A Right-Preconditioning Process for the Formal–Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets
Aright-preconditioning process for linear interval systems has been presented by Neumaier in 1987. It allows the construction of an outer estimate of the united solution set of a square linear interval system in the form of a parallelepiped. The denomination “right-preconditioning” is used to describe the preconditioning processes which involve the matrix product AC in contrast to the (usual) left-preconditioning processes which involve the matrix product AC, where A and C are respectively the interval matrix of the studied linear interval system and the preconditioning matrix.
The present paper presents a new right-preconditioning process similar to the one presented by Neumaier in 1987 but in the more general context of the inner and outer estimations of linear AEsolution sets. Following the spirit of the formal-algebraic approach to AE-solution sets estimation, summarized by Shary in 2002, the new right-preconditioning process is presented in the form of two new auxiliary interval equations. Then, the resolution of these auxiliary interval equations leads to inner and outer estimates of AE-solution sets in the form of parallelepipeds. This right-preconditioning process has two advantages: on one hand, the parallelepipeds estimates are often more precise than the interval vectors estimates computed by Shary. On the other hand, in many situations, it simplifies the formal algebraic approach to inner estimation of AE-solution sets. Therefore, some AE-solution sets which were almost impossible to inner estimate with interval vectors, become simple to inner estimate using parallelepipeds. These benefits are supported by theoretical results and by some experimentations on academic examples of linear interval systems.
Unable to display preview. Download preview PDF.
- 1.Beaumont, O.: Algorithmique pour les Intervalles, Ph.D. Thesis, Université de Rennes 1, 1999.Google Scholar
- 2.Beaumont, O.: Solving Interval Linear Systems with Oblique Boxes, Publication interne IRISA, 2000, http://www.irisa.fr/sage/publications/Category/RR.html.
- 3.Dimitrova, N. S., Markov, S. M., and Popova, E. D.: Extended Interval Aritrhmetics: New Results and Applications, in: Computer Arithmetics and Enclosure Methods, 1992, pp. 225–232.Google Scholar
- 4.Eijgenraam, P.: The Solution of Initial Value Problems Using Interval Arithmetic, Mathematical Centre Tracts 144, Stichting Mathematisch Centrum, Amsterdam, 1981.Google Scholar
- 5.Kaucher, E.: Interval Analysis in the Extended Interval Space, Computing, Suppl. 2 (1980), pp. 33–49.Google Scholar
- 6.Kaucher, E.: Uber metrische und algebraische Eigenschaften einiger beim numerischen Rechnen auftretender Raume, Ph.D. Thesis, Karlsruhe, 1973.Google Scholar
- 7.Lohner, R. J.: Enclosing the Solutions of Ordinary Initial and Boundary Value Problems, in: Computer Arithmetic: Scientific Computation and Programming Languages, Wiley-Teubner Series in Computer Science, Stuttgart, 1987, pp. 255–286.Google Scholar
- 8.Markov, S. M., Popova, E. D., and Ullrich, Ch.: On the Solution of Linear Algebraic Equations Involving Interval Coefficients, Iterative Methods in Linear Algebra, IMACS Series on Computational and Applied Mathematics 3 (1996), pp. 216–225.Google Scholar
- 10.Neumaier, A.: Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
- 18.Shary, S. P.: Algebraic Approach to the Interval Linear Static Identification, Tolerance and Control Problems, or One More Application of Kaucher Arithmetic, Reliable Computing 2 (1) (1996), pp. 3–33.Google Scholar
- 20.SIGLA/X group: Modal Intervals, Reliable Computing 7 (2) (2001), pp. 77–111.Google Scholar