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### Formally certified floating-point filters for homogeneous geometric predicates

Guillaume Melquiond, Sylvain Pion
2007 RAIRO - Theoretical Informatics and Applications
Floating-point arithmetic provides a fast but inexact way of computing geometric predicates.  ...  In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results.  ...  And we conclude with a comparison to other existing methods. ORIENTATION-2 PREDICATE This predicate is one of the most often encountered geometric predicates.  ...

### Indirect Predicates for Geometric Constructions

Marco Attene
2020 Computer-Aided Design
Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry.  ...  outperforms state-of-the-art solutions based on lazy exact intermediate representations.  ...  In our implementation we consider three computation models: floating point arithmetic (with semi-static filter), interval arithmetic (with dynamic filter), expansion arithmetic (exact).  ...

### The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra [chapter]

Jihun Yu, Chee Yap, Zilin Du, Sylvain Pion, Hervé Brönnimann
2010 Lecture Notes in Computer Science
This paper is concerned with one such approach called Exact Numeric Computation (ENC).  ...  The ENC approach to algebraic number computation is based on iterative verified approximations, combined with constructive zero bounds.  ...  Also known as Exact Geometric Computation (EGC) in the context of geometric applications.  ...

### A geometric construction of iterative functions of order three to solve nonlinear equations

Changbum Chun
2007 Computers and Mathematics with Applications
Some examples are given of deriving several third-order iteration methods, and several numerical results follow to illustrate the performance of the derived methods.  ...  In this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations.  ...  Acknowledgements The author would like to thank the referees for their useful comments and constructive suggestions which substantially improved the quality of this paper.  ...

### Geometric classification tests using interval arithmetic in b-rep solid modeling

M. de S. G. Tsuzuki, M. Shimada
2003 Journal of the Brazilian Society of Mechanical Sciences and Engineering
It is necessary to control the growth of the intervals based on the geometry and topology. This work will introduce the application of interval arithmetic to a B-Rep solid modeler.  ...  Another important step in the Boolean operation is the determination of intersection points where the use of interval arithmetic can have side effects as intervals with large dimensions, and may cause  ...  Acknowledgement This work was partially supported by FAPESP and CNPq under grant 300.224/96.  ...

### Algebraic methods and arithmetic filtering for exact predicates on circle arcs

Olivier Devillers, Alexandra Fronville, Bernard Mourrain, Monique Teillaud
2002 Computational geometry
The purpose of this paper is to present a new method to design exact geometric predicates in algorithms dealing with curved objects such as circular arcs.  ...  This method allows the use of efficient arithmetic and filtering techniques leading to fast implementation as shown by the experimental results.   ...  Acknowledgements The authors would like to thank Sylvain Pion for helpful discussions on arithmetic filters.  ...

### Analytic Root Clustering: A Complete Algorithm Using Soft Zero Tests [chapter]

Chee Yap, Michael Sagraloff, Vikram Sharma
2013 Lecture Notes in Computer Science
This is formalized as the root clustering problem, and we provide a complete (δ, ǫ)-exact algorithm based on soft zero tests. ⋆ This paper was presented at an invited Special Session on "Computational  ...  Zero tests are expensive and may be uncomputable. So we seek geometric algorithms based on a weak form of such tests, called soft zero tests.  ...  From the perspective of Exact Geometric Computation (EGC), current models of continua computing are lacking [22] . The touchstone is the Zero Problem, deciding if a real constant is zero.  ...

### Controlled perturbation for arrangements of circles

Dan Halperin, Eran Leiserowitz
2003 Proceedings of the nineteenth conference on Computational geometry - SCG '03
that the predicates involved in the construction can be safely computed with the given (limited) precision arithmetic.  ...  We implemented the perturbation scheme and the construction of the arrangement and we report on experimental results.  ...  To evaluate the bound on the error of an expression E, we compute an interval, which contains the exact value of E, and its length will be the bound on the error.  ...

### Controlled perturbation for arrangements of circles

Dan Halperin, Eran Leiserowitz
2003 Proceedings of the nineteenth conference on Computational geometry - SCG '03
that the predicates involved in the construction can be safely computed with the given (limited) precision arithmetic.  ...  We implemented the perturbation scheme and the construction of the arrangement and we report on experimental results.  ...  To evaluate the bound on the error of an expression E, we compute an interval, which contains the exact value of E, and its length will be the bound on the error.  ...

### Fast magnetic field manipulations and nonadiabatic geometric phases of nitrogen-vacancy center spin in diamond

Wen-Qi Fang, Bang-Gui Liu
2017 Journal of Physics D: Applied Physics
Here, we construct exact evolution operator of the nitrogen-vacancy-center (NV) spin in diamond under external magnetic fields and investigate the nonadiabatic geometric phases, both cyclic and non-cyclic  ...  It is believed that the nonadiabatic geometric phases can be measured in future experiments and these fast quantum manipulations can be useful in designing spin-based quantum applications.  ...  However, we notice that with Barnes's method [14] , it is difficult to obtain physically reasonable pulse which is zero in amplitude at initial and ending times.  ...

### Numerical integration over implicitly defined domains with topological guarantee [article]

Tianhui Yang, Ammar Qarariyah, Hongmei Kang, Jiansong Deng
2019 arXiv   pre-print
Numerical experiments are presented to demonstrate the accuracy and the potential of the proposed method.  ...  Furthermore, a geometry-based local error estimate is explored to guide the hierarchical subdivision and save the computation cost.  ...  Acknowledgment We would like to thank the anonymous reviewers and our laboratory group for helpful discussions and comments.  ...

### Assisted verification of elementary functions using Gappa

Florent de Dinechin, Christoph Quirin Lauter, Guillaume Melquiond
2006 Proceedings of the 2006 ACM symposium on Applied computing - SAC '06
The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function.  ...  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.  ...  This way of computing intervals was then fine for writing robust floating-point geometric predicates, but still not good enough for certify-1 http://lipforge.ens-lyon.fr/www/crlibm/ 2 http://lipforge.ens-lyon.fr  ...

### Analysis and algorithms for a regularized cauchy problem arising from a non-linear elliptic PDE for seismic velocity estimation

M.K. Cameron, S.B. Fomel, J.A. Sethian
2009 Journal of Computational Physics
(iv) the need to compute the solution only for a short interval of time.  ...  The key features of this scheme is the Lax-Friedrichs averaging and the wide stencil in space. The second approach is a spectral Chebyshev method with truncated series.  ...  Krasny and C. Morawetz for their attention, comments, and suggestions.  ...

### Determination of fractal dimensions for geometrical multifractals

Tamás Tél, Ágnes Fülöp, Tamás Vicsek
1989 Physica A: Statistical Mechanics and its Applications
Two independent approaches, the box counting and the sand box methods are used for the determinatiov of the generalized dimensions (Dq) associated with, the geometrical structure of growing deterministic  ...  Our study demonstrates that the above-mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres.  ...  In this "box counting" method the structure is covered by a grid with a mesh size equal to 1 and a box or lattice unit is considered as occupied if its intersection with the fractal is larger than zero  ...

### The Vasicek Model [chapter]

Simona Svoboda
2004 Interest Rate Modelling
And using a numerical method Euler Maruyama and computer simulation with the Maple software, for simulated data, gained averages and Standard deviations, confidence interval and their normal histogram  ...  Also, average of the solutions obtained from computer simulations is compared with real ones, and after analyzing and reviewing the results, performance of the Vasicek model will be measured, in interest  ...  We doing this job with the average of the solutions obtained on the computer simulations and compare it with real answers.  ...
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