On best approximations of polynomials in matrices in the matrix 2-norm

We show that certain matrix approximation problems in the matrix 2-norm have uniquely defined solutions, despite the lack of strict convexity of the matrix 2-norm. The problems we consider are generalizations of the ideal Arnoldi and ideal GMRES approximation problems introduced by Greenbaum and Trefethen [SIAM J. Sci. Comput., 15 (1994), pp. 359–368]. We also discuss general characterizations of best approximation in the matrix 2-norm and provide an example showing that a known sufficient condition for uniqueness in these characterizations is not necessary

On Chebyshev polynomials of matrices

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of $p(A)$ over all monic polynomials $p(z)$ of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well-known properties of Chebyshev polynomials of compact sets in the complex plane. We also derive explicit formulas of the Chebyshev polynomials of certain classes of matrices, and explore the relation between Chebyshev polynomials of one of these matrix classes and Chebyshev polynomials of lemniscatic regions in the complex plane

The Faber–Manteuffel theorem for linear operators

A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber–Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352–362]. Motivated by the description by J. Liesen and Z. Strakoš, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber–Manteuffel theorem for matrices

Design Crane Jib by Mechanized Welding Technology

Import 05/08/2014Bakalářská práce se zabývá svařováním vnějších podélných svárů na vnitřním výložníku jeřábu Reachstackeru. V úvodu je popsáno využití Reachstackeru v praxi a postup svařování výložníku. Na základě charakteristiky současného stavu výroby, jsou navrženy změny, které jsou podmíněny dalším úpravám, potřebným pro výrobu. Dále je pro každou navrženou změnu proveden rozbor. Na závěr je proveden ekonomický rozbor navrhovaných změn, který je zaměřený na úsporu času a nákladům při svařování výložníku.The bachelore thesis adresses welding of longitudinal welds on the inner boom of the crane Reachstekar. The introduction describes the use of the Reachsteaker in pracrice and procedure for welding of the boom. Based on the characteristics of the current state of production, are proposed changes, which are dependent for following adaptation, necessary for output. Furthermore, for each proposed change was conducted analysis. In conclusion, I have conducted an economic analysis of proposed changes, which are primarily focused on saving of time and overhead costs.345 - Katedra mechanické technologievelmi dobř

Properties of worst-case GMRES

In the convergence analysis of the GMRES method for a given matrix $A$, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step $k$, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for $A$ and $k$. We show that the worst-case behavior of GMRES for the matrices $A$ and $A^T$ is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we prove that such vectors satisfy a certain “cross equality.” We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is strict at all iteration steps $k\geq 3$. Finally, we give a complete characterization of how the values of the approximation problems change in the context of worst-case and ideal GMRES for a real matrix, when one considers complex (rather than real) polynomials and initial vectors