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We consider the Newton iteration for computing the principal matrix pth root, which is rarely used in the application for the bad convergence and the poor stability. We analyze the convergence conditions. In particular it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real part. Based on(More)
This concise and comprehensive treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is accessible to both practitioners and scholars. It is the first book in which nonsymmetric algebraic Riccati equations are treated in a clear and systematic way.(More)
Nonsymmetric algebraic Riccati equations for which the four coefficient matrices form an irreducible M-matrix M are considered. The emphasis is on the case where M is an irreducible singular M-matrix, which arises in the study of Markov models. The doubling algorithm is considered for finding the minimal nonnegative solution, the one of practical interest.(More)
Matrix fixed-point iterations z n+1 = ψ(zn) defined by a rational function ψ are considered. For these iterations a new proof is given that matrix convergence is essentially reduced to scalar convergence. It is shown that the principal Padé family of iterations for the matrix sign function and the matrix square root is a special case of a family of rational(More)
A special instance of the algebraic Riccati equation XCX − XE − AX + B = 0 where the n × n matrix coefficients A, B, C, E are rank structured matrices is considered. Relying on the structural properties of Cauchy-like matrices, an algorithm is designed for performing the customary Newton iteration in O(n 2) arithmetic operations (ops). The same technique is(More)
An algorithm for computing primary roots of a nonsingular matrix A is presented. In particular, it computes the principal root of a real matrix having no nonpositive real eigenvalues, using real arithmetic. The algorithm is based on the Schur decomposition of A and has an order of complexity lower than the customary Schur based algorithm, namely the Smith(More)
We survey 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's iteration is carried out in the cases of(More)
A new definition is introduced for the matrix geometric mean of a set of k positive definite n × n matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n 3 k 2) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando, Li and(More)