The geometric mean of two matrices from a computational viewpoint

The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different properties and representation of the geometric mean are discussed and analyzed and it is shown that most of them can be classified in terms of the rational approximations of the inverse square root functions. A review of the relevant applications is given

Palindromic linearization and numerical solution of nonsymmetric algebraic T-Riccati equations

We identify a relationship between the solutions of a nonsymmetric algebraic T Riccati equation (T -NARE) and the deflating subspaces of a palindromic matrix pencil, obtained by arranging the coefficients of the T -NARE. The interplay between T NAREs and palindromic pencils allows one to derive both theoretical properties of the solutions of the equation, and new methods for its numerical solution. In particular, we propose methods based on the (palindromic) QZ algorithm and the doubling algorithm, whose effectiveness is demonstrated by several numerical tests

Approximation and spectral analysis for large structured linear systems.

In this work we are interested in standard and less standard structured linear systems coming from applications in various _elds of computational mathematics and often modeled by integral and/or di_erential equations. Starting from classical Toeplitz and Circulant structures, we consider some extensions as g-Toeplitz and g-Circulants matrices appearing in several contexts in numerical analysis and applications. Then we consider special matrices arising from collocation methods for di_erential equations: also in this case, under suitable assumptions we observe a Toeplitz structure. More in detail we _rst propose a detailed study of singular values and eigenvalues of g-circulant matrices and then we provide an analysis of distribution of g-Toeplitz sequences. Furthermore, when possible, we consider Krylov space methods with special attention to the minimization of the computational work. When the involved dimensions are large, the Preconditioned Conjugate Gradient (PCG) method is recommended because of the much stronger robustness with respect to the propagation of errors. In that case, crucial issues are the convergence speed of this iterative solver, the use of special techniques (preconditioning, multilevel techniques) for accelerating the convergence, and a careful study of the spectral properties of such matrices. Finally, the use of radial basis functions allow of determining and studying the asymptotic behavior of the spectral radii of collocation matrices approximating elliptic boundary value problems