introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

an accelerated technique for solving one type of discrete time algebraic riccati equations abstract algebraic riccati equations are encountered in many applications of control and engineering problems e g lqg problems and h control theory in this work we study the properties of one type of discrete time algebraic riccati equations our contribution is twofold first we present sufficient conditions for the existence of a unique positive definite solution second we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order numerical experiments strongly support that our approach performs extremely well even in the almost critical case as a byproduct we show that this method is capable of computing the unique negative definite solution once it exists

newton s method for solving a class of nonlinear matrix equations in this paper a class of nonlinear matrix equations x q a h i x c 1a is considered an equivalent form of this equation is derived and the properties of the solutions are studied then newton s iterative method is applied to solve this equivalent one finally two numerical examples are given to illustrate the efficiency of newton s method

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

some interlacing properties of the schur complement of a hermitian matrix abstract for a hermitian matrix h with nonsingular principal submatrix a it is shown that the eigenvalues of the moore penrose inverse of the schur complement h a of a in h interlace the eigenvalues of the moore penrose inverse of h moreover if h is semidefinite it is shown that the eigenvalues of h a interlace the eigenvalues of h

new convergence results on functional iteration techniques for the numerical solution of m g 1 type markov chains by performing an accurate analysis of the convergence we give a complete theoretical explanation of the experimental behaviour of functional iteration techniques for the computation of the minimal nonnegative solution g of the matrix equation x sum i 0 infty x ia i arising in the numerical solution of m g 1 type markov chains here the a i s are nonnegative k times k matrices such that the matrix sum i 0 infty a i is column stochastic moreover we introduce a general class of functional iteration methods which includes the standard methods and we give an optimality convergence result in this class

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

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

the extremal solution of the matrix equation x s a x q a i in this paper the distribution of the positive definite solution of the nonlinear equation x s a x qa i is deduced the existence of the maximal solution and the minimal solution is discussed and two new iterative algorithms for obtaining these solutions are proposed these algorithms avoid matrix inversion

an accurate asymptotic approximation and precise numerical solution of highly sensitive troesch s problem abstract troesch s problem is a two point boundary value problem that arises in the confinement of a plasma column by radiation pressure and in the theory of gas porous electrodes it has attracted significant amount of interest for the last 50 years current numerical solvers developed so far suffers significantly for large values of the sensitivity parameter here we present a new scheme that can be shown to outperform every alternative numerical evaluation procedure and can provide accurate solutions for arbitrarily large values of the parameter our solution based on derivation of a new asymptotic form where the difference of solution and this approximation decays exponentially for moderate values of the parameter further we use a discontinuous galerkin method to calculate this difference we include a variety of numerical results that confirm that indeed our algorithm compares favorably with alternative methods

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

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

a note on the fixed point iteration for the matrix equations x a x 1a i the fixed point iteration is a simple method for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a i the plus minus equations the convergence of this method may be very slow if the initial matrix is not chosen carefully a strategy for choosing better initial matrices has been recently proposed by ivanov hasanov and uhlig they proved that this strategy can improve the convergence in general and observed from numerical experiments that dramatic improvement happens for the plus equation with some matrices a it turns out that the matrices a are normal for those examples in this note we prove a result that explains the dramatic improvement in convergence for normal and thus nearly normal matrices for the plus equation a similar result is also proved for the minus equation ams classification 15a24 65f30 65h10

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

a posteriori error estimates for a mixed fem formulation of a non linear elliptic problem abstract we consider the numerical solution via the mixed finite element method of a non linear elliptic partial differential equation in divergence form with dirichlet boundary conditions besides the temperature u and the flux we introduce u as a further unknown which yields a variational formulation with a twofold saddle point structure we derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem which does not need any equilibrium property for its solvability in addition for specific finite element subspaces of raviart thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems our approach does not need the exact finite element solution but any reasonable approximation of it such as for instance the one obtained with a fully discrete galerkin scheme in particular we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi linear forms involved finally several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions

numerical solution of a quadratic eigenvalue problem we consider the quadratic eigenvalue problem qep 2 m g k x 0 where m m t is positive denite k k t is negative denite and g g t the eigenvalues of the qep occur in quadruplets or in real or purely imaginary pairs we show that all eigenvalues of the qep can be found eciently and with the correct symmetry by nding a proper solvent x of the matrix equation mx 2 gx k 0 as long as the qep has no eigenvalues on the imaginary axis this solvent approach works well also for some cases where the qep has eigenvalues on the imaginary axis ams classication 15a18 15a24 65f30

on the tripling algorithm for large scale nonlinear matrix equations with low rank structure for the large scale nonlinear matrix equations with low rank structure the well developed doubling algorithm in low rank form da lr is known as an efficient method to compute the stabilizing solution by further analyzing the global efficiency index constructed in this paper we propose a tripling algorithm in low rank form ta lr from two points of view the cyclic reduction and the symplectic structure preservation the new presented algorithm shares the same pre processing complexity with that of da lr but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset under the solvability condition the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order numerical experiments including some from nano research show that the ta lr is highly efficient to compute the stabilizing solution of large scale nonlinear matrix equations with low rank structure

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

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

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

two structure preserving doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation in this paper we present two structure preserving doubling like algorithms for obtaining the positive definite solution of the nonlinear matrix equation x a h x 1 a q where x c n i n is an unknown matrix and q c n i n is a hermitian positive definite matrix we prove that the sequences generated by the algorithms converge to the positive definite solution of the considered matrix equation r quadratically in addition we also present some numerical results to illustrate the behavior of the considered algorithm

solution of a class of nonlinear matrix equations abstract in this paper we solve nonlinear matrix equations of the form x q i 1 p a i f i x a i r i and x q i 1 p a i f i x a i r i j 1 q b j g j x b j q j where 1 1 r i q j 1 1 q p n the collection of all n n hermitian positive definite matrices and a i b j s are n n matrices also f i g j s are monotone mappings from p n into p n examples are given to illustrate that the equations can not be solved by previously known theorems

solving large scale nonlinear matrix equations by doubling we consider the solution of the large scale nonlinear matrix equation x bx 1a q 0 with a b q x cn n and in some applications b a or h the matrix q is assumed to be nonsingular and sparse with its structure allowing the solution of the corresponding linear system qv r in o n computational complexity furthermore b and a are respectively of ranks ra rb n the type 2 structure preserving doubling algorithm by lin and xu 2006 24 is adapted with the appropriate applications of the sherman morrison woodbury formula and the low rank updates of various iterates two resulting large scale doubling algorithms have an o ra rb 3 computational complexity per iteration after some pre processing of data in o n computational complexity and memory requirement and converge quadratically these are illustrated by the numerical examples

on the hermitian positive definite solution of the nonlinear matrix equation x i 1mai x 1ai i abstract in this paper we study the matrix equation x i 1 m a i x 1 a i i where ai i 1 2 m is square matrices and obtain some conditions for the existence of positive definite solution of this equation two iterative algorithms to find the maximal positive definite solution are given some numerical results are reported to illustrate the effectiveness of the algorithms

the inverse mean problem of geometric mean and contraharmonic means abstract in this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices proposed in w n anderson m e mays t d morley g e trapp the contraharmonic mean of hsd matrices siam j algebra disc meth 8 1987 674 682 a x y 2 x 1 y 1 1 b x y by proving its equivalence to the well known nonlinear matrix equation x t bx 1b where t 1 2 a a a 8 ba 1 b is the unique positive definite solution of x a 2bx 1b the inverse mean problem is solvable if and only if b a

complex symmetric stabilizing solution of the matrix equation x a x 1a q abstract we study the matrix equation x a x 1 a q where a is a complex square matrix and q is complex symmetric special cases of this equation appear in green s function calculation in nano research and also in the vibration analysis of fast trains in those applications the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators the stabilizing solution is the solution of practical interest in this paper we provide an elementary proof of the existence for the general matrix equation under an assumption that is satisfied for the two special applications moreover our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part the unique stabilizing solution can be computed efficiently by the doubling algorithm