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National Repository of Grey Literature 2 records found  Search took 0.01 seconds. 
Numerical range of an interval matrix
 Ivičič, Michal ; Hladík, Milan (advisor) ; Tichý, Petr (referee)
The numerical range of a matrix is a set of complex numbers that contains all the eigen- values of the matrix. It is used for instance to estimate a matrix norm. This thesis is about the numerical range of an interval matrix. In the theoretical part, we examine its properties. We prove for example that it is NP-hard to find out whether a given point lies in the numerical range. On an example, we show that field of values of an interval matrix is not necessarily convex. The thesis contains descriptions of two algorithms for visualization of the convex hull of the numerical range. Both of them are only suitable for matrices of small sizes due to high time complexity. Therefore we also present a polyno- mial algorithm for computing the upper bound of the numerical range. In the practical part, we implement the algorithms as functions in the Matlab language. 1
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Interval linear and nonlinear systems
 Horáček, Jaroslav ; Hladík, Milan (advisor) ; Garloff, Jürgen (referee) ; Ratschan, Stefan (referee)
First, basic aspects of interval analysis, roles of intervals and their applications are addressed. Then, various classes of interval matrices are described and their relations are depicted. This material forms a prelude to the unifying theme of the rest of the work - solving interval linear systems. Several methods for enclosing the solution set of square and overdetermined interval linear systems are covered and compared. For square systems the new shaving method is introduced, for overdetermined systems the new subsquares approach is introduced. Detecting unsolvability and solvability of such systems is discussed and several polynomial conditions are compared. Two strongest condi- tions are proved to be equivalent under certain assumption. Solving of interval linear systems is used to approach other problems in the rest of the work. Computing enclosures of determinants of interval matrices is addressed. NP- hardness of both relative and absolute approximation is proved. New method based on solving square interval linear systems and Cramer's rule is designed. Various classes of matrices with polynomially computable bounds on determinant are characterized. Solving of interval linear systems is also used to compute the least squares linear and nonlinear interval regression. It is then applied to real...
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