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

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

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

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

An extended Hessenberg form for Hamiltonian matrices

A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the choice of Hamiltonian condensed forms, to be used within an extended Hamiltonian QR algorithm. Theoretical results identifying the structure of the extended Hamiltonian Hessenberg form and proofs of uniqueness of the reduction process are included. Numerical experiments confirm the validity of the approach

Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms

We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton\u27s iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence

On the Newton method for the matrix Pth root

Abstract. Stable versions of Newton’s iteration for computing the principal matrix pth root A1/p of an n × n matrix A are provided. In the case in which X0 is the identity matrix, it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real parts. Based on these results we provide a general algorithm for computing the principal pth root of any matrix A having no nonpositive real eigenvalues. The algorithm has quadratic convergence, is stable in a neighborhood of the solution, and has a cost of O(n3 log p) operations per step. Numerical experiments and comparisons are performed