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

2020
*
arXiv
*
pre-print

As we expect the reader to be familiar with the basics

arXiv:2007.06674v1
fatcat:o5bkpov6bfd6fborkmcujgrwpu
*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 ...##
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Computing the sign or the value of the determinant of an integer matrix, a complexity survey

2004
*
Journal of Computational and Applied Mathematics
*

Computation

doi:10.1016/j.cam.2003.08.019
fatcat:zlhj62sdsvcxdbczm4qzg6aija
*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. ...##
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Chapter 10: Algebraic Algorithms
[article]

2013
*
arXiv
*
pre-print

Diaz-Herrera, editors, covers

arXiv:1311.3731v1
fatcat:whtgwztbmbgqbl44s4e663oulu
*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*. ...##
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Design and Implementation of Numerical Linear Algebra Algorithms on Fixed Point DSPs

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. ...

##
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Computing floating-point Gröbner bases accurately

2011
*
ACM SIGSAM Bulletin
*

Then, we present

doi:10.1145/1940475.1940504
fatcat:e7orxey2xrbjbo2oniqzcx7gjm
*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. ...##
###
Development of a Stokes flow solver robust to large viscosity jumps using a Schur complement approach with mixed precision arithmetic

2011
*
Journal of Computational Physics
*

In our Krylov kernel, we

doi:10.1016/j.jcp.2011.09.007
fatcat:rjoblb7lrnfc5larevqdng5s4a
*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. ...##
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Algebraic and Numerical Algorithms
[chapter]

2009
*
Algorithms and Theory of Computation Handbook, Second Edition, Volume 1
*

In the customary

doi:10.1201/9781584888239-c17
fatcat:khegroceujdbpc3ukvlv6s3j4i
*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. ...##
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HONEI: A collection of libraries for numerical computations targeting multiple processor architectures

2009
*
Computer Physics Communications
*

We present HONEI, an open-source collection

doi:10.1016/j.cpc.2009.04.018
fatcat:aodts4clsvapfhwpwizmot4rha
*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. ...##
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Symbolic-Numeric Methods for Problem Solving in CPS (Dagstuhl Seminar 16491)

2017
*
Dagstuhl Reports
*

means

doi:10.4230/dagrep.6.12.1
dblp:journals/dagstuhl-reports/BogomolovFMR16
fatcat:sydj4slvefa7boehai7gpnfupm
*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. ...##
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Improving the energy efficiency of sparse linear system solvers on multicore and manycore systems

2014
*
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
*

For the particular domain

doi:10.1098/rsta.2013.0279
pmid:24842036
fatcat:kw7cnmvzrff6pmihhqenl53uwm
*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. ...##
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High-Precision Arithmetic in Mathematical Physics

2015
*
Mathematics
*

In other words, the judicious usage

doi:10.3390/math3020337
fatcat:ioayjlkv4ndsvp5yyw55zo2k4y
*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. ...##
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A Dynamic Dual Fixed-Point Arithmetic Architecture for FPGAs

2011
*
International Journal of Reconfigurable Computing
*

In this context,

doi:10.1155/2011/518602
fatcat:4ethppnukrd63jmqt3e3eu6omi
*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] . ...##
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Fourier's Method of Linear Programming and Its Dual

1986
*
The American mathematical monthly
*

value problems for systems

doi:10.2307/2322281
fatcat:yvhgyh2epbcwdoqdhuaopkcrue
*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*...##
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Arithmetical Algebra in the Islamic History of Mathematics and Its Peak in the 9th/15th Century: Ibn al-Hāʾim's al-Mumtiʿ

2017
*
Nazariyat İslam Felsefe ve Bilim Tarihi Araştırmaları Dergisi (Journal for the History of Islamic Philosophy and Sciences)
*

It is also probably

doi:10.12658/nazariyat.3.2.m0009en
fatcat:uakz7aldtzgpbc3rzaxqzxeosa
*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*." ...##
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Arithmetic Operations Beyond Floating Point Number Precision
[article]

2011
*
arXiv
*
pre-print

We present an undergraduate project that deals with such calculations beyond

arXiv:1009.5911v3
fatcat:lypqfyzeu5dhfmlmrcfhw46tyy
*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 ...
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