510 Mathematik
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- Bernstein Basis (1)
- Bernstein polynomial (2)
- Cauchon algorithm (3)
- Checkerboard partial ordering (1)
- Complex interval (1)
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Institute
This thesis considers bounding functions for multivariate polynomials and rational functions over boxes and simplices. It also considers the synthesis of polynomial Lyapunov functions for obtaining the stability of control systems. Bounding the range of functions is an important issue in many areas of mathematics and its applications like global optimization, computer aided geometric design, robust control etc.
A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inclusion isotonicity of the related enclosure function is shown.
This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.