly utilize the matrix multiplication routine (GEMM) of level-3 BLAS that is performed with standard floating-point arithmetic. Consequently, we can achieve a significant performance improvement by avoiding the performance overhead of DSA operations as well as by exploiting the speed of highly-optimized BLAS implementations. Finally, we demonstrate the performance gain using Intel MKL routines compared against the DSA version of BLAS routines.

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“…A straightforward extension to this work would be the control of accuracy in quadruple precision parallel programs. CADNA can be used for the numerical validation of single or double precision parallel programs based on OpenMP [5] or MPI [17]. CADNA could be improved to enable the control of quadruple precision parallel programs: the main new features would be, with OpenMP the reduction operations with quadruple precision stochastic variables, and with MPI the exchange of this kind of variables between processors.…”

Section: Discussionmentioning

confidence: 99%

Numerical validation in quadruple precision using stochastic arithmetic

Graillat¹,

Jézéquel²,

Picot³

et al.

Kalpa Publications in Computing

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Discrete Stochastic Arithmetic (DSA) enables one to estimate rounding errors and to detect numerical instabilities in simulation programs. DSA is implemented in the CADNA library that can analyze the numerical quality of single and double precision programs. In this article, we show how the CADNA library has been improved to enable the estimation of rounding errors in programs using quadruple precision floating-point variables, i.e. having 113-bit mantissa length. Although an implementation of DSA called SAM exists for arbitrary precision programs, a significant performance improvement has been obtained with CADNA compared to SAM for the numerical validation of programs with 113-bit mantissa length variables. This new version of CADNA has been successfully used for the control of accuracy in quadruple precision applications, such as a chaotic sequence and the computation of multiple roots of polynomials. We also describe a new version of the PROMISE tool, based on CADNA, that aimed at reducing in numerical programs the number of double precision variable declarations in favor of single precision ones, taking into account a requested accuracy of the results. The new version of PROMISE can now provide type declarations mixing single, double and quadruple precision.

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Estimation of Round-off Errors in OpenMP Codes

Cited by 4 publications

References 10 publications

A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

Can We Avoid Rounding-Error Estimation in HPC Codes and Still Get Trustworthy Results?

Numerical validation in quadruple precision using stochastic arithmetic

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