on the matrix equation x a x na i the nonlinear matrix equation x a x na i is discussed n is a positive integer convergence of the iterative methods proposed by el sayed comput math appl 41 2001 579 588 is proved under weaker restrictions for the matrix a and an arbitrary integer n

positive definite solutions of the matrix equations x a x qa q abstract in this paper we investigate nonlinear matrix equations x a x q a q and x a x q a q where q 0 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x q a q to have two different positive definite solutions and a sufficient condition for the equation x a x q a q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions for these equations

comments on stability margin evaluation for uncertain linear systems the bound derived in the above paper by gong thompson ibid vol 39 no 3 p 548 50 1994 is shown via two new stability criteria not to be the tightest possible for all unstructured perturbations a minor error of their theorem is also corrected

on the matrix equation x a tx 12ma i in this paper we study iterative methods for finding the extremal positive definite solutions of the matrix equation x a tx 1ma i first a condition on the existence of a positive definite solution of this matrix equation is given then the existence as well as the rate of convergence of some proposed algorithms to obtain the extremal positive definite solutions of this equation is presented moreover a generalization of computationally simple and efficient known algorithm is applied for obtaining the extremal positive definite solutions in addition both the necessary and sufficient conditions for this matrix equation to have positive definite solution are presented numerical examples are given to illustrate the performance and the effectiveness of the algorithms

on the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous saint venant beam theory abstract the relationship between twist and shear centres in an orthotropic s aint v enant beam with fiberwise homogeneous elastic moduli and constant poisson ratios is investigated arbitrary cross sections are considered as a new result the relative position of these points is expressed in terms of the scalar potential whose gradient is the rotated field of twist tangential stresses its evaluation requires the solution of n 1 boundary value problems being n 0 the number of holes in the cross section in an isotropic and homogeneous beam the potential is p randtl stress function and known formulae providing the relative position of twist and shear centres are recovered explicit expressions of sliding torsional compliance blocks for t imoshenko beams defined by an energy condition of equivalence with the orthotropic and fiberwise homogeneous s aint v enant theory are provided coincidence of twist centre and t imoshenko shear centre is proven numerical computations on homogeneous and composite orthotropic l sections are performed

iterative methods for nonlinear matrix equations x a x αa i abstract the present paper treats iterative methods for a class of nonlinear matrix equations x a x a i with two cases 1 and 0 a is nonsingular we will derive the necessary and sufficient conditions for the existence of positive definite solutions

perturbation analysis of the matrix equation x a x pa q abstract consider the nonlinear matrix equation x a x p a q with 0 p 1 this paper shows that there exists a unique positive definite solution to the equation a perturbation bound and the backward error of an approximate solution to this solution is evaluated we also obtain explicit expressions of the condition number for the unique positive definite solution the theoretical results are illustrated by numerical examples

improved perturbation estimates for the matrix equations x a x 1a q abstract we give new and improved perturbation estimates for the solution of the matrix quadratic equations x a x 1 a q some of the estimates depend and some do not depend on knowledge of the exact solution x these bounds are compared numerically against other known bounds from the literature

on the positive definite solutions of nonlinear matrix equation x a x δa q abstract in this paper we consider the positive definite solutions of nonlinear matrix equation x a x a q where 0 1 which appears for the first time in s m el sayed a c m ran on an iteration methods for solving a class of nonlinear matrix equations siam j matrix anal appl 23 2001 632 645 the necessary and sufficient conditions for the existence of a solution are derived an iterative algorithm for obtaining the positive definite solutions of the equation is discussed the error estimations are found

on the positive definite solutions of the matrix equations xs atx ta in abstract the two matrix equations x s a t x t a i n and x s a t x t a i n are studied based on the fixed point theory the existence of the symmetric positive definite solutions are proved sensitivity analysis of the maximal solution is presented some elegant estimates of the positive definite solutions are obtained three iterative methods for computing the positive solutions are proposed

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

a new inversion free method for a rational matrix equation abstract motivated by the classical newton schulz method for finding the inverse of a nonsingular matrix we develop a new inversion free method for obtaining the minimal hermitian positive definite solution of the matrix rational equation x a x 1 a i where i is the identity matrix and a is a given nonsingular matrix we present convergence results and discuss stability properties when the method starts from the available matrix aa we also present numerical results to compare our proposal with some previously developed inversion free techniques for solving the same rational matrix equation

on the nonlinear matrix equation x i 1mai xδiai q abstract based on fixed point theorems for monotone and mixed monotone operators in a normal cone we prove that the nonlinear matrix equation x i 1 m a i x i a i q 0 i 1 always has a unique positive definite solution a conjecture which is proposed in x g liu h gao on the positive definite solutions of the matrix equation x s a t x t a i n linear algebra appl 368 2003 83 97 is solved multi step stationary iterative method is proposed to compute the unique positive definite solution numerical examples show that this iterative method is feasible and effective

on two perturbation estimates of the extreme solutions to the equations x a x 1a q abstract two perturbation estimates for maximal positive definite solutions of equations x a x 1 a q and x a x 1 a q are considered these estimates are proved in hasanov et al improved perturbation estimates for the matrix equations x a x 1 a q linear algebra appl 379 2004 113 135 we derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations the theoretical results are illustrated by numerical examples

on the nonlinear matrix equation x a f x a q solutions and perturbation theory abstract in this paper the nonlinear matrix equation x a f x a q is discussed sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived also conditions are given under which the solution depends continuously on the matrices a and q

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

stability margin evaluation for uncertain linear systems a sufficient stability condition for parameter variation in a perturbed continuous time multivariable linear system represented by a state space model is presented starting with the existence of an algebraic riccati equation a stability bound is derived from the polar decomposition of the nominal system matrix unlike previous work the results are not dependent on the solution of the lyapunov equation and consequently not a function of an arbitrarily selected positive definite matrix in addition the bound would appear to be the tightest possible in that violation of the presented inequality can be shown to lead to instability

on positive definite solutions of the family of matrix equations x a x n a q the nonlinear matrix equation x a x na q and properties of its positive definite solutions are studied sufficient conditions for existence the minimal xs and special x1 positive definite solutions are derived and iterative procedures for computing these solution are discussed

improved methods and starting values to solve the matrix equations x a x 1a i iteratively the two matrix iterations x k 1 i a x 1 k a are known to converge linearly to a positive definite solution of the matrix equations x a x 1 a i respectively for known choices of x 0 and under certain restrictions on a the convergence for previously suggested starting matrices x 0 is generally very slow this paper explores different initial choices of x 0 in both iterations that depend on the extreme singular values of a and lead to much more rapid convergence further the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use newton s method in part

contractive maps on normed linear spaces and their applications to nonlinear matrix equations in this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space these conditions are already well known for maps on intervals in r using the conditions and banach s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space the fixed point theorem will be applied to the matrix equation x in a f x a where f is a map on the set of positive definite matrices induced by a real valued map on 0 this will give conditions on a and f under which the equation has a unique solution in a certain set we will consider two examples of f in detail in one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on a

notes on two perturbation estimates of the extreme solutions to the equations x a x 1 a q two perturbation estimates of the maximal positive definite solutions to the matrix equations x a x 1a q and x a x 1a q are considered these estimates are like to the estimates discussed by hasanov and ivanov 8 the conditions x l 1a 2 1 and x 1a 2 1 in 8 are not always satisfied we replace this conditions by px l 1ap 1 2 1 and px 1ap email protected 2 1 respective where p is positive definite matrix the theoretical results are illustrated by numerical examples

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

perturbation analysis of the maximal solution of the matrix equation x a x 1a p consider the nonlinear matrix equation x a x 1 a p where a p are n x n complex matrices with p hermitian positive definite and a denotes the conjugate transpose of a matrix a in this pape

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

positive definite solutions of the matrix equations x a x q a q q 1 in this paper we investigate nonlinear matrix equations x a x qa q where q 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x qa q to have two different positive definite solutions and several sufficient conditions for the equation x a x qa q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions of these equations numerical examples are given to illustrate the effectiveness of the methods

on hermitian positive definite solutions of matrix equation x a x 2a i abstract the hermitian positive definite solutions of the matrix equation x a x 2 a i are studied a necessary and sufficient condition for existence of solutions is given in case a is normal the basic fixed point iterations for the equation in case a is nonnormal with a 2 3 3 are discussed in some detail some of ivanov s hasanov s and minchev s results in linear algebra appl 326 2001 27 are improved