12,143 research outputs found
Properties of the subtraction valid for any floating point system
International audienceWe start in this text with a very generic definition of floating point systems. We show that just a few very natural necessary conditions are sufficient to focus down to two classes of implemented floating point arithmetic. Later, we prove that, for all the existing implementations, high level properties such as Sterbenz's theorem are satisfied. We finish this text by focusing on the differences between an IEEE-754 compatible unit and Texas Instrument TMS/SMJ 320C3x digital signal processing circuit that is recommended for avionics and military applications. The results presented in this text have been validated by the Coq automatic proof checker to build confidence for later implementations in critical systems such as an aircraft flight control primary or secondary computer
Interval Slopes as Numerical Abstract Domain for Floating-Point Variables
The design of embedded control systems is mainly done with model-based tools
such as Matlab/Simulink. Numerical simulation is the central technique of
development and verification of such tools. Floating-point arithmetic, that is
well-known to only provide approximated results, is omnipresent in this
activity. In order to validate the behaviors of numerical simulations using
abstract interpretation-based static analysis, we present, theoretically and
with experiments, a new partially relational abstract domain dedicated to
floating-point variables. It comes from interval expansion of non-linear
functions using slopes and it is able to mimic all the behaviors of the
floating-point arithmetic. Hence it is adapted to prove the absence of run-time
errors or to analyze the numerical precision of embedded control systems
Is Your Model Susceptible to Floating-Point Errors?
This paper provides a framework that highlights the features of computer models that make them especially vulnerable to floating-point errors, and suggests ways in which the impact of such errors can be mitigated. We focus on small floating-point errors because these are most likely to occur, whilst still potentially having a major influence on the outcome of the model. The significance of small floating-point errors in computer models can often be reduced by applying a range of different techniques to different parts of the code. Which technique is most appropriate depends on the specifics of the particular numerical situation under investigation. We illustrate the framework by applying it to six example agent-based models in the literature.Floating Point Arithmetic, Floating Point Errors, Agent Based Modelling, Computer Modelling, Replication
Certifying floating-point implementations using Gappa
High confidence in floating-point programs requires proving numerical
properties of final and intermediate values. One may need to guarantee that a
value stays within some range, or that the error relative to some ideal value
is well bounded. Such work may require several lines of proof for each line of
code, and will usually be broken by the smallest change to the code (e.g. for
maintenance or optimization purpose). Certifying these programs by hand is
therefore very tedious and error-prone. This article discusses the use of the
Gappa proof assistant in this context. Gappa has two main advantages over
previous approaches: Its input format is very close to the actual C code to
validate, and it automates error evaluation and propagation using interval
arithmetic. Besides, it can be used to incrementally prove complex mathematical
properties pertaining to the C code. Yet it does not require any specific
knowledge about automatic theorem proving, and thus is accessible to a wide
community. Moreover, Gappa may generate a formal proof of the results that can
be checked independently by a lower-level proof assistant like Coq, hence
providing an even higher confidence in the certification of the numerical code.
The article demonstrates the use of this tool on a real-size example, an
elementary function with correctly rounded output
Environmentally Friendly Renormalization
We analyze the renormalization of systems whose effective degrees of freedom
are described in terms of fluctuations which are ``environment'' dependent.
Relevant environmental parameters considered are: temperature, system size,
boundary conditions, and external fields. The points in the space of \lq\lq
coupling constants'' at which such systems exhibit scale invariance coincide
only with the fixed points of a global renormalization group which is
necessarily environment dependent. Using such a renormalization group we give
formal expressions to two loops for effective critical exponents for a generic
crossover induced by a relevant mass scale . These effective exponents are
seen to obey scaling laws across the entire crossover, including hyperscaling,
but in terms of an effective dimensionality, d\ef=4-\gl, which represents the
effects of the leading irrelevant operator. We analyze the crossover of an
model on a dimensional layered geometry with periodic, antiperiodic
and Dirichlet boundary conditions. Explicit results to two loops for effective
exponents are obtained using a [2,1] Pad\'e resummed coupling, for: the
``Gaussian model'' (), spherical model (), Ising Model (),
polymers (), XY-model () and Heisenberg () models in four
dimensions. We also give two loop Pad\'e resummed results for a three
dimensional Ising ferromagnet in a transverse magnetic field and corresponding
one loop results for the two dimensional model. One loop results are also
presented for a three dimensional layered Ising model with Dirichlet and
antiperiodic boundary conditions. Asymptotically the effective exponents are in
excellent agreement with known results.Comment: 76 pages of Plain Tex, Postscript figures available upon request from
[email protected], preprint numbers THU-93/14, DIAS-STP-93-1
Recursive algorithms for the elimination of redundant paths in spatial lag operators
Recursive algorithms for the elimination of redundant paths in spatial lag operators are introduced. It is shown that these algorithms have superior computational properties in comparison with the cumbersome procedure proposed by Ross and Harary (1952). A rigorous definition of spatial lag operators is given, while a number of mathematical results and properties are derived. Theoretical and empirical results regarding the performance of the proposed algorithms are presented
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