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Visualizing parametric solution sets. (English) Zbl 1144.65031

The authors consider the parameter-dependent linear system
\[ A(p)x=b(p), \]
where the components of the \(n \times n\) matrix \(A\) and the \(n\)-vector \(b\) are affine-linear functions of a parameter vector \(p \in\mathbb R^k,\) which is required to lie in a vector interval \([p].\) In applications, the parameter ranges might either be real ranges from the model, or reflect error bounds, uncertainty etcetera.
The set of all solutions is called the parametric solution set and denoted by \(\Sigma^p\); its boundary is denoted by \(\partial \Sigma^p.\) The authors give a brief survey of the existing methods for characterizing \(\Sigma^p\) and \(\partial \Sigma^p.\) These methods are based on inequalities and some Mathematica tools exist to plot them; the authors claim that the quality of the produced solution set image has some deficiences. In their own approach, the authors do not use inequalities. They use pieces of parametric hypersurfaces to characterize \(\partial \Sigma^p.\) This is particularly useful for drawing \(\partial \Sigma^p.\)
The paper is illustrated by many examples and the authors discuss the use of Maple and Mathematica to draw the plots. The authors also provide electronic supplementary material, namely a pdf file with graphics and an active Mathematica notebook to allow to manipulate the graphics for one of the examples.

MSC:

65F30 Other matrix algorithms (MSC2010)
68W30 Symbolic computation and algebraic computation
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References:

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