Transformations of solution semantics of interval linear equations system. (English) Zbl 07903610
Summary: A solution \(x\) to a system of interval linear equations \(\boldsymbol{A}x = \boldsymbol{b}\) possesses its own semantics, which may involve combinations of tolerance, control, left- and right-localized semantics. In scenarios where the need arises to persist with a solution \(x\) despite changes in its semantics, corresponding adjustments in the interval information of the system become necessary. Our focus is on perturbing the original interval matrix \(\boldsymbol{A}\) to produce a transformed matrix \(\boldsymbol{A}^\prime\), ensuring that the solution \(x\) to \(\boldsymbol{A}^\prime x = \boldsymbol{b}\) aligns with the new semantics. We present a series of theorems using quadratic programming concepts to determine \(\boldsymbol{A}^\prime\) in a manner that closely approximates \(\boldsymbol{A}\). Several applications are provided to illustrate the practical utility of our approach.
MSC:
| 65F05 | Direct numerical methods for linear systems and matrix inversion |
| 15A06 | Linear equations (linear algebraic aspects) |
Keywords:
interval perturbation; interval linear equations system; tolerance-control-localized solution; weak solutionReferences:
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