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### Modern Computer Arithmetic (version 0.5.1) [article]

Richard P. Brent, Paul Zimmermann
2010 arXiv   pre-print
This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers.  ...  Chapter titles are: 1, Integer Arithmetic; 2, Modular Arithmetic and the FFT; 3, Floating-Point Arithmetic; 4, Elementary and Special Function Evaluation; 5 (Appendix), Implementations and Pointers.  ...  n arithmetic, then the relative error is of order 4 k 2 −n ; and (b) if a recurrence based on (4.60) is used, then the relative error is O(k 2 2 −n ).  ...

### II. Arithmetic [chapter]

Alexandre Borovik, Tony Gardiner
2019 The Essence of Mathematics Through Elementary Problems
The next few problems have been stripped of any context, leaving us free to focus on the underlying structure in a purely algebraic, or arithmetical, spirit.  ...  The next problem invites you to devise divisibility tests for integers written in base 2 like those for base 10 (that is, tests which implement some check involving the base 2 digits in place of carrying  ...  -If a direct method of paying k cents involves at least one 3 cent coin, then we can replace one 3 cent coin by a 4 cent coin to produce a way of paying k`1 cents.  ...

### What every agent-based modeller should know about floating point arithmetic

J. Gary Polhill, Luis R. Izquierdo, Nicholas M. Gotts
2006 Environmental Modelling & Software
Floating point arithmetic is a subject all too often ignored, yet, for agent-based models in particular, it has the potential to create misleading results, and even to influence emergent outcomes of the  ...  Using a simple demonstration model, this paper illustrates the problems that accumulated floating point errors can cause, and compares a number of techniques that might be used to address them.  ...  Discussion Evaluation of techniques The use of tolerance windows is possibly one of the most popular techniques for coping with floating point arithmetic errors, and they are discussed by Knuth (1998  ...

### Industrial application of exact Boolean operations for meshes

Martin Schifko, Bert Jüttler, Bernhard Kornberger
2010 Proceedings of the 26th Spring Conference on Computer Graphics - SCCG '10
In order to avoid potential robustness problems, which may be caused by (almost) degenerate triangles or by intersections of nearly co-planar triangles, we use filtered exact arithmetic, based on the libraries  ...  CGAL and GNU Multi Precision Arithmetic Library.  ...  Moreover, implementations of geometric algorithms are prone to fail if they rely on predicates, that are evaluated using inexact arithmetic, because a wrong result can lead a program into a wrong state  ...

### Exact computation of the medial axis of a polyhedron

Tim Culver, John Keyser, Dinesh Manocha
2004 Computer Aided Geometric Design
These algorithms include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, culling operations, and floating-point filters.  ...  The algorithm has been implemented and we highlight its performance on a number of examples.  ...  Techniques based on modular arithmetic, floating-point filters, lazy evaluation, and spatial subdivision combine to improve the efficiency of the system by orders of magnitude.  ...

### Towards exact geometric computation

Chee-Keng Yap
1997 Computational geometry
This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative.  ...  Such implementations have many well-known problems, here informally called "robustness issues".  ...  Acknowledgements I would like to thank my colleague and collaborator Tom Dub6 for many discussions on these issues. His insights into implementations and number packages have been invaluable.  ...

### Handling Degeneracies in Exact Boundary Evaluation [article]

K. Ouchi, J. Keyser
2004 Solid Modeling
We present a method for dealing with degenerate situations in an exact boundary evaluation system. We describe the possible degeneracies that can arise and how to detect them.  ...  We then present a numeric perturbation method that is simpler to implement within a complex system than symbolic perturbation methods.  ...  We have developed and implemented a method based on the Rational Univariate Reduction which allows this. The details are omitted here.  ...

### Counterexample guided inductive optimization based on satisfiability modulo theories

Rodrigo F. Araújo, Higo F. Albuquerque, Iury V. de Bessa, Lucas C. Cordeiro, João E. Chaves Filho
2018 Science of Computer Programming
In particular, CEGIO relies on iterative executions to constrain a verification procedure, in order to perform inductive generalization, based on counterexamples extracted from SMT solvers.  ...  This paper describes three variants of a counterexample guided inductive optimization (CEGIO) approach based on Satisfiability Modulo Theories (SMT) solvers.  ...  The ABsolver uses a nonlinear optimization tool for Boolean and polynomial arithmetic and a lazy SMT procedure to perform a faster satisfiability checking.  ...

### Exact, robust, and efficient regularized Booleans on general 3D meshes

Hichem Barki, Gaël Guennebaud, Sebti Foufou
2015 Computers and Mathematics with Applications
Our method is based on a few geometric and topological predicates that allow to handle all input/output cases considered as degenerate in existing solutions, such as voids, non-manifold, disconnected,  ...  Our experimentation showed that our more general approach is also more robust and more efficient than Maya's implementation (×3), CGAL's robust Nef polyhedra (×5), and recent plane-based approaches.  ...  Acknowledgment This publication was made possible by NPRP grant #09-906-1-137 from the Qatar National Research Fund (a member of Qatar Foundation).  ...

### Optimising finite-difference methods for PDEs through parameterised time-tiling in Devito [article]

Nicholas Sim
2018 arXiv   pre-print
In a large problem set, approximations can take days or weeks to evaluate, yet the bulk of computation may occur within a single loop nest.  ...  We present an estimator for arithmetic intensity under time-tiling and a model to predict runtime improvements in stencil computations.  ...  In particular, it gives upper bounds for performance based on memory bandwidth and CPU performance, and describes the bottlenecks that a program would encounter based on its arithmetic intensity   ...

### An overview of the JPEG 2000 still image compression standard

Majid Rabbani, Rajan Joshi
2002 Signal processing. Image communication
In this paper, a technical description of Part 1 of the JPEG 2000 standard is provided, and the rationale behind the selected technologies is explained.  ...  In 1996, the JPEG committee began to investigate possibilities for a new still image compression standard to serve current and future applications.  ...  The authors would also like to thank Brian Banister for generating the bitplane results in Section 5.3, and Scott Houchin for providing the material on file format in Section 6.3.  ...

### A Scientific Calculator for Exact Real Number Computation Based on LRT, GMP and FC++

Alejandra Lucatero, J. Raymundo Marcial-Romero, J. A. Hernández
2012 Acta Universitaria
Its lazy evaluation mechanism (also called call-by-need) and its infinite list requirement, make the language appropriate to be implemented in a functional programming language such as Haskell.  ...  We show that a direct translation of LRT in FC++ results in a faster scientific calculator than the one presented in Haskell.  ...  In Marcial et. al  an implementation of a basic calculator based on LRT and programmed using FC++ and GMP was presented.  ...

### A Lagrangian Approach to Dynamic Interfaces through Kinetic Triangulation of the Ambient Space

J.-P. Pons, J.-D. Boissonnat
2007 Computer graphics forum (Print)
The principle of our approach is to maintain a two-dimensional triangulation which embeds the one-dimensional polygonal description of the interfaces.  ...  In this paper, we propose a robust and efficient Lagrangian approach for modeling dynamic interfaces between different materials undergoing large deformations and topology changes, in two dimensions.  ...  Consequently, in our implementation, for efficiency purposes, we have used a fixedprecision floating-point arithmetic both for the solver and for the lazy kinetic triangulation.  ...

### Analysis of boundary representation model rectification

Guoling Shen, Takis Sakkalis, Nicholas Patrikalakis
2001 Proceedings of the sixth ACM symposium on Solid modeling and applications - SMA '01
In so-called lazy arithmetic, evaluations are performed using error-bounded methods, and switched to rational arithmetic when approaching degeneracies.  ...  Typical examples are integer and rational arithmetic (viewed here as approximations of real arithmetic), and interval and lazy arithmetic.  ...  Appendix B Boxes on an interval B-spline surface Evaluation of a B-spline surface P(u, v) at a certain point (uo, vo) in the parameter domain using interval arithmetic creates a box guaranteed to containing  ...

### Approximate counting in SMT and value estimation for probabilistic programs

Dmitry Chistikov, Rayna Dimitrova, Rupak Majumdar
2017 Acta Informatica
In this paper, we show a reduction from an approximate version of #SMT to SMT. We focus on the theories of integer arithmetic and linear real arithmetic.  ...  We have implemented the algorithms and used them to solve the value problem for a model of loop-free probabilistic programs with nondeterminism.  ...  Our #SMT procedure for linear real arithmetic allows an additive error in the approximation; it is known that the volume of a polytope does not always have a small exact representation as a rational number  ...
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