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

The Pad\'e iterations for the matrix sign function and their reciprocals are optimal

It is proved that among the rational iterations locally converging with order s>1 to the sign function, the Pad\'e iterations and their reciprocals are the unique rationals with the lowest sum of the degrees of numerator and denominator

Solvability and uniqueness criteria for generalized Sylvester-type equations

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $\star$-Sylvester equations.Comment: This new version corrects some inaccuracies in corollaries 7 and

The palindromic cyclic reduction and related algorithms

The cyclic reduction algorithm is specialized to palindromic matrix polynomials and a complete analysis of applicability and convergence is provided. The resulting iteration is then related to other algorithms as the evaluation/interpolation at the roots of unity of a certain Laurent matrix polynomial, the trapezoidal rule for a certain integral and an algorithm based on the finite sections of a tridiagonal block Toeplitz matrix

Palindromic matrix polynomials, matrix functions and integral representations

AbstractWe study the properties of palindromic quadratic matrix polynomials φ(z)=P+Qz+Pz2, i.e., quadratic polynomials where the coefficients P and Q are square matrices, and where the constant and the leading coefficients are equal. We show that, for suitable choices of the matrix coefficients P and Q, it is possible to characterize by means of φ(z) well known matrix functions, namely the matrix square root, the matrix polar factor, the matrix sign and the geometric mean of two matrices. Finally we provide some integral representations of these matrix functions

[Budget impact analysis of empagliflozin for the treatment of type 2 diabetes in Italy]

BACKGROUND: Empagliflozin is the most recent molecule in the SGLT-2i class, new antidiabetic drugs that reduce renal glucose absorption by determining the excretion in the urine. The high prevalence of T2D, the chronicity of the condition and the severe economic and social burden caused by the disease, impose the need for a careful health economic assessment on each therapeutic innovation in this area.AIM: The aim of this study was to assess the budget impact of adopting empagliflozin in the diabetic population currently treated with sulfonylureas and potentially eligible for treatment with SGLT-2i.METHODS: The budget impact analysis was conducted from the perspective of the Italian National Health Service over a period of three years, through an analytic model developed in MS Excel. The target population was estimated in about 170,000 patients currently treated with sulfonylureas, based on the growth forecasts of the Italian population, epidemiological estimates and drug-use information available in the literature on diabetes in Italy. In the base case was assumed a replacement rate of sulfonylureas equal to 10%, 20% and 30% respectively at the first, second and third year of analysis. A scenario analysis was considered assuming a constant uptake of 100% since the first year. The following direct healthcare costs were considered: 1) acquisition of antidiabetic drugs as the main therapy and as rescue therapy; 2) self-monitoring of blood glucose; 3) management of severe hypoglycemic events and 4) management of major cardiovascular events. Clinical effectiveness data was based on the published literature and unit costs were derived from current prices and tariffs. Oneway sensitivity analysis was developed to assess the impact of input's uncertainty on the overall result.RESULTS: The base case analysis presented a substantially neutral impact on the budget. The 3-year cumulative impact was -454.337 €, corresponding to a 0.1% saving. This means that the replacement of sulphonylureas with empagliflozin determines an increase in acquisition costs of drugs, which is entirely offset by the reduction in costs of self-monitoring of blood glucose, management of hypoglycemic events and cardiovascular events. The scenario analysis, based on the assumption of a full substitution of sulphonylureas with empagliflozin at the first year, yielded a more enhanced savings. The cumulative impact was -2.269.517 €, corresponding to a 0.6% saving.CONCLUSIONS: The present study shows that the replacement of sulfonylureas (a class of generic products) with empagliflozin, motivated by the advantageous efficacy and safety profile, can take place optimizing healthcare expenditure for the management of DT2.[In Italian