. La-division-de-kaucher-x, y est dénie pour des intervalles généralisés x et y qui vérient 0 / ? (pro y) Elle peut être caractérisée par x/y = x × (1/y

A. Goldsztejn, A mean-value extension to generalized intervals
URL : https://hal.archives-ouvertes.fr/hal-00990048

A. Goldsztejn, Modal intervals revisited
URL : https://hal.archives-ouvertes.fr/hal-00294219

A. Goldsztejn, Correction of coercion to interpretability theorem in modal intervals theory, 2002.

A. Goldsztejn, Modal intervals revisited : a mean-value extension to generalized intervals, Proceedings of QCP-2005 (Quantication in Constraint Programming), 2005.
URL : https://hal.archives-ouvertes.fr/hal-00990048

A. Goldsztejn, A Right-Preconditioning Process for the Formal???Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets, Reliable Computing, vol.7, issue.2, p.443478, 2005.
DOI : 10.1007/s11155-005-0404-x

A. Neumaier, Tolerance Analysis with Interval Arithmetic. Freiburger Intervall- Berichte, p.519, 1986.

A. Neumaier, Overestimation in Linear Interval Equations, SIAM Journal on Numerical Analysis, vol.24, issue.1, p.207214, 1987.
DOI : 10.1137/0724017

A. Neumaier, Interval Methods for Systems of Equations, 1990.
DOI : 10.1017/CBO9780511526473

A. Neumaier, The wrapping eect, ellipsoid arithmetic, stability and condence regions, Computing Supplementum, vol.9, p.175190, 1993.

A. Neumaier, Taylor forms -use and limits, Reliable Computing, vol.9, p.4379, 2002.

E. Edited-by-hansen, Topics in Interval Analysis, 1969.

C. Bliek, Computer Methods for Design Automation, 1992.

D. Goldberg, What every computer scientist should know about oating-point arithmetic, Computing Surveys, 1991.

G. E. Collins, Quantier elimination by cylindrical algebraic decompositiontwenty years of progress, Quantier Elimination and Cylindrical Algebraic Decomposition, p.823, 1998.

E. Gardenyes, H. Mielgo, and A. Trepat, Modal intervals : Reason and ground semantics, Interval Mathematics, Lecture Notes in Computer Science, vol.212, p.2735, 1985.
DOI : 10.1007/3-540-16437-5_4

E. Hansen, On solving systems of equations using interval arithmetic, Mathematics of Computation, vol.22, issue.102, p.374384, 1968.
DOI : 10.1090/S0025-5718-1968-0229411-4

E. Hansen, Global Optimization Using Interval Analysis, Second Edition, Revised and Expanded, 1992.

E. Hansen and S. Sengupta, Bounding solutions of systems of equations using interval analysis, BIT, vol.17, issue.2, p.203211, 1981.
DOI : 10.1007/BF01933165

E. Kaucher, Uber metrische und algebraische Eigenschaften einiger beim numerischen Rechnen auftretender Raume, 1973.

E. Kaucher, Interval Analysis in the Extended Interval Space IR, Computing, issue.2, p.3349, 1980.
DOI : 10.1007/978-3-7091-8577-3_3

E. Loh and G. W. Walster, Rump's example revisited, Reliable Computing, vol.8, issue.3, pp.245-248, 2002.
DOI : 10.1023/A:1015569431383

E. D. Popova, Multiplication distributivity of proper and improper intervals, Reliable computing, vol.7, issue.2, p.129140, 2001.

E. R. Hansen, Bounding the Solution of Interval Linear Equations, SIAM Journal on Numerical Analysis, vol.29, issue.5, p.14931503, 1992.
DOI : 10.1137/0729086

R. B. Kearfott, Standardized notation in interval analysis, 2002.

E. W. Weisstein, Quantier elimination. From MathWorld A Wolfram Web Resource

F. Benhamou and F. Goualard, Universally quantied interval constraints, Principles and Practice of Constraint Programming -CP 2000, 2000.

F. Messine, Méthodes d'Optimisation Globale basées sur l'Analyse d'Intervalle pour la Résolution de Problèmes avec Contraintes, 1997.

F. N. Ris, Interval analysis and applications to linear algebra, 1972.

F. P. Preparata and M. I. Shamos, Computational geometry, an introduction, 1985.

G. Alefeld and J. Herzberger, Introduction to Interval Computations, Computer Science and Applied Mathematics, 1974.

G. Birkho and S. Maclane, A Survey of Modern Algebra. rev, 1953.

G. Mayer and I. Warnke, On the Fixed Points of the Interval Function f, Linear Algebra and its Applications, vol.363, p.201216, 2003.

J. Garlo, Convergent bounds for the range of multivariate polynomials, Interval Mathematics, p.3756, 1985.

G. H. Walster, The extended real interval system Available on the internet, 1998.

G. H. Walster and . Impty-intervali, Available on the internet, 1998.

S. Group, Modal intervals (basic tutorial) Applications of Interval Analysis to Systems and Control, Proceedings of MISC'99), p.157227, 1999.

H. Collavizza, F. Delobel, and M. Rueher, Comparing Partial Consistencies, Reliable Computing, vol.1, p.116, 1999.
DOI : 10.1007/978-94-017-1247-7_17

H. Collavizza, F. Delobel, and M. Rueher, Extending consistent domains of numeric csp, Proceedings of IJCAI-99, 1999.

H. Ratschek, Teilbarkeitskriterien der Intervallarithmetik Journal für die reine und angewandte Mathematik, Bd, p.128138, 1972.

H. Ortolf, Eine Verallgemeinerung der Intervallarithmetik, Geselschaft fuer Mathematik und Datenverarbeitung, p.171, 1969.

T. Hickey, Q. Ju, and M. H. Van-emden, Interval arithmetic: From principles to implementation, Journal of the ACM, vol.48, issue.5, p.10381068, 2001.
DOI : 10.1145/502102.502106

H. Hong and V. Stahl, Bernstein Form ist inklusionsmonoton, Computing, vol.28, issue.1, p.4353, 1995.
DOI : 10.1007/BF02238236

J. Armengol, J. Vehí, L. Travé-massuyès, and S. M. , Application of modal intervals to the generation of error-bounded envelopes, Reliable Computing, vol.7, issue.2, p.171185, 2001.

J. Davenport and J. Heintz, Real quantier elimination is doubly exponential, J. Symb. Comput, vol.5, p.2935, 1988.

J. Dixmier, Cours de mathématiques du premier cycle -1re année. Dunod, 1994.

J. Dixmier, Cours de mathématiques du premier cycle -2e année. Dunod, 1994.

J. Vehí, M. A. Sainz, J. Armengol, and I. Ferrer, Applications of modal interval analysis to systems and control, Current trends in Qualitative Reasoning and Applications, p.4964, 2000.

J. Wol-von-gudenberg, Determination of minimum sets of the set of zeros of a function. computing, p.203212, 1980.

J. M. Muller, Ordinateurs en quête d'arithmétique, La Recherche, vol.278, p.772777, 1995.

J. M. Muller, Ordinateurs en quête d'arithmétique. La Recherche, hors-série Août, p.9096, 1999.

W. Kühn, Rigorously Computed Orbits of Dynamical Systems Without the Wrapping Eect, Computing, vol.61, p.4767, 1998.

L. Kupriyanova, ???????????? ???????? ?????????????????????????? ?????????????????? ?????????????? ?????????????????????? ?????????????????? ???????????????????????????? ????????????, Reliable Computing, vol.4, issue.1, p.1531, 1995.
DOI : 10.1007/BF02390519

L. Jaulin, M. Kieer, O. Didrit, and E. Walter, Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics, 2001.
URL : https://hal.archives-ouvertes.fr/hal-00845131

M. Christie, Spécication de trajectoires de caméra sous contraintes, 2003.

M. Jirstand, Nonlinear control system design by quantier elimination, Journal of Symbolic Computation, vol.24, issue.2, p.137152, 1997.

M. Warmus, Calculus of Approximations, p.253257, 1956.

M. A. Sainz, E. Gardenyes, and L. Jorba, Formal Solution to Systems of Interval Linear or Non-Linear Equations, Reliable computing, vol.8, issue.3, p.189211, 2002.

M. A. Sainz, E. Gardenyes, and L. Jorba, Interval Estimations of Solutions Sets to Real-Valued Systems of Linear or Non-Linear Equations, Reliable computing, vol.8, issue.4, p.283305, 2002.

M. A. Sainz, J. Armengol, and J. Vehí, Fault detection and isolation of the three-tank system using the modal interval analysis, Journal of Process Control, vol.12, issue.2, pp.325-338, 2002.
DOI : 10.1016/S0959-1524(01)00033-6

S. Markov, On the algebra of intervals and convex bodies, J. UCS, vol.4, issue.1, pp.34-47, 1998.

K. Nickel, How to ght the wrapping eect, Proceedings of the International Symposium on interval mathematics on Interval mathematics 1985, p.121132, 1985.

N. S. Dimitrova, S. M. Markov, and P. E. , Extended interval aritrhmetics : New results and applications, Computer Arithmetics and Enclosure Methods, p.225232, 1992.

O. Beaumont, Algorithmique pour les Intervalles, 1999.

O. Beaumont, Solving interval linear systems with oblique boxes Publication interne IRISA, 2000.

O. Knueppel and . Profil, PROFIL/BIAS???Eine schnelle Intervallbibliothek, Computing, vol.5, issue.3-4, pp.277-287, 1994.
DOI : 10.1007/BF02307379

P. Herrero, M. A. Sainz, J. Vehí, and L. Jaulin, Quantied set inversion with applications to control, IEEE International Symposium on Computer Aided Control Systems Design, 2004.

P. Herrero, M. A. Sainz, J. Vehí, and L. Jaulin, Quantied set inversion algorithm with applications to control, Proceedings of Interval Mathematics and Constrained Propagation methods) of Reliable Computing, 2004.

P. Schodl, Multi-valued mappings and their zeros, 2005.

R. Krawczyk and A. Neumaier, Interval Slopes for Rational Functions and Associated Centered Forms, SIAM Journal on Numerical Analysis, vol.22, issue.3, p.604616, 1985.
DOI : 10.1137/0722037

S. Ratschan, Approximate quantied constraint solving by cylindrical box decomposition, Reliable Computing, vol.8, issue.1, p.2142, 2002.

S. Ratschan, Continuous rst-order constraint satisfaction with equality and disequality constraints, Proc. 8th International Conference on Principles and Practice of Constraint Programming, number 2470 in Lecture Notes in Computer Science, p.680685, 2002.

S. Ratschan, Solving existentially quantied constraints with one equality and arbitrarily many inequalities, Proceedings of the Ninth International Conference on Principles and Practice of Constraint Programming, number 2833 in Lecture Notes in Computer Science, p.615633, 2003.

R. B. Kearfott, Interval computation : Introduction, uses, and resources, Euromath, Bulletin, vol.2, issue.1, p.95112, 1996.

R. B. Kearfott, Einschlie??ungen nicht-glatter Funktionen f??r Codes zur globalen Optimierung und zur L??sung nichtlinearer Systeme, Computing, vol.20, issue.2, p.149162, 1996.
DOI : 10.1007/BF02276877

R. B. Kearfott, Rigorous Global Search : Continuous Problems, 1996.
DOI : 10.1007/978-1-4757-2495-0

R. B. Kearfott, M. Novoa, and C. Hu, A Review of Preconditioners for the Interval Gauss-Seidel Method, Interval Computations, vol.1, issue.1, p.5985, 1991.

R. G. Gelbaum and J. M. Olmsted, Counterexamples in Analysis, 2003.

S. Ratschan, Continuous rst-order constraint satisfaction, Proceedings of Articial Intelligence and Symbolic Computation, 2002.

S. Zuhe and M. A. Wolfe, On interval enclosures using slope arithmetic, Applied Mathematics and Computation, vol.39, issue.1, p.89105, 1990.
DOI : 10.1016/0096-3003(90)90124-L

S. P. Shary, On Controlled Solution set of Interval Algebraic Systems, Interval Computations, vol.4, issue.6, p.6675, 1992.

S. M. Markov, Extended interval arithmetic and some applications. Freiburger Intervall-Berichte, p.112, 1978.

S. M. Markov, Isomorphic embeddings of abstract interval systems, Reliable Computing, vol.3, 1997.

S. M. Markov, On the algebraic properties of convex bodies. Pliska Stud, Math. Bulgar, vol.12, p.119132, 1998.

S. M. Markov, E. D. Popova, and U. Ch, On the solution of linear algebraic equations involving interval coecients. Iterative Methods in Linear Algebra, IMACS Series on Computational and Applied Mathematics, vol.3, p.216225, 1996.

S. M. Rump, Algorithms for veried inclusions -theory and practice, Perspectives in Computing, vol.19, p.109126, 1988.
DOI : 10.1016/b978-0-12-505630-4.50012-2

URL : http://tubdok.tub.tuhh.de/bitstream/11420/318/1/Ru88a.pdf

S. P. Shary, Algebraic approach to the interval linear static identication, tolerance and control problems, or one more application of Kaucher arithmetic, Reliable computing, vol.2, p.333, 1996.

S. P. Shary, Algebraic Solutions to Interval Linear Equations and their Application, Proceedings of the IMACS -GAMM International Symposium on Numerical Methods and Error Bounds, 1995.

S. P. Shary, Algebraic approach in the "outer problem" for interval linear equations. Reliable computing, p.103135, 1997.

S. P. Shary, Interval Gauss-Seidel method for generalized solution sets to interval linear systems, Reliable computing, vol.7, p.141155, 2001.
DOI : 10.1007/978-94-017-1247-7_25

S. P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity. Reliable computing, p.321418, 2002.

T. Sunaga, Theory of an interval algebra and its application to numerical analysis, RAAG Memoirs, Ggujutsu Bunken Fukuy-kai, p.2946, 1958.
DOI : 10.1007/BF03186528

V. Ménissier-morain, Arbitrary precision real arithmetic: design and algorithms, The Journal of Logic and Algebraic Programming, vol.64, issue.1, 1996.
DOI : 10.1016/j.jlap.2004.07.003

V. Stahl, Interval Methods for Bounding the Range of Polynomials and Solving Systems of Nonlinear Equations, 1995.

V. I. Istratescu, Fixed point theory : an introduction, Mathematics and its applications . D. Reidel, 1981.
DOI : 10.1007/978-94-009-8177-5