Abstract
We characterize the boundary ∂Σp of the solution set Σp of a parametric linear system A(p)x=b(p) where the elements of the n×n matrix and the right-hand side vector depend on a number of parameters p varying within interval bounds. The characterization of ∂Σp is by means of pieces of parametric hypersurfaces, the latter represented by their coordinate functions depending on corresponding subsets of n-1 parameters. The presented approach has a direct application for efficient visualization of parametric solution sets by utilizing some plotting functions supported by Mathematica and Maple.
Similar content being viewed by others
References
J. Ackermann, A. Bartlett, D. Kesbauer, W. Sienel, and R. Steinhauser, Robust Control: Systems with Uncertain Physical Parameters, Springer, Berlin, 1994.
B. R. Barmish, New Tools for Robustness of Linear Systems, MacMillan, New York, 1994.
M. S. Baouendi, P. Ebenfelt, L. P. Rothschild, Real Submanifolds in Complex Space and Their Mappings, Princeton Univ. Press, Princeton, New Jersey, 1999.
G. Alefeld, V. Kreinovich, and G. Mayer, On the shape of the symmetric, persymmetric, and skew-symmetric solution set, SIAM J. Matrix Anal. Appl., 18(3) (1997), pp. 693–705.
G. Alefeld, V. Kreinovich, and G. Mayer, The shape of the solution set for systems of interval linear equations with dependent coefficients, Math. Nachr., 192 (1998), pp. 23–36.
G. Alefeld, V. Kreinovich, and G. Mayer, On symmetric solution sets, Comput. Suppl., 16 (2002), pp. 1–22.
G. Alefeld, V. Kreinovich, and G. Mayer, On the solution sets of particular classes of linear interval systems, J. Comput. Appl. Math., 152 (2003), pp. 1–15.
S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
G. E. Collins, Quantifier elimination by cylindrical algebraic decomposition – twenty years of progress, in B. F. Caviness and J. R. Johnson (eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer, New York, 1998, pp. 8–23.
A. Ganesan, S. Ross, B. Ross Barmish, An extreme point result for convexity, concavity and monotonicity of parametrized linear equation solutions, Linear Algebra Appl., 390 (2004), pp. 61–73.
M. Kofler, Maple: An Introduction and Reference, Addison-Wesley, Reading, MA, 1997.
W. Krämer, Computing and visualizing the exact solution set of systems of equations with interval coefficients, in 12th International Conference on Applications of Computer Algebra (ACA’06), June 26–29, 2006, Varna, Bulgaria.
M. Kraus, LiveGraphics3D: a Java applet which can display Mathematica graphics, http://www.vis.uni-stuttgart.de/∼kraus/LiveGraphics3D/.
R. L. Muhannah, R. L. Mullen (eds.), in Proceedings of the NSF Workshop on Reliable Engineering Computing, Svannah, Georgia USA, Feb. 22–24, 2006, http://www.gtsav.gatech.edu/workshop/rec06/proceedings.html.
W. Oettli, W. Prager, Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math., 6 (1964), pp. 405–409.
K. Okumura, An application of interval operations to electric network analysis, Bull. Japan Soc. Industr. Appl. Math., 2 (1993), pp. 115–127.
E. Popova, On the solution of parametrised linear systems, in W. Krämer, J. Wolff von Gudenberg (eds.), Scientific Computing, Validated Numerics, Interval Methods, Kluwer Acad. Publishers, Boston, Dordrecht, London, 2001, pp. 127–138.
E. Popova, Web-accessible tools for interval linear systems, Proc. Appl. Math. Mech. (PAMM), 5(1) (2005), pp. 713–714.
M. Spivak, A Comprehensive Introduction to Differential Geometry, four volumes, 2nd edn., Publish or Perish Inc., Berkeley, 1979.
S. Wolfram, The Mathematica Book, 4th edn., Wolfram Media/Cambridge U. Press, New York, 1999.
Wolfram Res. Inc., The Mathematica Player, 2007 (http://www.wolfram.com/products/player).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Lars Eldén.
AMS subject classification (2000)
15A06, 65G99, 65S05, 68U05
Electronic Supplementary Material
This file is unfortunately not in the Publisher's archive anymore: 3D Example 1 800KB
Rights and permissions
About this article
Cite this article
Popova, E., Krämer, W. Visualizing parametric solution sets. Bit Numer Math 48, 95–115 (2008). https://doi.org/10.1007/s10543-007-0159-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-007-0159-3