On preconditioning and solving an extended class of interval parametric linear systems
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On preconditioning and solving an extended class of interval parametric linear systems Iwona Skalna1 · Milan Hlad´ık2 Received: 9 March 2020 / Accepted: 22 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously, we want this enclosure to be tight and cheap to compute; unfortunately, these two objectives are conflicting. The review of the available literature shows that in order to make a system more tractable, most of the solution methods use left preconditioning of the system by the midpoint inverse. Surprisingly, and in contrast to standard interval linear systems, our investigations have shown that double preconditioning can be more efficient than a single one, both in terms of checking the regularity of the system matrix and enclosing the solution set, which is demonstrated by numerical examples. Consequently, right (which was hitherto mentioned in the context of checking regularity of interval parametric matrices) and double preconditioning together with the p-solution concept enable us to solve a larger class of interval parametric linear systems than most existing methods. The applicability of the proposed approach to solving interval parametric linear systems is illustrated by several numerical examples. Keywords Preconditioning · Revised affine forms · Interval parametric linear systems · Parametric solution Mathematics Subject Classification (2010) 15A06 · 15B99 · 65G40 · 68U99
Iwona Skalna
[email protected] Milan Hlad´ık [email protected] 1
AGH University of Science and Technology, Krak´ow, Poland
2
Department of Applied Mathematics, Charles University, Malostransk´e n´am. 25, 118 00, Prague, Czech Republic
Numerical Algorithms
1 Introduction Solving systems of parametric linear equations with parameters varying within prescribed intervals is an important part of many scientific and engineering computations [1–5]. The reason is that (parametric) linear systems are prevalence in virtually all areas of science and engineering, and uncertainty is a ubiquitous aspect of most real-world problems. Consider the following family of systems of parametric linear equations: {A(p)x = b(p), p ∈ p},
(1)
where A(p) ∈ Rn×n , b(p) ∈ Rn , and p is an ordered K-tuple of intervals (i.e., an interval vector). The entries of A(p) and b(p) are assumed, in general case, to be real-valued continuous functions1 of a vector of parameters p, i.e.: Aij , bi : p → R, i, j = 1, . . . , n. A particular form of (1) arises when there are affine-linear dependencies. This means that the entries of A(p) and b(p) depend linearly on p = (p1 , . . . , pK ), that is, A(p) and b(p) have, respectively, the following form: A(p) = A(0) +
K
A(k) pk ,
b(p) = b(0) +
k=1
K
b(k) pk .
k=1
The family (1) is often written as: A(p)x = b(p)
(2)
to underline its strong relationship with interval linea
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