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A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic [article]

Ahmad Abdelfattah, Hartwig Anzt, Erik G. Boman, Erin Carson, Terry Cojean, Jack Dongarra, Mark Gates, Thomas Grützmacher, Nicholas J. Higham, Sherry Li, Neil Lindquist, Yang Liu (+13 others)
2020 arXiv   pre-print
As we expect the reader to be familiar with the basics of numerical linear algebra, we refrain from providing a detailed background on the algorithms themselves but focus on how mixed- and multiprecision  ...  To start the multiprecision focus effort, we survey the numerical linear algebra community and summarize all existing multiprecision knowledge, expertise, and software capabilities in this landscape analysis  ...  Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc  ... 
arXiv:2007.06674v1 fatcat:o5bkpov6bfd6fborkmcujgrwpu

Computing the sign or the value of the determinant of an integer matrix, a complexity survey

Erich Kaltofen, Gilles Villard
2004 Journal of Computational and Applied Mathematics  
Computation of the sign of the determinant of a matrix and the determinant itself is a challenge for both numerical and exact methods.  ...  We survey the complexity of existing methods to solve these problems when the input is an n×n matrix A with integer entries.  ...  Numerical methods must deal with condition numbers that in uence the precision of the computations.  ... 
doi:10.1016/ fatcat:zlhj62sdsvcxdbczm4qzg6aija

Chapter 10: Algebraic Algorithms [article]

Ioannis Z. Emiris, Victor Y. Pan, Elias P. Tsigaridas
2013 arXiv   pre-print
Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical, for matrix computations and root-finding for polynomials and systems of polynomials equations.  ...  To meet space limitation we cite books, surveys, and comprehensive articles with pointers to further references, rather than including all the original technical papers.  ...  The alternative numerical methods rely on operations with binary or decimal numbers truncated or rounded to a fixed precision.  ... 
arXiv:1311.3731v1 fatcat:whtgwztbmbgqbl44s4e663oulu

Design and Implementation of Numerical Linear Algebra Algorithms on Fixed Point DSPs

Zoran Nikolić, Ha Thai Nguyen, Gene Frantz
2007 EURASIP Journal on Advances in Signal Processing  
Numerical linear algebra algorithms use the inherent elegance of matrix formulations and are usually implemented using C/C++ floating point representation.  ...  We transformed selected linear algebra algorithms from floating point to fixed point arithmetic, and compared real-time requirements and performance between the fixed point DSP and floating point DSP algorithm  ...  C/C++ is still the most popular method for describing numerical linear algebra algorithms.  ... 
doi:10.1155/2007/87046 fatcat:n6ufbfoncbfihgaszrkew4sfim

Computing floating-point Gröbner bases accurately

Tateaki Sasaki, Fujio Kako
2011 ACM SIGSAM Bulletin  
Then, we present a new method, which is robust for numerical errors and gives bases of high quality. The idea is to improve the accuracy by a numerical method.  ...  We discuss the term cancellation which makes the floatingpoint Gröbner basis computation unstable, and shows that error accumulation is never negligible in our previous method.  ...  In Sect. 4, we explain our new method which utilizes celebrated numerical methods to improve the accuracy. Finally, in Sect. 5, we describe our new algorithm.  ... 
doi:10.1145/1940475.1940504 fatcat:e7orxey2xrbjbo2oniqzcx7gjm

Development of a Stokes flow solver robust to large viscosity jumps using a Schur complement approach with mixed precision arithmetic

Mikito Furuichi, Dave A. May, Paul J. Tackley
2011 Journal of Computational Physics  
In our Krylov kernel, we utilize quad precision arithmetic which is emulated via the double-double precision method.  ...  In addition, our mixed precision method produces improved convergence rates of Arnoldi type Krylov subspace methods without a drastic increasing the computational time.  ...  All of the numerical calculations presented in this paper were performed by Earth Simulator of Japan Agency for Marine-Earth Science and Technology.  ... 
doi:10.1016/ fatcat:rjoblb7lrnfc5larevqdng5s4a

Algebraic and Numerical Algorithms [chapter]

Ioannis Emiris, Victor Pan, Elias Tsigaridas
2009 Algorithms and Theory of Computation Handbook, Second Edition, Volume 1  
In the customary arithmetic filtering approach, one applies numerical methods as long as they work and, in the rare cases when they fail, shifts to the slower algebraic methods.  ...  Introduction Arithmetic manipulation with matrices and polynomials is a common subject for algebraic (or symbolic) and numerical computing.  ... 
doi:10.1201/9781584888239-c17 fatcat:khegroceujdbpc3ukvlv6s3j4i

HONEI: A collection of libraries for numerical computations targeting multiple processor architectures

Danny van Dyk, Markus Geveler, Sven Mallach, Dirk Ribbrock, Dominik Göddeke, Carsten Gutwenger
2009 Computer Physics Communications  
We present HONEI, an open-source collection of libraries offering a hardware oriented approach to numerical calculations.  ...  We demonstrate the flexibility and performance of our approach with two test applications, a Finite Element multigrid solver for the Poisson problem and a robust and fast simulation of shallow water waves  ...  Acknowledgements Parts of this work were supported by the German Science Foundation (DFG), projects TU102/22-1 and TU102/22-2. We thank all participants of PG512 at TU Dortmund for initial support.  ... 
doi:10.1016/j.cpc.2009.04.018 fatcat:aodts4clsvapfhwpwizmot4rha

Symbolic-Numeric Methods for Problem Solving in CPS (Dagstuhl Seminar 16491)

Sergiy Bogomolov, Martin Fränzle, Kyoko Makino, Nacim Ramdani, Marc Herbstritt
2017 Dagstuhl Reports  
means of error-propagation analysis; numerical and/or symbolic methods such as verified integrations, interval methods and arithmetic constraint solving; reactive and in-advance planning and optimization  ...  Reflecting the fundamental role numeric and mixed symbolic-numeric arguments play in the analysis, decision making, and control of cyber-physical processes, this seminar promoted crossfertilization between  ...  The non-standard topology of a modular robot requires numeric or semi-algebraic solutions for the IK.  ... 
doi:10.4230/dagrep.6.12.1 dblp:journals/dagstuhl-reports/BogomolovFMR16 fatcat:sydj4slvefa7boehai7gpnfupm

Improving the energy efficiency of sparse linear system solvers on multicore and manycore systems

H. Anzt, E. S. Quintana-Orti
2014 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
For the particular domain of sparse linear algebra, we analyse the energy efficiency of a broad collection of hardware architectures and investigate how algorithmic and implementation modifications can  ...  improve the energy performance of sparse linear system solvers, without negatively impacting their performance.  ...  This work summarizes some key insights gained from a collaboration during the last few years. We thank the long list of colleagues that were involved in this cooperation.  ... 
doi:10.1098/rsta.2013.0279 pmid:24842036 fatcat:kw7cnmvzrff6pmihhqenl53uwm

High-Precision Arithmetic in Mathematical Physics

David Bailey, Jonathan Borwein
2015 Mathematics  
In other words, the judicious usage of double-double arithmetic in a critical summation completely eliminates the numerical non-reproducibility, with only a minor increase in run time.  ...  A larger example of this sort arose in an atmospheric model (a component of large climate model).  ...  Author Contributions Each of the authors made major contributions both to the research and the writing of this article. Conflicts of Interest The authors declare no conflict of interest.  ... 
doi:10.3390/math3020337 fatcat:ioayjlkv4ndsvp5yyw55zo2k4y

A Dynamic Dual Fixed-Point Arithmetic Architecture for FPGAs

G. Alonzo Vera, Marios Pattichis, James Lyke
2011 International Journal of Reconfigurable Computing  
In this context, a designer is left with the task of implementing several arithmetic cores for parallel processing while supporting high numerical precision with finite logical resources.  ...  In FPGA embedded systems, designers usually have to make a compromise between numerical precision and logical resources.  ...  A numerical algorithm's precision and convergence characteristics can benefit from a variable or mixed arithmetic precision implementation [3] [4] [5] .  ... 
doi:10.1155/2011/518602 fatcat:4ethppnukrd63jmqt3e3eu6omi

Fourier's Method of Linear Programming and Its Dual

H. P. Williams
1986 The American mathematical monthly  
value problems for systems of mixed type 35M33 Initial-boundary value problems for systems of mixed type 35M85 Linear unilateral problems and variational inequalities of mixed type [See also 35R35  ...  solutions (perturbation methods, asymptotic methods, series, etc.) 74G15 Numerical approximation of solutions 74G20 Local existence of solutions (near a given solution) 74G25 Global existence of  ...  ; algebraically special solutions, metrics with symmetries 83C22 Einstein-Maxwell equations 83C25 Approximation procedures, weak fields 83C27 Lattice gravity, Regge calculus and other discrete methods  ... 
doi:10.2307/2322281 fatcat:yvhgyh2epbcwdoqdhuaopkcrue

Arithmetical Algebra in the Islamic History of Mathematics and Its Peak in the 9th/15th Century: Ibn al-Hāʾim's al-Mumtiʿ

Elif Baga
2017 Nazariyat İslam Felsefe ve Bilim Tarihi Araştırmaları Dergisi (Journal for the History of Islamic Philosophy and Sciences)  
It is also probably a novel aspect that this supporter of arithmetical algebra completely ignored geometry for the proof of mixed equations and instead provided numerical causes. 75 Nevertheless, with  ...  , became known as "analytical/arithmetical/numerical algebra."  ... 
doi:10.12658/nazariyat.3.2.m0009en fatcat:uakz7aldtzgpbc3rzaxqzxeosa

Arithmetic Operations Beyond Floating Point Number Precision [article]

Chih-Yueh Wang, Chen-Yang Yin, Hong-Yu Chen, Yung-Ko Chen
2011 arXiv   pre-print
We present an undergraduate project that deals with such calculations beyond a machine's numerical limit, known as arbitrary precision arithmetic.  ...  In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine.  ...  Given the subtraction method of division, numerous mathematical functions as simple as x 3.0 = 100 and algebraic equations of the form f (x) = 0 can be solved to a high precision via Newton-Raphson iteration  ... 
arXiv:1009.5911v3 fatcat:lypqfyzeu5dhfmlmrcfhw46tyy
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