eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
15
10.4230/DagSemProc.05391.1
article
05391 Abstracts Collection – Algebraic and Numerical Algorithms and Computer-assisted Proofs
Buchberger, Bruno
Oishi, Shin'ichi
Plum, Michael
Rump, Siegfried M.
From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs'' 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.
Links to extended abstracts or full papers are provided, if available.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.1/DagSemProc.05391.1.pdf
Self-validating methods
computer algebra
computer-assisted proofs
real number algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
5
10.4230/DagSemProc.05391.2
article
05391 Executive Summary – Numerical and Algebraic Algorithms and Computer-assisted Proofs
Buchberger, Bruno
Jansson, Christian
Oishi, Shin'ichi
Plum, Michael
Rump, Siegfried M.
The common goal of self-validating methods and computer
algebra methods is to solve mathematical problems with complete rigor
and with the aid of computers. The seminar focused on several aspects
of such methods for computer-assisted proofs.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.2/DagSemProc.05391.2.pdf
Self-validating methods
computer algebra
computer-assisted proofs
real number algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
0
10.4230/DagSemProc.05391.3
article
Compensated Horner Scheme
Langlois, Philippe
Graillat, Stef
Louvet, Nicolas
Using error-free transformations, we improve the classic Horner Scheme (HS)
to evaluate (univariate) polynomials in floating point arithmetic.
We prove that this Compensated Horner Scheme (CHS) is as accurate as HS
performed with twice the working precision.
Theoretical analysis and experiments exhibit a reasonable running time
overhead being also more interesting than double-double implementations.
We introduce a dynamic and validated error bound of the CHS computed value.
The talk presents these results together with a survey about error-free
transformations and related hypothesis.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.3/DagSemProc.05391.3.pdf
Polynomial evaluation
Horner scheme
error-free transformation
floating point arithmetic
accuracy
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
5
10.4230/DagSemProc.05391.4
article
Enclosure for the Biharmonic Equation
Fazekas, Borbála
Plum, Michael
Wieners, Christian
In this paper we give an enclosure for the solution of the biharmonic problem and also for its gradient and Laplacian in the $L_2$-norm, respectively.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.4/DagSemProc.05391.4.pdf
Biharmonic problem
enclosure
finite elements
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
17
10.4230/DagSemProc.05391.5
article
Integration of reliable algorithms into modeling software
Luther, Wolfram
Haßlinger, Gerhard
Auer, Ekaterina
Dyllong, Eva
Traczinski, Daniela
Traczinski, Holger
In this note we discuss strategies that would enhance modern modeling and simulation software (MSS) with reliable routines using validated data types, controlled rounding, algorithmic differentiation and interval equation or initial value problem solver. Several target systems are highlighted. In stochastic traffic modeling, the computation of workload distributions plays a prominent role since they influence the quality of service parameters. INoWaTIV is a workload analysis tool that uses two different techniques: the polynomial factorization approach and the Wiener-Hopf factorization to determine the work-load distributions of GI/GI/1 and SMP/GI/1 service systems accurately. Two extensions of a multibody modeling and simulation software were developed to model kinematic and dynamic properties of multibody systems in a validated way. Furthermore, an interface was created that allows the computation of convex hulls and reliable lower bounds for the distances between subpav-ing-encoded objects constructed with SIVIA (Set Inverter Via Interval Analysis).
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.5/DagSemProc.05391.5.pdf
Reliable algorithms
stochastic traffic modeling
multibody modeling tools
geometric modeling
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
11
10.4230/DagSemProc.05391.6
article
Lurupa - Rigorous Error Bounds in Linear Programming
Keil, Christian
Linear Programming has numerous applications, e.g., operations research,
relaxations in global optimization, computational geometry. Recently it has
been shown that many real world problems exhibit numerical difficulties due to
ill-conditioning.
Lurupa is a software package for computing rigorous optimal value bounds. The
package can handle point and interval problems. Numerical experience with the
Netlib lp library is given.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.6/DagSemProc.05391.6.pdf
Linear programming
rigorous error bounds
netlib
interval arithmetic
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
8
10.4230/DagSemProc.05391.7
article
Rigorous Results in Combinatorial Optimization
Jansson, Christian
Many current deterministic solvers for NP-hard
combinatorial optimization problems are based on nonlinear
relaxation techniques that use floating point arithmetic.
Occasionally, due to solving these relaxations, rounding errors
may produce erroneous results, although the deterministic
algorithm should compute the exact solution in a finite number of
steps. This may occur especially if the relaxations are
ill-conditioned or ill-posed, and if Slater's constraint
qualifications fail. We show how exact solutions can be obtained
by rigorously bounding the optimal value of semidefinite
relaxations, even in the ill-posed case. All rounding errors due
to floating point arithmetic are taken into account.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.7/DagSemProc.05391.7.pdf
Combinatorial Optimization
Semidefinite Programming
Ill-posed Problems
Verification Methods
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
15
10.4230/DagSemProc.05391.8
article
Toward accurate polynomial evaluation in rounded arithmetic (short report)
Demmel, James
Dumitriu, Ioana
Holtz, Olga
Given a multivariate real (or complex) polynomial $p$ and a domain $cal D$,
we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately
for all $x in {cal D}$ using rounded real (or complex) arithmetic.
Here ``accurately'' means with relative error less than 1, i.e., with some correct
leading digits. The answer depends on the model of rounded arithmetic:
We assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or
$a cdot b$, its computed value is $op(a,b) cdot (1 + delta)$,
where $| delta |$ is bounded by some constant $epsilon$ where $0 < epsilon ll 1$,
but $delta$ is otherwise arbitrary. This model is the traditional one used to analyze
the accuracy of floating point algorithms.
Our ultimate goal is to establish a decision procedure that, for any $p$ and $cal D$,
either exhibits an accurate algorithm or proves that none exists. In contrast to the
case where numbers are stored and manipulated as finite bit strings (e.g., as floating
point numbers or rational numbers) we show that some polynomials $p$ are impossible to
evaluate accurately. The existence of an accurate algorithm will depend not just
on $p$ and $cal D$, but on which arithmetic operators and constants are available
to the algorithm and whether branching is permitted in the algorithm.
Toward this goal, we present necessary conditions on $p$ for it to be
accurately evaluable on open real or complex domains ${cal D}$.
We also give sufficient conditions, and describe progress toward
a complete decision procedure. We do present a complete
decision procedure for homogeneous polynomials $p$ with integer coefficients,
${cal D} = C^n$, using only arithmetic operations
$+$, $-$ and $cdot$.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.8/DagSemProc.05391.8.pdf
Accurate polynomial evaluation
models or rounded arithmetic
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-01-31
5391
1
21
10.4230/DagSemProc.05391.9
article
Verification of Solutions for Almost Linear Complementarity Problems
Alefeld, Götz
Wang, Zhengyu
We present a computational enclosure method for the solution of a class of nonlinear complementarity problems.
The procedure also delivers a proof for the uniqueness of the solution.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol05391/DagSemProc.05391.9/DagSemProc.05391.9.pdf
Complementarity problems
verification of solutions