Numerical Validation in Current Hardware Architectures: International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 6-11, 2008. Revised Papers PDF
Preview Numerical Validation in Current Hardware Architectures: International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 6-11, 2008. Revised Papers
Lecture Notes in Computer Science 5492 CommencedPublicationin1973 FoundingandFormerSeriesEditors: GerhardGoos,JurisHartmanis,andJanvanLeeuwen EditorialBoard DavidHutchison LancasterUniversity,UK TakeoKanade CarnegieMellonUniversity,Pittsburgh,PA,USA JosefKittler UniversityofSurrey,Guildford,UK JonM.Kleinberg CornellUniversity,Ithaca,NY,USA AlfredKobsa UniversityofCalifornia,Irvine,CA,USA FriedemannMattern ETHZurich,Switzerland JohnC.Mitchell StanfordUniversity,CA,USA MoniNaor WeizmannInstituteofScience,Rehovot,Israel OscarNierstrasz UniversityofBern,Switzerland C.PanduRangan IndianInstituteofTechnology,Madras,India BernhardSteffen UniversityofDortmund,Germany MadhuSudan MassachusettsInstituteofTechnology,MA,USA DemetriTerzopoulos UniversityofCalifornia,LosAngeles,CA,USA DougTygar UniversityofCalifornia,Berkeley,CA,USA GerhardWeikum Max-PlanckInstituteofComputerScience,Saarbruecken,Germany Annie Cuyt Walter Krämer Wolfram Luther Peter Markstein (Eds.) Numerical Validation in Current Hardware Architectures International Dagstuhl Seminar Dagstuhl Castle, Germany, January 6-11, 2008 Revised Papers 1 3 VolumeEditors AnnieCuyt UniversiteitAntwerpen DepartmentofMathematicsandComputerScience Middelheimlaan1,2020Antwerpen,Belgium E-mail:[email protected] WalterKrämer BergischeUniversitätWuppertal WissenschaftlichesRechnen/Softwaretechnologie Gaussstrasse20,42097Wuppertal,Germany E-mail:[email protected] WolframLuther UniversitätDuisburg-Essen,FakultätIngenieurwissenschaften AbteilungInformatikundAngewandteKognitionswissenschaft 47048Duisburg,Germany E-mail:[email protected] PeterMarkstein 160RedlandRoad,Woodside,CA94062,USA E-mail:[email protected] LibraryofCongressControlNumber:Appliedfor CRSubjectClassification(1998):G.1,G.4,D.2,F.2.1,D.3 LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues ISSN 0302-9743 ISBN-10 3-642-01590-5SpringerBerlinHeidelbergNewYork ISBN-13 978-3-642-01590-8SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliable toprosecutionundertheGermanCopyrightLaw. springer.com ©Springer-VerlagBerlinHeidelberg2009 PrintedinGermany Typesetting:Camera-readybyauthor,dataconversionbyScientificPublishingServices,Chennai,India Printedonacid-freepaper SPIN:12659739 06/3180 543210 Preface The major emphasis ofthe Dagstuhl Seminar on“NumericalValidation inCur- rent Hardware Architectures” lay on numerical validation in current hardware architecturesandsoftwareenvironments.Thegeneralideawastobringtogether experts who are concerned with computer arithmetic in systems with actual processor architectures and scientists who develop, use, and need techniques from verified computation in their applications. Topics of the seminar therefore included: – The ongoingrevisionof the IEEE754/854standardfor floating-pointarith- metic – Feasible ways to implement multiple precision (multiword) arithmetic and to compute the actual precision at run-time according to the needs of input data – The achievement of a similar behavior of fixed-point, floating-point and in- terval arithmetic across language compliant implementations – The designofrobustandefficientnumericalprogramsportable fromdiverse computers to those that adhere to the IEEE standard – The development and propagation of validated special-purpose software in different application areas – Error analysis in several contexts – Certification of numerical programs, verification and validation assessment Computer arithmetic plays an important role at the hardware and software level, when microprocessors,embedded systems, or grids are designed. The reli- abilityofnumericalsoftwarestronglydependsonthecompliancewiththecorre- sponding floating-pointnorms.StandardCISC processorsfollow the 1985IEEE norm 754, which is currently under revision, but the new highly performing CELL processor is not fully IEEE compliant. The draft standard IEEE 754r guarantees that systems perform floating-point computation that yields results independent of whether the processing is done in hardware, software,or a com- bination of the two. For operations specified in this standard, numerical results andexceptionsareuniquelydeterminedbythevaluesoftheinputdata,sequence ofoperations,anddestinationformats,allunderusercontrol.Therewasabroad consensus that the standard should include interval arithmetic. Thediscussionfocusedondecimalnumberformats,faithfulrounding,higher mantissa length, high precise standard functions, and linking of numerical and symbolic algebraic computation. This work is accompanied by new vector, ma- trix, and elementary or special function libraries (i.e., complex functions and continued fractions) with guaranteed precision within their domains. Functions should be correctly rounded and even last bit accuracyis available for standard functions. Important discussion points were additional features such as fast in- terval arithmetic, staggered correction arithmetic, or a fast and accurate mul- VI tiply and accumulate instruction by pipelining. The latter is the key operation for a fast multiple precision arithmetic. An exact dot product (implemented in pipelinedhardware)forfloatingpointvectorswouldprovideoperationsinvector spaceswithaccurateresultswithoutanytimepenalty.Correctlyroundedresults in these vector spaces go hand in hand with correctly roundedelementary func- tions.The newnormwillbebasedonsolidandinterestingtheoreticalstudies in integrated or reconfigurablecircuit design, mathematics, and computer science. Inparalleltothe ongoingIEEEcommittee discussions,the seminaraimedat highlightingdesigndecisionsonfloating-pointcomputationsatrun-timeoverthe whole execution process under the silent consensus that there are features de- finedby the hardwarestandard,bylanguage,ordeferredto the implementation forseveralreasons.Hardwareandsoftwareshouldsupportseveral(user-defined) number types, i.e., fixed width, binary, or decimal floating-point numbers or interval arithmetic. A serious effort is being made to standardize the use of in- tervals especially in the programming language C. Developers are prompted to write efficient numerical programs easily portable with small revisions to other platforms. C-XSC is a C++ Library for Extended Scientific Computing for de- velopingnumericalsoftwarewith resultverification.This libraryis permanently enlarged and enhanced as highlighted in several talks. However,depending on the requirements on speed, input, and output range, orprecision,special-purposeprocessorsalsouseothernumbersystems,i.e.,fixed- pointorlogarithmicnumbersystemsandnon-standardmantissalength.Interest- ingexamplesreportedwerea16-bitintervalarithmeticontheFPGA(FieldPro- grammableGate Array)-basedNIOS-II softprocessorandanonline-arithmetic withrationalnumbers.Therefore,researchonreliablecomputingincludesawide range of current hardware and software platforms and compilers. Standardization is also asked for in inhomogeneous computer networks. The verification step should validate the partial results coming from the owners of parcels before combining them to the final result. Our insights and the implemented systems should be used by people with various numerical problems to solve these problems in a comprehensible and reliable way and by people incorporating validated software tools into other systems or providing interfaces to these tools. Thus, we want raise awareness of interval tools, the validated modeling and simulation systems, and computer-based proofs in science and engineering. Thisbookcontainstheproceedings(revisedpapers)oftheDagstuhlSeminar 08021 “Numerical Validation in Current Hardware Architectures” held during January 6-11,2008. The 16 contributions are grouped into four sections: 1. Standardization/Hardware 2. Tools 3. Applications 4. Linear Systems All papers were refereedby atleasttwo of the refereeslisted here.We would like to thank all the referees for doing a very good job. We would also like to VII thankallauthorsforprovidinguswiththeirexcellentcontributionsandfortheir willingnesstojoiningroupstopresentacoherentdescriptionofrelatedresearch and software tools. We are also grateful to Springer for the fruitful cooperation when preparing this volume and, last but not least, to Michael Zimmer, for compiling the final version of the text. January 2009 Annie Cuyt Walter Kra¨mer Wolfram Luther Peter Markstein Organization Organizers Annie Cuyt University of Antwerp Walter Kra¨mer University of Wuppertal Wolfram Luther University of Duisburg-Essen Peter Markstein Woodside California Referees G. Alefeld U. Klatte S. Oishi E. Auer W. Kra¨mer E. Popova G. Bohlender V. Kreinovich J. Pryce A. Cuyt U. Kulisch A. Rauh A. Frommer B. Lang N. Revol M. Grimmer W. Luther S.M. Rump M. Hochbruck G. Melquiond B. Tibken R.B. Kearfott J.M. Muller J. Wolff von Gudenberg M. Kieffer M. Neher Additional online information available from the Dagstuhl server Dagstuhl Seminar Proceedings 08021 (DROPS), Numerical Validation in Current Hardware Architectures. A. Cuyt, W. Kra¨mer, W. Luther, P. Markstein (Eds.), ISSN 1862 - 4405,see http://drops.dagstuhl.de/portals/index.php?semnr=08021 Table of Contents Standardization/Hardware Discussions on an Interval Arithmetic Standard at Dagstuhl Seminar 08021 .......................................................... 1 R. Baker Kearfott, John Pryce, and Nathalie Revol Complete Interval Arithmetic and Its Implementation on the Computer....................................................... 7 Ulrich W. Kulisch Tools Continued Fractions for Special Functions: Handbook and Software .... 27 Annie Cuyt, Franky Backeljauw, Stefan Becuwe, Michel Colman, Tom Docx, and Joris Van Deun A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range ................................... 41 Frithjof Blomquist, Werner Hofschuster, and Walter Kra¨mer C-XSC and Closely Related Software Packages....................... 68 Werner Hofschuster, Walter Kra¨mer, and Markus Neher Extending the Range of C-XSC: Some Tools and Applications for the Use in Parallel and Other Environments ............................ 103 Markus Grimmer Mathematica Connectivity to Interval Libraries filib++ and C-XSC ..... 117 Evgenija D. Popova Applications Some Applications of Interval Arithmetic in Hierarchical Solid Modeling ....................................................... 133 Eva Dyllong Numerical Verification Assessment in Computational Biomechanics..... 145 Ekaterina Auer and Wolfram Luther Robustness of Boolean Operations on Subdivision-Surface Models...... 161 Di Jiang and Neil F. Stewart XII Table of Contents Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering.............................................. 175 Andreas Rauh, Johanna Minisini, and Eberhard P. Hofer Distributed Bounded-Error Parameter and State Estimation in Networks of Sensors .............................................. 189 Michel Kieffer Error Bounds for Lanczos Approximations of Rational Functions of Matrices ........................................................ 203 Andreas Frommer and Valeria Simoncini Linear Systems Error-FreeTransformation in Rounding Mode toward Zero ............ 217 Stef Graillat, Jean-Luc Lamotte, and Diep Nguyen Hong Fast(Parallel)Dense Linear SystemSolversinC-XSC Using ErrorFree Transformations and BLAS ....................................... 230 Walter Kr¨amer and Michael Zimmer A Note on Solving Problem 7 of the SIAM 100-Digit Challenge Using C-XSC ......................................................... 250 Mariana Kolberg, Walter Kra¨mer, and Michael Zimmer Author Index.................................................. 263 Discussions on an Interval Arithmetic Standard at Dagstuhl Seminar 08021 R. Baker Kearfott1, John Pryce2, and Nathalie Revol3 1 University of Louisiana at Lafayette Box 4-1010 Lafayette, Louisiana 70504-1010 USA 2 Department of Informatics and Simulation Royal Military College of Science Shrivenham Wilts SN6 8LA England 3 Projet Arenaire Laboratoire del’Informatique duParall´elisme E´cole Normale Sup´erieure deLyon 46 all´ee d’Italie 69364 Lyon Cedex 07 France 1 Background Effortshavebeenmadetostandardizeintervalarithmetic(IA)foroveradecade. Thereasonshavebeentoenablemorewidespreaduseofthetechnology,toenable morewidespreadsharingandcollaborationamongresearchersanddevelopersof the technology, and to enable easier checking that computer codes have been correctly programmed. During the late 1990’s, the first author of this report led such a project to introduce an interval data type into the Fortranlanguage. One reason for failure of that effort was the Fortran language standardization committee’slackoffamiliaritywithintervaltechnologyandconsequentcaution. AnotherwasmisunderstandingbetweentheFortranstandardizationcommittee’s basic tenets on standardizing interline optimization and some views expressed by members ofthe intervalanalysiscommunity.A thirdwasconfusionoverhow extended IA (arithmetic dealing with division by intervals that contain zero) should be handled. This was coupled with a heavy committee load associated with other projects, such as standardizing an interface for interoperability with “C” language programs. Sincethen,theintervalanalysiscommunityhasstudiedandgainedadditional understanding of extended IA. One such study is [10], a systematization of the options.Another,withaparticularpointofview,isProf.Kulisch’scontribution to this volume. Extended arithmetic remains a controversial part of efforts to standardize the arithmetic, particularly whether the underlying model should consider −∞ and ∞ to be numbers in their own right or if −∞ and ∞ should A.Cuytetal.(Eds.):NumericalValidation,LNCS5492,pp.1–6,2009. (cid:2)c Springer-VerlagBerlinHeidelberg2009
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