Abstract
We discuss the use of interval arithmetic in the computation of the convex hull of n points in D dimensions. Convex hull algorithms rely on simple geometric tests, such as “does some point p lie in a certain half-space or affine subspace?” to determine the structure of the hull. These tests in turn can be carried out by solving appropriate (not necessarily square) linear systems. If interval-based methods are used for the solution of these systems then the correct hull can be obtained at lower cost than with exact arithmetic. In addition, interval-based methods can determine the topological structure of the convex hull even if the position of the points is not known exactly. In the present paper we compare various interval linear solvers with respect to their ability to handle close-to-pathological situations. This property determines how often interval arithmetic cannot provide the required information and therefore some computations must be redone with exact arithmetic.
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References
G. Alefeld and J. Herzberger, Introduction to Interval Computations (Academic Press, New York, 1983).
G. Alefeld and G. Mayer, The Cholesky method for interval data, Linear Algebra Appl. 194 (1993) 161–182.
C.B. Barber, D.P. Dobkin and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22(4) (1996) 469–483.
A. Deif, Singular values of an interval matrix, Linear Algebra Appl. 151 (1991) 125–133.
I.Z. Emiris, A complete implementation for computing general-dimensional convex hulls, Internat. J. Comput. Geom. Appl. 8(2) (1998) 223–253.
C. Jansson, Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in right-hand side, Computing 46(3) (1991) 265–274.
S. Krivsky, Verifizierte Berechnung konvexer Hüllen und Lösung linearer Gleichungssysteme, Diplomarbeit, Fachbereich Mathematik, Bergische Universität GH Wuppertal (1998).
S. Krivsky and B. Lang, Verified computation of higher-dimensional convex hulls and the solution of linear systems, Electron. J. Math. Comput. (2003) 21–35.
A. Neumaier, Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications (Cambridge Univ. Press, Cambridge, 1990).
J. O'Rourke, Computational Geometry in C, 2nd ed. (Cambridge Univ. Press, Cambridge, 1998).
F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, 2nd ed. (Springer, New York, 1988).
G. Rex and J. Rohn, Sufficient conditions for regularity and singularity of interval matrices, SIAM J. Matrix Anal. Appl. 20(2) (1999) 437–445.
J. Rohn, Enclosing solutions of overdetermined systems of linear interval equations, Reliable Comput. 2(2) (1996) 167–171.
S.M. Rump, Solving algebraic problems with high accuracy, In: A New Approach to Scientific Computation, eds. U.W. Kulisch and W.L. Miranker, Notes and Reports in Computer Science and Applied Mathematics (Academic Press, New York, 1983) pp. 51–120.
J.R. Shewchuk, Adaptive precision floating-point arithmetic and fast robust geometric predicates, Discrete Comput. Geom. 18(3) (1997) 305–363.
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Krivsky, S., Lang, B. Using Interval Arithmetic for Determining the Structure of Convex Hulls. Numerical Algorithms 37, 233–240 (2004). https://doi.org/10.1023/B:NUMA.0000049470.99748.8d
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DOI: https://doi.org/10.1023/B:NUMA.0000049470.99748.8d