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Dagstuhl Seminar Proceedings, Volume 8021



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  • published at: 2008-04-22
  • Publisher: Schloss-Dagstuhl - Leibniz Zentrum für Informatik

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Document
08021 Abstracts Collection – Numerical Validation in Current Hardware Architectures

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


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.

Cite as

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
08021 Summary – Numerical Validation in Current Hardware Architectures

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


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

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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
A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range

Authors: Frithjof Blomquist, Werner Hofschuster, and Walter Krämer


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
A Note on Solving Problem 7 of the SIAM 100-Digit Challenge Using C-XSC

Authors: Mariana Kolberg, Walter Krämer, and Michael Zimmer


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.

Cite as

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
A Note on Some Applications of Interval Arithmetic in Hierarchical Solid Modeling

Authors: Eva Dyllong


Abstract
Techniques of reliable computing like interval arithmetic can be used to guarantee a reliable solution even in the presence of numerical round-off errors. The need to trace bounds for the error function separately can be eliminated using these techniques. In this talk, we focus on some demonstrations how the techniques and algorithms of reliable computing can be applied to the construction and further processing of hierarchical solid representations using the octree model as an example. An octree is a common hierarchical data structure to represent 3D geometrical objects in solid modeling systems or to reconstruct a real scene. The solid representation is based on recursive cell decompositions of the space. Unfortunately, the data structure may require a large amount of memory when it uses a set of very small cubic nodes to approximate a solid. In this talk, we present a novel generalization of the octree model created from a CSG object that uses interval arithmetic and allows us to extend the tests for classifying points in space as inside, on the boundary or outside the object to handle whole sections of the space at once. Tree nodes with additional information about relevant parts of the CSG object are introduced in order to reduce the depth of the required subdivision. Furthermore, this talk is concerned with interval-based algorithms for reliable proximity queries between the extended octrees and with further processing of the structure. We conclude the talk with some examples of implementations.

Cite as

Eva Dyllong. A Note on Some Applications of Interval Arithmetic in Hierarchical Solid Modeling. 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{dyllong:DagSemProc.08021.5,
  author =	{Dyllong, Eva},
  title =	{{A Note on Some Applications of Interval Arithmetic in Hierarchical Solid Modeling}},
  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.08021.5},
  URN =		{urn:nbn:de:0030-drops-14408},
  doi =		{10.4230/DagSemProc.08021.5},
  annote =	{Keywords: Reliable solid modeling, hierarchical data structure}
}
Document
A Software Library for Reliable Online-Arithmetic with Rational Numbers

Authors: Gregorio de Miguel Casado and Juan Manuel García Chamizo


Abstract
An overview of a novel calculation framework for scientific computing in integrable spaces is introduced. This paper discusses some implementation issues adopted for a software library devoted to exact rational online-arithmetic operators for periodic rational operands codified in fractional positional notation.

Cite as

Gregorio de Miguel Casado and Juan Manuel García Chamizo. A Software Library for Reliable Online-Arithmetic with Rational Numbers. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{demiguelcasado_et_al:DagSemProc.08021.6,
  author =	{de Miguel Casado, Gregorio and Garc{\'\i}a Chamizo, Juan Manuel},
  title =	{{A Software Library for Reliable Online-Arithmetic with Rational Numbers}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--3},
  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.08021.6},
  URN =		{urn:nbn:de:0030-drops-14392},
  doi =		{10.4230/DagSemProc.08021.6},
  annote =	{Keywords: Computable analysis, online-arithmetic, rational numbers}
}
Document
C-XSC and Closely Related Software Packages

Authors: Werner Hofschuster, Walter Krämer, and Markus Neher


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
Complete Interval Arithmetic and its Implementation

Authors: Ulrich Kulisch


Abstract
A Complete Interval Arithmetic and its Implementation is discussed.

Cite as

Ulrich Kulisch. Complete Interval Arithmetic and its Implementation. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{kulisch:DagSemProc.08021.8,
  author =	{Kulisch, Ulrich},
  title =	{{Complete Interval Arithmetic and its Implementation}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--12},
  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.08021.8},
  URN =		{urn:nbn:de:0030-drops-14461},
  doi =		{10.4230/DagSemProc.08021.8},
  annote =	{Keywords: Interval Arithmetic, implementation}
}
Document
Distributed parameter and state estimation in a network of sensors

Authors: Michel Kieffer


Abstract
In this paper, we have considered distributed bounded-error state estimation applied to the problem of source tracking with a network of wireless sensors. Estimation is performed in a distributed context, emph{i.e.}, each sensor has only a limited amount of measurements available. A guaranteed set estimator is put at work. At each time instant, any sensor of the node has its own set estimate of the location of the source.

Cite as

Michel Kieffer. Distributed parameter and state estimation in a network of sensors. 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{kieffer:DagSemProc.08021.9,
  author =	{Kieffer, Michel},
  title =	{{Distributed parameter and state estimation in a network of sensors}},
  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.08021.9},
  URN =		{urn:nbn:de:0030-drops-14440},
  doi =		{10.4230/DagSemProc.08021.9},
  annote =	{Keywords: Parameter estimation, state estimation, bounded errors, nonlinear estimation}
}
Document
Extending the Range of C-XSC: Some Tools and Applications for the use in Parallel and other Environments

Authors: Markus Grimmer


Abstract
There is a broad range of packages and libraries for verified numerical computation. C-XSC is a library combining one of the most extensive sets of functions and operations on the one hand with a wide range of applications and special features on the other hand. As such it is an important task both to make use of its existing capabilities in applications and to develop further extensions giving access to additional areas and environments. In this talk, we present some examples of extensions for C-XSC that have been developed lately. Among these are extensions that give access to further hardware and software environments as well as applications making use of these possibilities. Software libraries for interval computation always imply great computation effort: One way to reduce computation times is the development of parallel methods to make use of parallel hardware. For this, it is important that the features and data types of the used library can be easily used in parallel programs. An MPI package for C-XSC data types allows to easily use C-XSC in parallel programs without bothering about the internal structure of data types. Another extension of C-XSC, the C-XSC Taylor arithmetic, is also covered by the MPI package. Parallel verified linear system solvers based on the package are available as well, and further development has been and is being done to integrate more efficient methods for interval linear system solution. One application making use of the mentioned extensions is a parallel verified Fredholm integral equation solver. Some results are given to demonstrate the reduction of computation time and, at the same time, the accuracy gain that can be obtained using the increased computation power. Naturally, hardware interval support would offer still more possibilities towards optimal performance of verified numerical software. Another possibility to extend the range of C-XSC is to make results available for further computations in other software environments as, for example, computer algebra packages. An example of this is presented for the Maple interval package intpakX. This kind of interfaces also allows the user to get access to further platforms like operating systems, compilers or even hardware. References: [1] ALiCEnext: http://www.alicenext.uni-wuppertal.de. [2] Blomquist, F.; Hofschuster, W.; Kraemer, W.: Real and Complex Taylor Arithmetic in C-XSC. Preprint BUW-WRSWT 2005/4, University of Wuppertal, 2005. [3] Grimmer, M.; Kraemer, W.: An MPI Extension for Verified Numerical Computations in Parallel Environments. In: Int. Conf. on Scientific Computing (CSC’07, Worldcomp’07) Las Vegas, June 25-28, 2007, Proceedings pp. 111-117, Arabnia et al. (eds.), 2007. [4] Grimmer, M.: An MPI Extension for the Use of C-XSC in Parallel Environments. Preprint BUW-WRSWT 2005/3, University of Wuppertal, 2005. [5] Grimmer, M.: Selbstverifizierende mathematische Softwarewerkzeuge im High Performance Computing. Dissertation, Logos Verlag, Berlin, 2007. [6] Grimmer, M.: Interval Arithmetic in Maple with intpakX. In: PAMM - Proceedings in Applied Mathematics and Mechanics, Vol. 2, Nr. 1, p. 442-443, Wiley-InterScience, 2003. [7] 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. [8] Klein, W.: Enclosure Methods for Linear and Nonlinear Systems of Fredholm Integral Equations of the Second Kind. In: Adams, Kulisch: Scientific Computing with Result Verification, Academic Press, 1993.

Cite as

Markus Grimmer. Extending the Range of C-XSC: Some Tools and Applications for the use in Parallel and other Environments. 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{grimmer:DagSemProc.08021.10,
  author =	{Grimmer, Markus},
  title =	{{Extending the Range of C-XSC: Some Tools and Applications for the use in Parallel and other Environments}},
  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.08021.10},
  URN =		{urn:nbn:de:0030-drops-14416},
  doi =		{10.4230/DagSemProc.08021.10},
  annote =	{Keywords: C-XSC, Integral Equations, Interval Arithmetic, Maple, MPI, Parallel Environment, Taylor Arithmetic, Verified Linear System Solver.}
}
Document
Fast (Parallel) Dense Linear Interval Systems Solvers in C-XSC Using Error Free Transformations and BLAS

Authors: Michael Zimmer and Walter Krämer


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.}
}
Document
Implementation of the reciprocal square root in MPFR

Authors: Paul Zimmermann


Abstract
We describe the implementation of the reciprocal square root --- also called inverse square root --- as a native function in the MPFR library. The difficulty is to implement Newton's iteration for the reciprocal square root on top's of GNU MP's extsc{mpn} layer, while guaranteeing a rigorous $1/2$ ulp bound on the roundoff error.

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Paul Zimmermann. Implementation of the reciprocal square root in MPFR. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{zimmermann:DagSemProc.08021.12,
  author =	{Zimmermann, Paul},
  title =	{{Implementation of the reciprocal square root in MPFR}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--3},
  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.08021.12},
  URN =		{urn:nbn:de:0030-drops-14357},
  doi =		{10.4230/DagSemProc.08021.12},
  annote =	{Keywords: Multiple precision, floating-point, inverse square root, correct rounding, MPFR library}
}
Document
Improving the Performance of a Verified Linear System Solver Using Optimized Libraries and Parallel Computation

Authors: Mariana Kolberg, Gerd Bohlender, and Dalcidio Claudio


Abstract
A parallel version of the self-verified method for solving linear systems was presented on PARA and VECPAR conferences in 2006. In this research we propose improvements aiming at a better performance. The idea is to implement an algorithm that uses technologies as MPI communication primitives associated to libraries as LAPACK, BLAS and C-XSC, aiming to provide both self-verification and speed-up at the same time. The algorithms should find an enclosure even for very ill-conditioned problems. In this scenario, a parallel version of a self-verified solver for dense linear systems appears to be essential in order to solve bigger problems. Moreover, the major goal of this research is to provide a free, fast, reliable and accurate solver for dense linear systems.

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Mariana Kolberg, Gerd Bohlender, and Dalcidio Claudio. Improving the Performance of a Verified Linear System Solver Using Optimized Libraries and Parallel Computation. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{kolberg_et_al:DagSemProc.08021.13,
  author =	{Kolberg, Mariana and Bohlender, Gerd and Claudio, Dalcidio},
  title =	{{Improving the Performance of a Verified Linear System Solver Using Optimized Libraries and Parallel Computation}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--5},
  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.08021.13},
  URN =		{urn:nbn:de:0030-drops-14386},
  doi =		{10.4230/DagSemProc.08021.13},
  annote =	{Keywords: Linear systems, result verification, parallel computing}
}
Document
Interval Arithmetic and Standardization

Authors: Jürgen Wolff von Gudenberg


Abstract
Interval arithmetic is arithmetic for continuous sets. Floating-point intervals are intervals of real numbers with floating-point bounds. Operations for intervals can be efficiently implemented. There is an unanimous agreement, how to define the basic operations, if we exclude division by an interval containing zero. Hence, it should be standardized. For division by zero, two options are possible, the clean exception free interval arithmetic or the containment arithmetic. They can be standardized as options. Elementary functions for intervals can be defined. In some application areas loose evaluation of functions, i.e. evaluation over an interval which is not completely contained in the function domain, is recommended, In this case, however, a discontinuity flag has to be set to inform that Brouwer's fixed point theorem is no longer applicable in that case.

Cite as

Jürgen Wolff von Gudenberg. Interval Arithmetic and Standardization. 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{wolffvongudenberg:DagSemProc.08021.14,
  author =	{Wolff von Gudenberg, J\"{u}rgen},
  title =	{{Interval Arithmetic and Standardization}},
  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.08021.14},
  URN =		{urn:nbn:de:0030-drops-14342},
  doi =		{10.4230/DagSemProc.08021.14},
  annote =	{Keywords: Intervals, containment sets, IEEE754r}
}
Document
Numerical Verification Assessment in Computational Biomechanics

Authors: Ekaterina Auer and Wolfram Luther


Abstract
In this paper, we present several aspects of the recent project PROREOP, in which a new prognosis system is developed for optimizing patient-specific preoperative surgical planning for the human skeletal system. We address verification and validation assessment in PROREOP with special emphasis on numerical accuracy and performance. To assess numerical accuracy, we propose to employ graded instruments, including accuracy tests and error analysis. The use of such instruments is exemplified for the process of accurate femur reconstruction. Moreover, we show how to verify the simulation results and take into account measurement uncertainties for a part of this process using tools and techniques developed in the project TellHIM&S.

Cite as

Ekaterina Auer and Wolfram Luther. Numerical Verification Assessment in Computational Biomechanics. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{auer_et_al:DagSemProc.08021.15,
  author =	{Auer, Ekaterina and Luther, Wolfram},
  title =	{{Numerical Verification Assessment in Computational Biomechanics}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--15},
  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.08021.15},
  URN =		{urn:nbn:de:0030-drops-14374},
  doi =		{10.4230/DagSemProc.08021.15},
  annote =	{Keywords: Numerical verification assessment, validation, uncertainty, result verification}
}
Document
On the Interoperability between Interval Software

Authors: Evgenija D. Popova


Abstract
The increased appreciation of interval analysis as a powerful tool for controlling round-off errors and modelling with uncertain data leads to a growing number of diverse interval software. Beside in some other aspects, the available interval software differs with respect to the environment in which it operates and the provided functionality. Some specific software tools are built on the top of other more general interval software but there is no single environment supporting all (or most) of the available interval methods. On another side, most recent interval applications require a combination of diverse methods. It is difficult for the end-users to combine and manage the diversity of interval software tools, packages, and research codes, even the latter being accessible. Two recent initiatives: [1], directed toward developing of a comprehensive full-featured library of validated routines, and [3] intending to provide a general service framework for validated computing in heterogeneous environment, reflect the realized necessity for an integration of the available methods and software tools. It is commonly understood that quality comprehensive libraries are not compiled by a single person or small group of people over a short time [1]. Therefore, in this work we present an alternative approach based on interval software interoperability. While the simplest form of interoperability is the exchange of data files, we will focus on the ability to run a particular routine executable in one environment from within another software environment, and vice-versa, via communication protocols. We discuss the motivation, advantages and some problems that may appear in providing interoperability between the existing interval software. Since the general-purpose environments for scientific/technical computing like Matlab, Mathematica, Maple, etc. have several features not attributable to the compiled languages from one side and on another side most problem solving tools are developed in some compiled language for efficiency reasons, it is interesting to study the possibilities for interoperability between these two kinds of interval supporting environments. More specifically, we base our presentation on the interoperability between Mathematica [5] and external C-XSC programs [2] via MathLink communication protocol [4]. First, we discuss the portability and reliability of interval arithmetic in Mathematica. Then, we present MathLink technology for building external MathLink-compatible programs. On the example of a C-XSC function for solving parametric linear systems, called from within a Mathematica session, we demonstrate some advantages of interval software interoperability. Namely, expanded functionality for both environments, exchanging data without using intermediate files and without any conversion but under dynamics and interactivity in the communication, symbolic manipulation interfaces for the compiled language software that often make access to the external functionality from within Mathematica more convenient even than from its own native environment. Once established, MathLink connection to external interval libraries or problem-solving software opens up an array on new possibilities for the latter. References: [1] G. Corliss, R. B. Kearfott, N. Nedialkov, S. Smith: Towards an Interval Subroutine Library, Workshop on Reliable Engineering Computing, Svannah, Georgia, USA, Feb. 22-24, 2006. [2] W. Hofschuster: C-XSC: Highlights and new developments. In: Numerical Validation in Current Hardware Architectures. Number 08021 Dagstuhl Seminar, Internationales Begegnungs- und Forschungszentrum f"ur Informatik, Schloss Dagstuhl, Germany, 2008. [3] W. Luther, W. Kramer: Accurate Grid Computing, 12th GAMM-IMACS Int. Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), Duisburg, Sept. 26-29, 2006. [4] Ch. Miyaji, P. Abbot eds.: Mathlink: Network Programming with Mathematica, Cambridge Univ. Press, Cambridge, 2001. [5] Wolfram Research Inc.: Mathematica, Version 5.2, Champaign, IL, 2005.

Cite as

Evgenija D. Popova. On the Interoperability between Interval Software. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{popova:DagSemProc.08021.16,
  author =	{Popova, Evgenija D.},
  title =	{{On the Interoperability between Interval Software}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--13},
  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.08021.16},
  URN =		{urn:nbn:de:0030-drops-14501},
  doi =		{10.4230/DagSemProc.08021.16},
  annote =	{Keywords: Software interoperability, interfacing, interval software, C-XSC, MathLink, Mathematica}
}
Document
Robustness of Boolean operations on subdivision-surface models

Authors: Di Jiang and Neil Stewart


Abstract
This work was presented in two parts at Dagstuhl seminar 08021. The two presentations described work in progress, including a ``backward bound'' for a combined backward/forward error analysis for the problem mentioned in the title. We seek rigorous proofs that representations of computed sets, produced by algorithms to compute Boolean operations, are well formed, and that the algorithms are correct. Such proofs should eventually take account of the use of finite-precision arithmetic, although the proofs presented here do not. The representations studied are based on subdivision surfaces. Such representations are being used more and more frequently in place of trimmed NURBS representations, and the robustness analysis for these new representations is simpler than for trimmed NURBS. The particular subdivision-surface representation used is based on the Loop subdivision scheme. The analysis is broken into three parts. First, it is established that the input operands are well-formed two-dimensional manifolds without boundary. This can be done with existing methods. Secondly, we introduce the so-called ``limit mesh'', and view the limit meshes corresponding to the input sets as defining an approximate problem in the sense of a backward error analysis. The presentations mentioned above described a proof of the corresponding error bound. The third part of the analysis corresponds to the ``forward bound'': this remains to be done.

Cite as

Di Jiang and Neil Stewart. Robustness of Boolean operations on subdivision-surface models. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{jiang_et_al:DagSemProc.08021.17,
  author =	{Jiang, Di and Stewart, Neil},
  title =	{{Robustness of Boolean operations on subdivision-surface models}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--10},
  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.08021.17},
  URN =		{urn:nbn:de:0030-drops-14435},
  doi =		{10.4230/DagSemProc.08021.17},
  annote =	{Keywords: Robustness, finite-precision arithmetic, Boolean operations, subdivision surfaces}
}
Document
Second Note on Basic Interval Arithmetic for IEEE754R

Authors: John D. Pryce, George C. Corliss, R. Baker Kearfott, Ned S. Nedialkov, and Spencer Smith


Abstract
The IFIP Working Group 2.5 on Numerical Software (IFIPWG2.5) wrote on 5th Septem- ber 2007 to the IEEE Standards Committee concerned with revising the IEEE Floating- Point Arithmetic Standards 754 and 854 (IEEE754R), expressing the unanimous request of IFIPWG2.5 that the following requirement be included in the future computer arithmetic standard: For the data format double precision, interval arithmetic should be made available at the speed of simple floating-point arithmetic. IEEE754R (we believe) welcomed this development. They had before them a document defining interval arithmetic operations but, to be the basis of a standards document, it needed more detail. Members of the Interval Subroutine Library (ISL) team were asked to comment, in an email from Ulrich Kulisch that enclosed one from Jim Demmel to Van Snyder raising the issue. This paper provides ISL's comments.

Cite as

John D. Pryce, George C. Corliss, R. Baker Kearfott, Ned S. Nedialkov, and Spencer Smith. Second Note on Basic Interval Arithmetic for IEEE754R. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{pryce_et_al:DagSemProc.08021.18,
  author =	{Pryce, John D. and Corliss, George C. and Kearfott, R. Baker and Nedialkov, Ned S. and Smith, Spencer},
  title =	{{Second Note on Basic Interval Arithmetic for IEEE754R}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--8},
  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.08021.18},
  URN =		{urn:nbn:de:0030-drops-14517},
  doi =		{10.4230/DagSemProc.08021.18},
  annote =	{Keywords: Interval arithmetic, validated computation, floating point, standards, exceptions, not an interval}
}
Document
The CoStLy C++ Class Library

Authors: Markus Neher


Abstract
CoStLy (ul{Co}mplex ul{St}andard Functions ul{L}ibrarul{y}) has been developed as a C++ class library for the validated computation of function values and of ranges of complex standard functions. If performed in exact arithmetic, the inclusion functions for principal branches compute optimal range bounds. For the sake of accuracy, a major effort has been made in the implementation of the algorithms in floating point arithmetic to eliminate all intermediate expressions subject to numerical overflow, underflow, or cancellation. The CoStLy library has been extensively tested for arguments with absolute values ranging from 1.0E-300 to 1.0E+300. For most arguments, the computed bounds for function values are highly accurate. In many test cases, the observed precision of the result was about 50 correct bits (out of the 53 bits available in IEEE 754 floating point arithmetic) for point arguments.

Cite as

Markus Neher. The CoStLy C++ Class Library. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-6, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{neher:DagSemProc.08021.19,
  author =	{Neher, Markus},
  title =	{{The CoStLy C++ Class Library}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--6},
  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.08021.19},
  URN =		{urn:nbn:de:0030-drops-14495},
  doi =		{10.4230/DagSemProc.08021.19},
  annote =	{Keywords: Complex interval arithmetic, inclusion functions}
}
Document
The New IEEE-754 Standard for Floating Point Arithmetic

Authors: Peter Markstein


Abstract
The current IEEE-754 floating point standard was adopted 23 years ago. IEEE chartered a committee to revise the standard to include new common practice in floating point arithmetic, to incorporate decimal floating point into the standard, and to address the issue of reproducible results. This talk will visit these issues, based on the current work of the IEEE-754 revisions committee, which expects that a new standard will be adopted sometime in 2008.

Cite as

Peter Markstein. The New IEEE-754 Standard for Floating Point Arithmetic. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{markstein:DagSemProc.08021.20,
  author =	{Markstein, Peter},
  title =	{{The New IEEE-754 Standard for Floating Point Arithmetic}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--3},
  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.08021.20},
  URN =		{urn:nbn:de:0030-drops-14481},
  doi =		{10.4230/DagSemProc.08021.20},
  annote =	{Keywords: Floating point arithmetic, standards}
}
Document
Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering

Authors: Andreas Rauh, Johanna Minisini, and Eberhard P. Hofer


Abstract
Modern techniques for the design and analysis of control strategies for nonlinear dynamical systems are often based on the simulation of the open-loop as well as the closed-loop dynamical behavior of suitable mathematical models. In control engineering, continuous-time and discrete-time state-space representations are widely used which are given by sets of ordinary differential equations and difference equations, respectively. In addition to these representations, sets of differential algebraic equations are commonly used. Since we will focus on computational techniques which are applied for the design and mathematical verification of controllers for lumped parameter systems, i.e., systems which do not contain elements with distributed parameters, partial differential equations will not be considered in this talk. The prerequisite for the design and robustness analysis of each control system is the identification of mathematical models which describe the dynamics of the plant to be controlled as well as the available measurement devices with a sufficient accuracy. The model identification task comprises the derivation of physically motivated state equations, their parameterization based on measured data, as well as simplifications to apply specific approaches for controller design. In the design stage, both open-loop and closed-loop control strategies can be considered. Since dynamical system models are subject to uncertain parameters and uncertain initial conditions in most practical applications, detailed mathematical formulations of the desired dynamics of the controlled system are necessary. These specifications involve the definition of robustness with respect to the above-mentioned uncertainties. For linear system representations, robustness is commonly specified in terms of regions in the complex domain containing all admissible poles of the closed-loop transfer functions ($Gamma$-stability) or in terms of specifications of worst-case bounds for the frequency response ($mathcal{B}$-stability) [1]. However, these specifications do not allow for inclusion of bounds for the state variables which are often available in the time domain if controllers are designed for safety critical applications. Especially for nonlinear dynamical systems, pole assignment based on the linearization of nonlinear mathematical models generally leads to the necessity for the analysis of asymptotic stability of the resulting closed-loop dynamics. In this presentation, we will give an overview of the potential use of validated techniques for the analysis and design of controllers for nonlinear dynamical systems with uncertainties, where the systems under consideration will be subject to constraints for both state and control variables. As an application scenario the design of robust control strategies for a biological wastewater treatment process will be discussed. In the design and the verification process, constraints for both state and control variables which are given by guaranteed interval bounds in the time domain are taken into account. Suitable computational techniques are, for example, based on an extension of the validated initial value problem solver {sc ValEncIA-IVP} [2,6]. For that purpose, differential sensitivities of the trajectories of all state variables with respect to variations of the parameters of the mathematical system model as well as the adaptation of controller parameters are computed. This information can then be used for online identification and adaptation of parameters during the operation of a closed-loop controller as well as in offline design, verification, and optimization. Here, the interval arithmetic routines for sensitivity analysis allow to compute guaranteed differential sensitivity measures for system models with both nominal parameters and interval uncertainties. The presented interval arithmetic techniques are the basis for a general purpose tool for the analysis and the design of robust and optimal control strategies for uncertain dynamical systems. The presentation is concluded with an outlook on the formulation of control problems using sets of differential algebraic equations. Possibilities for the extension of {sc ValEncIA-IVP} to this type of system representation will be summarized. Relations between the presented interval arithmetic approach and methods for stabilizing control of nonlinear dynamical systems which make use of structural system properties such as differential flatness [3] and exact feedback linearization are highlighted [4,5]. In the latter case, input-output linearization as well as (in special cases) input-to-state linearization are of practical importance. References: [1] J. Ackermann, P. Blue, T. B"unte, L. G"uvenc, D. Kaesbauer, M. Kordt, M. Muhler, and D. Odenthal, {it{Robust Control: The Parameter Space Approach}}, Springer--Verlag, London, 2nd edition, 2002. [2] E. Auer, A. Rauh, E. P. Hofer, and W. Luther, {it{Validated Modeling of Mechanical Systems with {sc SmartMOBILE}: Improvement of Performance by {sc ValEncIA-IVP}}}, In Proceedings of Dagstuhl Seminar 06021: Reliable Implementation of Real Number Algorithms: Theory and Practice, Lecture Notes in Computer Science, Dagstuhl, Germany, 2006. In print. [3] M. Fliess, J. Lévine, P. Martin, and P. Rouchon, {it{Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples}}, International Journal of Control, vol. 61, pp. 1327--1361, 1995. [4] H. K. Khalil, {it{Nonlinear Systems}}, Prentice-Hall, Upper Saddle River, New Jersey, 3rd edition, 2002. [5] H. J. Marquez, {it{Nonlinear Control Systems}}, John Wiley & Sons, Inc., New Jersey, 2003. [6] A. Rauh and E. Auer, {{www.valencia-ivp.com}}.

Cite as

Andreas Rauh, Johanna Minisini, and Eberhard P. Hofer. Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering. In Numerical Validation in Current Hardware Architectures. Dagstuhl Seminar Proceedings, Volume 8021, pp. 1-10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)


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@InProceedings{rauh_et_al:DagSemProc.08021.21,
  author =	{Rauh, Andreas and Minisini, Johanna and Hofer, Eberhard P.},
  title =	{{Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering}},
  booktitle =	{Numerical Validation in Current Hardware Architectures},
  pages =	{1--10},
  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.08021.21},
  URN =		{urn:nbn:de:0030-drops-14529},
  doi =		{10.4230/DagSemProc.08021.21},
  annote =	{Keywords: Interval techniques, \{sc\{ValEncIA-IVP\}\}, controller design, robustness, validated integration of ODEs, parameter uncertainties, sensitivity analysis}
}

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