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Documents authored by Krämer, Walter

Document
DOI: 10.4230/DagSemProc.08021.1
08021 Abstracts Collection – Numerical Validation in Current Hardware Architectures

Authors: Wolfram Luther, Annie Cuyt, Walter Krämer, and Peter Markstein

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract
From 06.01. to 11.01.2008, the Dagstuhl Seminar 08021 ``Numerical Validation in Current Hardware Architectures'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
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Wolfram Luther, Annie Cuyt, Walter Krämer, and Peter Markstein. 08021 Abstracts Collection – Numerical Validation in Current Hardware Architectures. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{luther_et_al:DagSemProc.08021.1,
  author =	{Luther, Wolfram and Cuyt, Annie and Kr\"{a}mer, Walter and Markstein, Peter},
  title =	{{08021 Abstracts Collection – Numerical Validation in Current Hardware Architectures}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--31},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.1},
  URN =		{urn:nbn:de:0030-drops-14785},
  doi =		{10.4230/DagSemProc.08021.1},
  annote =	{Keywords: Computer arithmetic, arbitrary precision, floating-point arithmetic standardization, language support, reliable libraries,high-precision special functions, reliablealgorithms, reliable floating-point and interval computing on different platforms}
}
Document
DOI: 10.4230/DagSemProc.08021.2
08021 Summary – Numerical Validation in Current Hardware Architectures

Authors: Annie Cuyt, Walter Krämer, Wolfram Luther, and Peter Markstein

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract
Numerical validation in current hardware architectures - From embedded system to high-end computational grids Topics List of participants Schedule List of talks
Cite as

Annie Cuyt, Walter Krämer, Wolfram Luther, and Peter Markstein. 08021 Summary – Numerical Validation in Current Hardware Architectures. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{cuyt_et_al:DagSemProc.08021.2,
  author =	{Cuyt, Annie and Kr\"{a}mer, Walter and Luther, Wolfram and Markstein, Peter},
  title =	{{08021 Summary – Numerical Validation in Current Hardware Architectures}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.2},
  URN =		{urn:nbn:de:0030-drops-14334},
  doi =		{10.4230/DagSemProc.08021.2},
  annote =	{Keywords: Computer arithmetic, arbitrary precision, floating-point arithmetic standardization, language support, reliable libraries, high-precision special functions, reliablealgorithms, reliable floating-point and interval computing on different platforms}
}
Document
DOI: 10.4230/DagSemProc.08021.3
A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range

Authors: Frithjof Blomquist, Werner Hofschuster, and Walter Krämer

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract
A so called staggered precision arithmetic is a special kind of a multiple precision arithmetic based on the underlying floating point data format (typically IEEE double format) and fast floating point operations as well as exact dot product computations. Due to floating point limitations it is not an arbitrary precision arithmetic. However, it typically allows computations using several hundred mantissa digits. A set of new modified staggered arithmetics for real and complex data as well as for real interval and complex interval data with very wide exponent range is presented. Some applications show the increased accuracy of computed results compared to ordinary staggered interval computations. The very wide exponent range of the new arithmetic operations allows computations far beyond the IEEE data formats. The new arithmetics would be extremly fast, if an exact dot product was available in hardware (the fused accumulate and add instruction is only one step in this direction).
Cite as

Frithjof Blomquist, Werner Hofschuster, and Walter Krämer. A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{blomquist_et_al:DagSemProc.08021.3,
  author =	{Blomquist, Frithjof and Hofschuster, Werner and Kr\"{a}mer, Walter},
  title =	{{A Modified Staggered Correction Arithmetic  with Enhanced Accuracy and Very Wide Exponent Range}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--23},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.3},
  URN =		{urn:nbn:de:0030-drops-14454},
  doi =		{10.4230/DagSemProc.08021.3},
  annote =	{Keywords: Staggered correction, multiple precision, C-XSC, interval computation, wide exponent range, reliable numerical computations, complex interval funct}
}
Document
DOI: 10.4230/DagSemProc.08021.4
A Note on Solving Problem 7 of the SIAM 100-Digit Challenge Using C-XSC

Authors: Mariana Kolberg, Walter Krämer, and Michael Zimmer

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract
C-XSC is a powerful C++ class library which simplifies the development of selfverifying numerical software. But C-XSC is not only a development tool, it also provides a lot of predefined highly accurate routines to compute reliable bounds for the solution to standard numerical problems. In this note we discuss the usage of a reliable linear system solver to compute the solution of problem 7 of the SIAM 100-digit challenge. To get the result we have to solve a 20 000 × 20 000 system of linear equations using interval computations. To perform this task we run our software on the advanced Linux cluster engine ALiCEnext located at the University of Wuppertal and on the high performance computer HP XC6000 at the computing center of the University of Karlsruhe. The main purpose of this note is to demonstrate the power/weakness of our approach to solve linear interval systems with a large dense system matrix using C-XSC and to get feedback from other research groups all over the world concerned with the topic described. We are very much interested to see comparisons concerning different methods/algorithms, timings, memory consumptions, and different hardware/software environments. It should be easy to adapt our main routine (see Section 3 below) to other programming languages, and different computing environments. Changing just one variable allows the generation of arbitrary large system matrices making it easy to do sound (reproducible and comparable) timings and to check for the largest possible system size that can be handled successfully by a specific package/environment.
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Mariana Kolberg, Walter Krämer, and Michael Zimmer. A Note on Solving Problem 7 of the SIAM 100-Digit Challenge Using C-XSC. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kolberg_et_al:DagSemProc.08021.4,
  author =	{Kolberg, Mariana and Kr\"{a}mer, Walter and Zimmer, Michael},
  title =	{{A Note on Solving Problem 7 of the SIAM 100-Digit Challenge Using C-XSC}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.4},
  URN =		{urn:nbn:de:0030-drops-14479},
  doi =		{10.4230/DagSemProc.08021.4},
  annote =	{Keywords: C-XSC, reliable computing, 100-digit challenge, reliable linear system solver, high performance computing, large dense linear systems}
}
Document
DOI: 10.4230/DagSemProc.08021.7
C-XSC and Closely Related Software Packages

Authors: Werner Hofschuster, Walter Krämer, and Markus Neher

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract
C-XSC and Closely Related Software Packages
Cite as

Werner Hofschuster, Walter Krämer, and Markus Neher. C-XSC and Closely Related Software Packages. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{hofschuster_et_al:DagSemProc.08021.7,
  author =	{Hofschuster, Werner and Kr\"{a}mer, Walter and Neher, Markus},
  title =	{{C-XSC and Closely Related Software Packages}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.7},
  URN =		{urn:nbn:de:0030-drops-14425},
  doi =		{10.4230/DagSemProc.08021.7},
  annote =	{Keywords: Mathematical software, reliable computing, C-XSC, CoStLy, ACETAF}
}
Document
DOI: 10.4230/DagSemProc.08021.11
Fast (Parallel) Dense Linear Interval Systems Solvers in C-XSC Using Error Free Transformations and BLAS

Authors: Michael Zimmer and Walter Krämer

Published in: Dagstuhl Seminar Proceedings, Volume 8021, Numerical Validation in Current Hardware Architectures (2008)

Abstract
The traditional solver for linear interval systems available in C-XSC [6,1] is mathematically based on the Krawczyk[12] operator and modifications introduced by Rump[17]. The Krawczyk operator is composed of matrix/vector operations. These operations are realized in C-XSC with higest accuracy (only one final rounding) using a so called long accumulator (dotprecision variable). C-XSC dotprecision variables allow the error free computation of sums of floating point numbers as well as the error free computation of scalar products of floating point vectors. Thus, from a mathematical point of view these operations are perfect. Because actual hardware does not support these perfect scalar products all operations have to be realized by software. This fact leads to a tremendous time penalty (note: it has been shown that with modest additional hardware costs perfect scalar products can be made as fast as simple floating-point loops). To speed up the C-XSC scalar product software-operations we adapt the so called DotK algorithm as published in [14]. Error free transformations[14,3,4,10] are used as basic building blocks to develop summation and scalar product algorithms simulating a K-fold precision. Compared to the perfect C-XSC operations these operations are fast. They are more accurate than simple floating-point loops (but of course no longer perfect in the mathematical sense). The fast operations are available in C-XSC via the new data types DotK, IDotK, CDotk and CIDotK. These new data types are composed in such a way that traditional C-XSC code using dotprecision variables can be adapted with minimal effort. It is possible to switch (at runtime!) from perfect computations to fast operations using K-fold precision (K equal 0 means traditional dotprecision computations) and it is possible to hold intermediate results with corresponding error bounds for further summations or scalar product updates. The details are described in [19]. Additionaly, based on similar algorithms used in Intlab[16], BLAS and LAPACK libraries [2] are used in the O(n³) parts of the linear system solver. For matrix-matrix products, manipulation of the rounding mode of the processor is used to compute enclosures of the correct result. Comparing the traditional solver with the new version shows that the class of problems which are solvable with the new version is smaller than the class of problems which can be solved using the solver based on perfect operations. But it seems that for real world problems also the new solver is appropriate. Using the new solver based on BLAS and simulating a quadrupel precision (i.e. k==2) the speedup comes close to 200(!). The new solver is nearly as fast as the corresponding IntLab[16] solver verifylss. Solving a real linaer system of dimension 1000 on a Pentium 4 with 3.2GHz takes about 2.8 seconds. In all cases tested the accuracy of our new solver was better and in some cases significantly better than the accuracy of the corresponding IntLab results. The new solver also allows solving larger (dense) problems than its IntLab counterpart. We also show some examples where IntLab falls down whereas our new solver still works. A parallel version of this solver, based on ScaLAPACK, is also available. Unlike the previous parallel solver in C-XSC[5], this new solver does not depend on a root-node, which makes it possible to compute a verified solution even of very large linear systems. In the talk we will discuss the new data types in more detail, we will emphasize our modifications to the DotK algorithm taken from the literature [14,15], we will show time measurements and we will present results concerning the accuracy of the computed enclosures. Our results will also be compared to corresponding results computed with the IntLab package. We also will comment on hardware features and compiler options which can/should be used to get reliable results on different platforms efficiently. References: [1] Downloads: C-XSC library: http://www.math.uni-wuppertal.de/~xsc/xsc/cxsc.html Solvers: http://www.math.uni-wuppertal.de/~xsc/xsc/cxsc_software.html [2] L.S. Blackford, J. Demmel, J. Dongarra, I. Duff, S. Hammarling, G. Henry, M. Heroux, L. Kaufman, A. Lumsdaine, A. Petitet, R. Pozo, K. Remington, R. C. Whaley, An Updated Set of Basic Linear Algebra Subprograms (BLAS), ACM Trans. Math. Soft., 28-2 (2002), pp. 135--151. [3] Bohlender, G.; Walter, W.; Kornerup, P.; Matula, D.W.; Kornerup, P.; Matula, D.W.: Semantics for Exact Floating Point Operations. Proceedings, 10th IEEE Symposium on Computer Arithmetic, 26-28 June 1991, IEEE, 1991. [4] Dekker, T.J.: A floating-point technique for extending the available precision. Numer. Math., 18:224, 1971. [5] Grimmer, M.: Selbstverifizierende Mathematische Softwarewerkzeuge im High-Performance Computing. Konzeption, Entwicklung und Analyse am Beispiel der parallelen verifizierten Loesung linearer Fredholmscher Integralgleichungen zweiter Art. Logos Verlag, 2007. [6] Hofschuster, W.; Kraemer, W.: C-XSC 2.0: A C++ Library for Extended Scientific Computing. Numerical Software with Result Verification, Lecture Notes in Computer Science, Volume 2991/2004, Springer-Verlag, Heidelberg, pp. 15 - 35, 2004. [7] Kersten, Tim: Verifizierende rechnerinvariante Numerikmodule, Dissertation, University of Karlsruhe, 1998 [8] Klatte, Kulisch, Wiethoff, Lawo, Rauch: "C-XSC - A C++ Class Library for Extended Scientific Computing", Springer-Verlag, Heidelberg, 1993. Due to the C++ standardization (1998) and dramatic changes in C++ compilers over the last years this documentation describes no longer the actual C-XSC environment. Please refer to more accurate documentation (e.g.[1]) available from the web site of our research group: http... [9] Kirchner, R., Kulisch, U.: Hardware Support for Interval Arithmetic. Reliable Computing, Volume 12, Number 3, June 2006 , pp. 225-237(13). [10] Knuth, D.E.: The Art of Computer Programming: Seminumerical Algorithms. Addison Wesley, 1969, vol. 2. [11] Kulisch, U.: Computer Arithmetic and Validity - Theory, Implementation. To appear. [12] Krawczyk, R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing, 4:187-201, 1969. [13] Lerch, M.; Tischler, G.; Wolff von Gudenberg, J.; Hofschuster, W; Kraemer, W.: filib++, a Fast Interval Library Supporting Containment Computations. ACM TOMS, volume 32, number 2, pp. 299-324, 2006. [14] Ogita, T., Rump, S.M., Oishi, S.: Accurate sum and dot product. SIAM Journal on Scientific Computing, 26:6, 2005. [15] Oishi, S., Tanabe, K., Ogita, T., Rump, S.M., Yamanaka, N.: A Parallel Algorithm of Accurate Dot Product. Submitted for publication, 2007. [16] Rump, S.M.: Intlab - Interval Laboratory. Developments in Reliable Computing, pp. 77-104, 1999. [17] Rump, S.M.: Kleine Fehlerschranken bei Matrixproblemen, Dissertation, University of Karlsruhe, 1980 [18] Stroustrup, Bjarne: The C++-Programming Language, 3rd Edition, Addison-Wesley, 2000. [19] Zimmer, Michael: Laufzeiteffiziente, parallele Loeser fuer lineare Intervallgleichungssysteme in C-XSC, Master thesis, University of Wuppertal, 2007. AMS subject classification: 65H10, 15-04, 65G99, 65G10, 65-04
Cite as

Michael Zimmer and Walter Krämer. Fast (Parallel) Dense Linear Interval Systems Solvers in C-XSC Using Error Free Transformations and BLAS. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{zimmer_et_al:DagSemProc.08021.11,
  author =	{Zimmer, Michael and Kr\"{a}mer, Walter},
  title =	{{Fast (Parallel) Dense Linear Interval Systems Solvers in C-XSC Using Error Free Transformations and BLAS}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--20},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{8021},
  editor =	{Annie Cuyt and Walter Kr\"{a}mer and Wolfram Luther and Peter Markstein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08021.11},
  URN =		{urn:nbn:de:0030-drops-14365},
  doi =		{10.4230/DagSemProc.08021.11},
  annote =	{Keywords: Error-free transformations, K-fold accuracy, accurate dot product, C-XSC, high accuracy, dense linear systems, verified computation.}
}
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