Quadratic Matrix Equation Research Articles

General exact solutions of the second-order homogeneous algebraic differential equations

The second-order homogeneous algebraic differential equation is studied, where are known such that is non-singular for some . A formula for the general exact solutions of the above equation is provided under the condition that there exists , such that . For a special second-order homogeneous algebraic differential equation coming from a physics model investigated by Bhat and Bernstein, a detailed description is given about how to construct a particular solution of the latter quadratic matrix equation.

On Using the Shcur Method for Solving Unilateral Quadratic Matrix Equation

On Using the Shcur Method for Solving Unilateral Quadratic Matrix Equation

Quadratic equations over matrix skew series

We introduce notions of skew and dual-skew series of a set of square matrices, and in these terms we formulate deflation conditions for general matrix quadratic equations.

THE BINOMIAL METHOD FOR A MATRIX SQUARE ROOT

There are various methods for evaluating a matrix square root, which is a solvent of the quadratic matrix equation <TEX>$X^2-A=0$</TEX>. We consider new iterative methods for solving matrix square roots of M-matrices. Particulary we show that the relaxed binomial iteration is more efficient than Newton-Schulz iteration in some cases. And we construct a formula to find relaxation coefficients through statistical experiments.

Perturbation analysis of a quadratic matrix equation associated with an [formula omitted]-matrix

This paper is devoted to the perturbation analysis for the quadratic matrix equation X2−EX−F=0, where E is a diagonal matrix and F is an M-matrix. The quadratic matrix equation of this type arises in noisy Wiener–Hopf problems for Markov chains. The solution of practical interest is a particular M-matrix solution. In this paper the perturbation bound of the M-matrix solution is presented, meanwhile the residual bound for an approximate solution to the M-matrix solution is obtained. The theoretical results are illustrated by using simple numerical examples.

Block tridiagonal reduction of perturbed normal and rank structured matrices

It is well known that if a matrix A∈Cn×n solves the matrix equation f(A,AH)=0, where f(x,y) is a linear bivariate polynomial, then A is normal; A and AH can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.

NEWTON'S METHOD FOR SOLVING A QUADRATIC MATRIX EQUATION WITH SPECIAL COEFFICIENT MATRICES

We consider the iterative solution of a quadratic matrix equation with special coefficient matrices which arises in the quasibirth and death problem. In this paper, we show that the elementwise minimal positive solvent of the quadratic matrix equations can be obtained using Newton's method if there exists a positive solvent and the convergence rate of the Newton iteration is quadratic if the Frchet derivative at the elementwise minimal positive solvent is nonsingular. Although the Frchet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.

Assisted Research of the Robots Kinematics and Dynamics Behavior with LabVIEW Instrumentation

In the industrial robots research one of the most important problem is to know cinematic and dynamic behavior to use them in the optimization process of the 3D space trajectory. Now, in the world, the results of the dynamic behavior research where obtained by research without one complex matrix model and without LabVIEW virtual instrumentation. In this paper it is show one assisted method and mathematical modeling with quadratic 6x6 matrix equations in the theoretical and experimental cinematic and dynamic analyze of didactical robot. For that where realized and used many virtual LabVIEW instruments (VI) and one didactical robot. All VI-s work on-line with the possibility to change some constructive and functional parameters of the structure or of the command low and see what is happened with the position, velocities, accelerations, forces and moments in all robot’s joints It is possible to try different control low of the movements like: simultaneously, successive or successive and simultaneously in all joints. The most important of the results is the possibility to know in every moment what is the better velocity characteristic for the dynamic behavior, validate easily the dynamic model and to know what is the variation of the absolute joint’s moments, forces, accelerations and velocities to can calculate all mechanical parts in each joints.

NEWTON'S METHOD FOR SYMMETRIC AND BISYMMETRIC SOLVENTS OF THE NONLINEAR MATRIX EQUATIONS

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by <TEX>$$Q(X)=AX^2+BX+C=0$$</TEX>, where X is a <TEX>$n{\times}n$</TEX> unknown real matrix, and A, B and C are <TEX>$n{\times}n$</TEX> given matrices with real elements. Another one is the matrix polynomial <TEX>$$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m=0,\;X,\;A_i{\in}\mathbb{R}^{n{\times}n}$$</TEX>. Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations Q(X) and P(X). The method does not depend on the singularity of the Fr<TEX>$\acute{e}$</TEX>chet derivative. Finally, we give some numerical examples.

THE CONDITION NUMBERS OF A QUADRATIC MATRIX EQUATION

In this paper we consider the quadratic matrix equation which can be defined by <TEX>$$Q(X)=AX^2+BX+C=0$$</TEX>, where X is a <TEX>$n{\times}n$</TEX> unknown complex matrix, and A, B and C are <TEX>$n{\times}n$</TEX> given matrices with complex elements. We first introduce a couple of condition numbers of the equation Q(X) and present normwise condition numbers. Finally, we compare the results and some numerical experiments are given.

Two kinds of condition numbers for the quadratic matrix equation

In the situation that the quadratic matrix equation has the dominant or minimal solvents, the mixed and componentwise condition numbers of the quadratic matrix equation are considered. The explicit approximate expressions of two kinds of condition numbers are derived in the uniform manner, and the upper bounds are proposed. The theoretical results are illustrated by using some simple numerical examples.

Structured canonical forms for products of (skew-) symmetric matrices and the matrix equation XAX=B

The contragredient transformation A bar right arrow P-1 AP-(inverted perpendicular) , B bar right arrow P-inverted perpendicular BP of two matrices A, B effects simultaneous similarity transformations of the products AB and BA. This work provides structured canonical forms under this transformation for symmetric or skew-symmetric A, B. As an application, these forms are used to study the quadratic matrix equation XAX = B, where both A, B are skew-symmetric or symmetric matrices. Necessary and sufficient conditions for the existence of a (nonsingular) symmetric solution X are formulated in terms of the structured canonical form.

FINDING THE SKEW-SYMMETRIC SOLVENT TO A QUADRATIC MATRIX EQUATION

In this paper we consider the quadratic matrix equation which can be defined be <TEX>$$Q(X)=AX^2+BX+C=0$$</TEX>, where X is a <TEX>$n{\times}n$</TEX> unknown real matrix; A,B and C are <TEX>$n{\times}n$</TEX> given matrices with real elements. Newton's method is considered to find the skew-symmetric solvent of the nonlinear matrix equations Q(X). We also show that the method converges the skew-symmetric solvent even if the Fr<TEX>$\acute{e}$</TEX>chet derivative is singular. Finally, we give some numerical examples.

Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

The analysis of priority queues where both the arrival and the service processes are correlated does not have a long literature. Only a few results are known, that attack the problem with the matrix geometric machinery. Unfortunately, these results have some restrictions that limit their usability significantly, for example they require the calculation of infinite series of matrices and infinite summations, that can be implemented only by truncation.The method presented in this paper calculates only the queue length moments, but without relying on infinite series of matrices and provides procedures to calculate the arising infinite sums accurately in an efficient way by means of linear equations, matrix-quadratic equations and a coupled matrix-quadratic equation.The numerical examples demonstrate that the presented method is several orders of magnitudes faster than the existing ones. From the large number of queue length moments it is possible to obtain lower and upper bounds for the queue length distribution by using existing moment based distribution estimation results.

On iterative methods for the quadratic matrix equation with M-matrix

In this paper, we consider Newton’s method and Bernoulli’s method for a quadratic matrix equation arising from an overdamped vibrating system. By introducing M-matrix to this equation, we provide a sufficient condition for the existence of the primary solution. Moreover, we show that Newton’s method and Bernoulli’s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition.

Quadratic vector equations

We study in a unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses, by seeing them as special cases of the general problem Mx=a+b(x,x), where a and the unknown x are componentwise nonnegative vectors, M is a nonsingular M-matrix, and b is a bilinear map from pairs of nonnegative vectors to nonnegative vectors. Specific cases of this equation have been studied extensively in the past by several authors, and include unilateral matrix equations from queuing problems (Bini, Latouche, and Meini, 2005 [7]), nonsymmetric algebraic Riccati equations (Guo and Laub, 2000 [14]), and quadratic matrix equations encountered in neutron transport theory (Lu, 2005 [23,24]).We present a unified approach which treats the common aspects of their theoretical properties and basic iterative solution algorithms. This has interesting consequences: in some cases, we are able to derive in full generality theorems and proofs appeared in literature only for special cases of the problem; this broader view highlights the role of hypotheses such as the strict positivity of the minimal solution. In an example, we adapt an algorithm derived for one equation of the class to another, with computational advantage with respect to the existing methods. We discuss possible research lines, including the relationship among Newton-type methods and the cyclic reduction algorithm for unilateral quadratic equations.

On Determination of Solution of Unilateral Quadratic Matrix Equation

On Determination of Solution of Unilateral Quadratic Matrix Equation

Approximations to quasi-birth-and-death processes with infinite blocks

The numerical analysis of quasi-birth-and-death processes rests on the resolution of a matrix-quadratic equation for which efficient algorithms are known when the matrices have finite order, that is, when the number of phases is finite. In this paper we consider the case of infinitely many phases from the point of view of theoretical convergence of truncation and augmentation schemes, and we develop four different methods. Two methods rely on forced transitions to the boundary. In one of these methods, the transitions occur as a result of the truncation itself, while in the other method, they are artificially introduced so that the augmentation may be chosen to be as natural as possible. Two other methods rely on forced transitions within the same level. We conclude with a brief numerical illustration.

Approximations to quasi-birth-and-death processes with infinite blocks

The numerical analysis of quasi-birth-and-death processes rests on the resolution of a matrix-quadratic equation for which efficient algorithms are known when the matrices have finite order, that is, when the number of phases is finite. In this paper we consider the case of infinitely many phases from the point of view of theoretical convergence of truncation and augmentation schemes, and we develop four different methods. Two methods rely on forced transitions to the boundary. In one of these methods, the transitions occur as a result of the truncation itself, while in the other method, they are artificially introduced so that the augmentation may be chosen to be as natural as possible. Two other methods rely on forced transitions within the same level. We conclude with a brief numerical illustration.

Transforming algebraic Riccati equations into unilateral quadratic matrix equations

The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX 2 + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.