Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive-definite solutions of the matrix equation X^s+A*X^?tA=Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q=I is discussed. The error estimations are found.     

Stability of linear infinite-dimensional systems revisited

The paper is devoted to a study of stability questions for linear infinite-dimensional discrete-time and continuous-time systems. The concepts of power stability and l ^p Instability for a linear discrete-time system x _k+1 = Ax _k (where x _k ε X, X is a Banach space, A is linear and bounded) are introduced and studied. Relationships between these concepts and the inequality r(A) < 1, where r(A) denotes the spectral radius of A, are also given. The discrete-time results are used for a simple derivation of some well-known properties of exponentially stable and L^p-stable linear continuous-time systems described by [xdot](t) = Ax(t) (A generates here a strongly continuous semigroup of linear and bounded operators on X). Some remarks on norms related to stable systems are also included.     

On Nonnegative matrices generating a finite multiplicative monoid

Square nonnegative matrices with the property that the multiplicative monoid M(A) generated by the matrix A is finite are characterized in several ways. At first, the least general upper bound for the cardinality of M(A) is derived. Then it is shown that any square nonnegative matrix is cogredient to a lower triangular block form with the diagonal consisting of three blocks L, A ₀, and M where L and M are strictly lower triangular, A ₀ has no zero rows or columns, and M(A) is finite if and only if. M(A ₀) is so. Several criteria for, M(A ₀) to be finite are presented. One of the normal forms of A applies very well to the characterization of the nonnegative solutions of each of the matrix equations X ^k = 0, X ^k = 1, X ^k = X, and X ^k = X ^T where k > 1 is an integer. It also leads to a polynomial time algorithm for deciding whether or not M(A) is finite, if the entries of A are nonnegative rationals.     

Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equation X+A T X −2 A=I

Necessary and sufficient conditions for the matrix equation X+A ^T X ^?2 A=I to have a real symmetric positive definite solution X are derived. Based on these conditions, some properties of the matrix A as well as relations between the solution X and A are derived.     

Convergence acceleration for the iterative solution of the equations X = AX + f

A method is presented for accelerating the convergent iterative procedures of solving the system of linear equations X = AX + f. The method is also applicable to divergent iterative schemes if the number of eigenvalues of A that are greater in absolute value than unity is not very large. The method is particularly advantageous if the matrix A has not been explicitly constructed because of extensive storage requirements and if it is not possible to use the alhorithms (such as the Chebyshev and Lanzcos polynomial methods) which are designed with respect to the position of eigenvalues of A in the complex plane.     

Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =?η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =?A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .     

New Finite Algorithm for Solving the Generalized Nonhomogeneous Yakubovich‐Transpose Matrix Equation

In this paper, the development of the conjugate direction (CD) method is constructed to solve the generalized nonhomogeneous Yakubovich‐transpose matrix equation AXB + CX^TD + EYF = R. We prove that the constructed method can obtain the (least Frobenius norm) solution pair (X,Y) of the generalized nonhomogeneous Yakubovich‐transpose matrix equation for any (special) initial matrix pair within a finite number of iterations in the absence of round‐off errors. Finally, two numerical examples show that the constructed method is more efficient than other similar iterative methods in practical computation.     

An algorithm for computing positive definite solutions of the nonlinear matrix equation X + A*X −1 A = I

In this paper, an algorithm for obtaining the Hermitian positive definite solutions of the nonlinear matrix equation X?+?A*X ^?1 A?=?I is considered and discussed. The convergence of the algorithm is proved. Numerical experiments to illustrate the behavior of the algorithm are executed.     

Quotient convergence and multi-splitting methods for solving singular linear equations

Abstract   In this paper, we use the group inverse to characterize the quotient convergence of an iterative method for solving consistent singular linear systems, when the matrix index equals one. Next, we show that for stationary splitting iterative methods, the convergence and the quotient convergence are equivalent, which was first proved in [7]. Lastly, we propose a (multi-)splitting iterative method A=F–G, where the splitting matrix F may be singular, endowed with group inverse, by using F ^# as a solution tool for any iteration. In this direction, sufficient conditions for the quotient convergence of these methods are given. Then, by using the equivalence between convergence and quotient convergence, the classical convergence of these methods is proved. These latter results generalize Cao’s result, which was given for nonsingular splitting matrices F. Keywords: Group inverse, singular linear equations, iterative method, P-regular splitting, Hermitian positive definite matrix, multi-splitting, quotient convergence AMS Classification: 15A09, 65F35     

Monotonicity of maximal solutions of algebraic Riccati equations

A monotonicity result for the maximal solution of the equation XBB^*X − A^*X − XA − Q = 0, Q = Q^*, (A, B) stabilizable, is proved.     

On generalized language equations

A system of generalized language equations over an alphabet A is a set of n equations in n variables: X_i = G_i(X₁,..., X_n), i = 1,...,n, where the G_i are functions from [P(A^*)]ⁿ into P(A^*), i=1,..., n, P(A^*) denoting the set of all languages over A. Furthermore the G_i are expressible in terms of set-operations, concatenations, and stars which involve the variable X_i as well as certain mixed languages. In this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations. Furthermore we show that a solution is regular if all fixed languages are regular.     

Positive expansiveness versus network dimension in symbolic dynamical systems

A symbolic dynamical system is a continuous transformation Φ:X?X of closed subset X⊆A^V, where A is a finite set and V is countable (examples include subshifts, odometers, cellular automata, and automaton networks). The function Φ induces a directed graph (‘network’) structure on V, whose geometry reveals information about the dynamical system (X,Φ). The dimensiondim(V) is an exponent describing the growth rate of balls in this network as a function of their radius. We show that, if X has positive entropy and dim(V)>1, and the system (A^V,X,Φ) satisfies minimal symmetry and mixing conditions, then (X,Φ) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Hölder-continuous.     

Granulation of a fuzzy set: Nonspecificity

Granulation (decomposition) of a fuzzy set A defined on a finite set of objects X is studied. Two types of decomposition are considered: external granulation determined by a given equivalence relation on X and internal granulation created by clusters of elements from X with similar membership grades in A. Axiomatic definitions of measures of granular nonspecificity and granular specificity are proposed. Some general approaches to the construction of measures of granular nonspecificity (specificity) are suggested. Relationship between granular nonspecificity, roughness and nonspecificity of a fuzzy set is discussed.     

Approximating the inverse of a matrix for use in iterative algorithms on vector processors

Most iterative techniques for solving the symmetric positive-definite systemAx=b involve approximating the matrixA by another symmetric positive-definite matrixM and then solving a system of the formMz=d at each iteration. On a vector machine such as the CDC-STAR-100, the solution of this new system can be very time consuming. If, however, an approximationM ^?1 can be given toA ^?1, the solutionz=M ^?1 d can be computed rapidly by matrix multiplication, a fast operation on the STAR. Approximations using the Neumann expansion of the inverse ofA give reasonable forms forM ^?1 and are presented. Computational results using the conjugate gradient method for the “5-point” matrixA are given.     

An iterative scheme for a geodynamo system

In this paper we propose an iterative scheme for a nonlinear spherical geodynamo model and carry out convergence analysis of this scheme. Our model is a coupled system of fluid velocity u , total kinematic pressure P and magnetic field B . They are governed by the Navier–Stokes equations and the dynamo equations. Using the mathematical induction method and energy estimates, it is concluded that our iterative scheme converges in the L ^∞(0, T; H ¹) sense and in the L ²(0, T; H ²) sense.     

Systolic networks for iterative methods for linear equations with cyclic reduction II

This paper presents systolic networks for the application of cyclic reduction to iterative methods for the solution of a linear system of equations A x=b where A is a p-cyclic matrix derived from multi-colouring ordered difference schemes on a regular mesh.     

Stability radii of differential algebraic equations with structured perturbations

This paper deals with a formula for computing stability radii of a differential algebraic equation of the form AX^′(t)−BX(t)=0, where A,B are constant matrices. A computable formula for the complex stability radius is given and a key difference between the ordinary differential equation (ODEs for short) and the differential algebraic equation (DAEs for short) is pointed out. A special case where the real stability radius and the complex one are equal is considered.     

Two iteration processes for computing positive definite solutions of the equation X - A*X−nA = Q

In the present paper, we suggest two iteration methods for obtaining positive definite solutions of nonlinear matrix equation X - A^*X⁻ⁿA = Q, for the integer n ≥ 1. We obtain sufficient conditions for existence of the solutions for the matrix equation. Finally, some numerical examples to illustrate the effectiveness of the algorithms and some remarks.     

Ein Iterationsverfahren,welches die optimale Intervall-Einschließung des Inversen eines M-Matrixintervalls liefert

Let beX the set of all inverse matricesA ^?1, whereA is contained in a given M-matrixinterval. Then using some properties of M-matrices it will be proved, that an interval version of the Schulz-method produces universally — i. e. without any restrictive condition for convergence — the best possible interval inclusion of the setX.     

A posteriori-Fehlerabschätzungen für die Pseudoinverse und die Lösung minimaler Länge

The pseudoinverseA ^I of a matrixA is characterized through two inA ^I linear equations and rank (A ^I )≤rank(A). A posteriori error bounds are developped for the derivation of an approximationX ofA ^I and the errors of the residuesAA ^I -AX andA ^I A-XA. The results are extended to the best least squares solution. A numerical example illustrates the technique.