The Generic Multiple-Precision Floating-Point Addition With Exact Rounding (as in the MPFR Library)

We study the multiple-precision addition of two positive floating-point numbers in base 2, with exact rounding, as specified in the MPFR library, i.e. where each number has its own precision. We show how the best possible complexity (up to a constant factor that depends on the implementation) can be obtain.Comment: Conference website at http://cca-net.de/rnc6

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

Correctly Rounded Arbitrary-Precision Floating-Point Summation

International audienceWe present a fast algorithm together with its low-level implementation of correctly rounded arbitrary-precision floating-point summation. The arithmetic is the one used by the GNU MPFR library: radix 2; no subnormals; each variable (each input and the output) has its own precision. We also describe how the implementation is tested

Reproducibility Strategies for Parallel Preconditioned Conjugate Gradient

The Preconditioned Conjugate Gradient method is often used in numerical simulations. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we aim at a twofold goal: enhance the accuracy of the solver but also ensure its reproducibility in a message-passing implementation. We design and employ various strategies starting from the ExBLAS approach (through preserving every bit of information until final rounding) to its more lightweight performance-oriented variant (through expanding the intermediate precision). These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these strategies on modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of only 29 % (ExBLAS) and 4 % (lightweight) on 768 processes

Reproducibility strategies for parallel preconditioned Conjugate Gradient

[EN] The Preconditioned Conjugate Gradient method is often used in numerical simulations. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we aim at a twofold goal: enhance the accuracy of the solver but also ensure its reproducibility in a message-passing implementation. We design and employ various strategies starting from the ExBLAS approach (through preserving every bit of information until final rounding) to its more lightweight performance-oriented variant (through expanding the intermediate precision). These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these strategies on modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of only 29% (ExBLAS) and 4% (lightweight) on 768 processes.To begin with, we would like to thank the reviewers for their thorough reading of the article as well as their valuable comments and suggestions. This research was partially supported by the European Union's Horizon 2020 research, innovation programme under the Marie Sklodowska-Curie grant agreement via the Robust project No. 842528 as well as the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the H2020 EC RIA Programme; in particular, the author gratefully acknowledges the support of Vicenc Beltran and the computer resources and technical support provided by BSC. The researchers from Universitat Jaume I (UJI) and Universidad Politecnica de Valencia (UPV) were supported by MINECO, Spain project TIN2017-82972-R. Maria Barreda was also supported by the POSDOC-A/2017/11 project from the Universitat Jaume I, Spain.Iakymchuk, R.; Barreda, M.; Wiesenberger, M.; Aliaga, JI.; Quintana Ortí, ES. (2020). Reproducibility strategies for parallel preconditioned Conjugate Gradient. Journal of Computational and Applied Mathematics. 371:1-13. https://doi.org/10.1016/j.cam.2019.112697S113371Lawson, C. L., Hanson, R. J., Kincaid, D. R., & Krogh, F. T. (1979). Basic Linear Algebra Subprograms for Fortran Usage. ACM Transactions on Mathematical Software, 5(3), 308-323. doi:10.1145/355841.355847Dongarra, J. J., Du Croz, J., Hammarling, S., & Duff, I. S. (1990). A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software, 16(1), 1-17. doi:10.1145/77626.79170Demmel, J., & Nguyen, H. D. (2015). Parallel Reproducible Summation. IEEE Transactions on Computers, 64(7), 2060-2070. doi:10.1109/tc.2014.2345391Iakymchuk, R., Graillat, S., Defour, D., & Quintana-Ortí, E. S. (2019). Hierarchical approach for deriving a reproducible unblocked LU factorization. The International Journal of High Performance Computing Applications, 33(5), 791-803. doi:10.1177/1094342019832968Iakymchuk, R., Defour, D., Collange, S., & Graillat, S. (2016). Reproducible and Accurate Matrix Multiplication. Lecture Notes in Computer Science, 126-137. doi:10.1007/978-3-319-31769-4_11Rump, S. M., Ogita, T., & Oishi, S. (2009). Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest. SIAM Journal on Scientific Computing, 31(2), 1269-1302. doi:10.1137/07068816xBurgess, N., Goodyer, C., Hinds, C. N., & Lutz, D. R. (2019). High-Precision Anchored Accumulators for Reproducible Floating-Point Summation. IEEE Transactions on Computers, 68(7), 967-978. doi:10.1109/tc.2018.2855729D. Mukunoki, T. Ogita, K. Ozaki, Accurate and reproducible BLAS routines with Ozaki scheme for many-core architectures, in: Proc. International Conference on Parallel Processing and Applied Mathematics, PPAM2019, 2019, accepted.Ogita, T., Rump, S. M., & Oishi, S. (2005). Accurate Sum and Dot Product. SIAM Journal on Scientific Computing, 26(6), 1955-1988. doi:10.1137/030601818Kulisch, U., & Snyder, V. (2010). The exact dot product as basic tool for long interval arithmetic. Computing, 91(3), 307-313. doi:10.1007/s00607-010-0127-7Boldo, S., & Melquiond, G. (2008). Emulation of a FMA and Correctly Rounded Sums: Proved Algorithms Using Rounding to Odd. IEEE Transactions on Computers, 57(4), 462-471. doi:10.1109/tc.2007.70819Wiesenberger, M., Einkemmer, L., Held, M., Gutierrez-Milla, A., Sáez, X., & Iakymchuk, R. (2019). Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures. Computer Physics Communications, 238, 145-156. doi:10.1016/j.cpc.2018.12.006Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., & Zimmermann, P. (2007). MPFR. ACM Transactions on Mathematical Software, 33(2), 13. doi:10.1145/1236463.1236468J. Demmel, H.D. Nguyen, Fast reproducible floating-point summation, in: Proceedings of ARITH-21, 2013, pp. 163–172.Ozaki, K., Ogita, T., Oishi, S., & Rump, S. M. (2011). Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications. Numerical Algorithms, 59(1), 95-118. doi:10.1007/s11075-011-9478-1Carson, E., & Higham, N. J. (2018). Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions. SIAM Journal on Scientific Computing, 40(2), A817-A847. doi:10.1137/17m114081

Analysing and bounding numerical error in spiking neural network simulations

This study explores how numerical error occurs in simulations of spiking neural network models, and also how this error propagates through the simulation, changing its observed behaviour. The issue of non-reproducibility in parallel spiking neural network simulations is illustrated, and a method to bound all possible trajectories is discussed. The base method used in this study is known as mixed interval and affine arithmetic (mixed IA/AA), but some extra modifications are made to improve the tightness of the error bounds. I introduce Arpra, my new software, which is an arbitrary precision range analysis library, based on the GNU MPFR library. It improves on other implementations by enabling computations in custom floating-point precisions, and reduces the overhead rounding error of mixed IA/AA by computing in extended precision internally. It also implements a new error trimming technique, which reduces the error term whilst preserving correct boundaries. Arpra also implements deviation term condensing functions, which can reduce the number of floating-point operations per function significantly. Arpra is tested by simulating the Hénon map dynamical system, and found to produce tighter ranges than those of INTLAB, an alternative mixed IA/AA implementation. Arpra is used to bound the trajectories of fan-in spiking neural network simulations. Despite performing better than interval arithmetic, the mixed IA/AA method used by Arpra is shown to be inadequate for bounding the simulation trajectories, due to the highly nonlinear nature of spiking neural networks. A stability analysis of the neural network model is performed, and it is found that error boundaries are moderately tight in non-spiking regions of state space, where linear dynamics dominate, but error boundaries explode in spiking regions of state space, where nonlinear dynamics dominate

MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding

This paper presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend ideas from the IEEE-754 standard to arbitrary precision, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved | with no signicant slowdown with respect to other tools | and discuss a few applications where such a library can be useful

The functions erf and erfc computed with arbitrary precision and explicit error bounds

The version available on the HAL server is slightly different from the published version because it contains full proofs.International audienceThe error function erf is a special function. It is widely used in statistical computations for instance, where it is also known as the standard normal cumulative probability. The complementary error function is defined as erfc(x)=erf(x)-1. In this paper, the computation of erf(x) and erfc(x) in arbitrary precision is detailed: our algorithms take as input a target precision t' and deliver approximate values of erf(x) or erfc(x) with a relative error bounded by 2^(-t'). We study three different algorithms for evaluating erf and erfc. These algorithms are completely detailed. In particular, the determination of the order of truncation, the analysis of roundoff errors and the way of choosing the working precision are presented. The scheme used for implementing erf and erfc and the proofs are expressed in a general setting, so they can directly be reused for the implementation of other functions. We implemented the three algorithms and studied experimentally what is the best algorithm to use in function of the point x and the target precision t'

Reproducibility of parallel preconditioned conjugate gradient in hybrid programming environments

[EN] The Preconditioned Conjugate Gradient method is often employed for the solution of linear systems of equations arising in numerical simulations of physical phenomena. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we propose two algorithmic solutions that originate from the ExBLAS project to enhance the accuracy of the solver as well as to ensure its reproducibility in a hybrid MPI + OpenMP tasks programming environment. One is based on ExBLAS and preserves every bit of information until the final rounding, while the other relies upon floating-point expansions and, hence, expands the intermediate precision. Instead of converting the entire solver into its ExBLAS-related implementation, we identify those parts that violate reproducibility/non-associativity, secure them, and combine this with the sequential executions. These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these approaches on two modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of less than 37.7% on 768 cores.The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was partially supported by the European Union's Horizon 2020 research, innovation program under the Marie Sklodowska-Curie grant agreement via the Robust project No. 842528 as well as the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the H2020 EC RIA Programme; in particular, the author gratefully acknowledges the support of Vicenc comma Beltran and the computer resources and technical support provided by BSC. The researchers from Universitat Jaume I (UJI) and Universitat Polit ' ecnica de Valencia (UPV) were supported by MINECO project TIN2017-82972-R. Maria Barreda was also supported by the POSDOC-A/2017/11 project from the Universitat Jaume I.Iakymchuk, R.; Barreda Vayá, M.; Graillat, S.; Aliaga, JI.; Quintana Ortí, ES. (2020). Reproducibility of parallel preconditioned conjugate gradient in hybrid programming environments. International Journal of High Performance Computing Applications. 34(5):502-518. https://doi.org/10.1177/1094342020932650S502518345Aliaga, J. I., Barreda, M., Flegar, G., Bollhöfer, M., & Quintana-Ortí, E. S. (2017). Communication in task-parallel ILU-preconditioned CG solvers using MPI + OmpSs. Concurrency and Computation: Practice and Experience, 29(21), e4280. doi:10.1002/cpe.4280Bailey, D. H. (2013). High-precision computation: Applications and challenges [Keynote I]. 2013 IEEE 21st Symposium on Computer Arithmetic. doi:10.1109/arith.2013.39Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., … van der Vorst, H. (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. doi:10.1137/1.9781611971538Burgess, N., Goodyer, C., Hinds, C. N., & Lutz, D. R. (2019). High-Precision Anchored Accumulators for Reproducible Floating-Point Summation. IEEE Transactions on Computers, 68(7), 967-978. doi:10.1109/tc.2018.2855729Carson, E., & Higham, N. J. (2018). Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions. SIAM Journal on Scientific Computing, 40(2), A817-A847. doi:10.1137/17m1140819Collange, S., Defour, D., Graillat, S., & Iakymchuk, R. (2015). Numerical reproducibility for the parallel reduction on multi- and many-core architectures. Parallel Computing, 49, 83-97. doi:10.1016/j.parco.2015.09.001Dekker, T. J. (1971). A floating-point technique for extending the available precision. Numerische Mathematik, 18(3), 224-242. doi:10.1007/bf01397083Demmel, J., & Hong Diep Nguyen. (2013). Fast Reproducible Floating-Point Summation. 2013 IEEE 21st Symposium on Computer Arithmetic. doi:10.1109/arith.2013.9Demmel, J., & Nguyen, H. D. (2015). Parallel Reproducible Summation. IEEE Transactions on Computers, 64(7), 2060-2070. doi:10.1109/tc.2014.2345391Dongarra, J. J., Du Croz, J., Hammarling, S., & Duff, I. S. (1990). A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software, 16(1), 1-17. doi:10.1145/77626.79170Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., & Zimmermann, P. (2007). MPFR. ACM Transactions on Mathematical Software, 33(2), 13. doi:10.1145/1236463.1236468Hida, Y., Li, X. S., & Bailey, D. H. (s. f.). Algorithms for quad-double precision floating point arithmetic. Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001. doi:10.1109/arith.2001.930115Hunold, S., & Carpen-Amarie, A. (2016). Reproducible MPI Benchmarking is Still Not as Easy as You Think. IEEE Transactions on Parallel and Distributed Systems, 27(12), 3617-3630. doi:10.1109/tpds.2016.2539167IEEE Computer Society (2008) IEEE Standard for Floating-Point Arithmetic. Piscataway: IEEE Standard, pp. 754–2008.Kulisch, U., & Snyder, V. (2010). The exact dot product as basic tool for long interval arithmetic. Computing, 91(3), 307-313. doi:10.1007/s00607-010-0127-7Kulisch, U. (2013). Computer Arithmetic and Validity. doi:10.1515/9783110301793Lawson, C. L., Hanson, R. J., Kincaid, D. R., & Krogh, F. T. (1979). Basic Linear Algebra Subprograms for Fortran Usage. ACM Transactions on Mathematical Software, 5(3), 308-323. doi:10.1145/355841.355847Lutz, D. R., & Hinds, C. N. (2017). High-Precision Anchored Accumulators for Reproducible Floating-Point Summation. 2017 IEEE 24th Symposium on Computer Arithmetic (ARITH). doi:10.1109/arith.2017.20Mukunoki, D., Ogita, T., & Ozaki, K. (2020). Reproducible BLAS Routines with Tunable Accuracy Using Ozaki Scheme for Many-Core Architectures. Lecture Notes in Computer Science, 516-527. doi:10.1007/978-3-030-43229-4_44Nguyen, H. D., & Demmel, J. (2015). Reproducible Tall-Skinny QR. 2015 IEEE 22nd Symposium on Computer Arithmetic. doi:10.1109/arith.2015.28Ogita, T., Rump, S. M., & Oishi, S. (2005). Accurate Sum and Dot Product. SIAM Journal on Scientific Computing, 26(6), 1955-1988. doi:10.1137/030601818Ozaki, K., Ogita, T., Oishi, S., & Rump, S. M. (2011). Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications. Numerical Algorithms, 59(1), 95-118. doi:10.1007/s11075-011-9478-1Priest, D. M. (s. f.). Algorithms for arbitrary precision floating point arithmetic. [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic. doi:10.1109/arith.1991.145549Rump, S. M., Ogita, T., & Oishi, S. (2008). Accurate Floating-Point Summation Part I: Faithful Rounding. SIAM Journal on Scientific Computing, 31(1), 189-224. doi:10.1137/050645671Rump, S. M., Ogita, T., & Oishi, S. (2009). Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest. SIAM Journal on Scientific Computing, 31(2), 1269-1302. doi:10.1137/07068816xRump, S. M., Ogita, T., & Oishi, S. (2010). Fast high precision summation. Nonlinear Theory and Its Applications, IEICE, 1(1), 2-24. doi:10.1587/nolta.1.2Saad, Y. (2003). Iterative Methods for Sparse Linear Systems. doi:10.1137/1.9780898718003Wiesenberger, M., Einkemmer, L., Held, M., Gutierrez-Milla, A., Sáez, X., & Iakymchuk, R. (2019). Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures. Computer Physics Communications, 238, 145-156. doi:10.1016/j.cpc.2018.12.00