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In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian R -conjugate matrices given by A 1 V B 1 + C 1 W D 1 = E 1 V ¯ F 1 + G 1 and A 2 V B 2 + C 2 W D 2 = E 2 V ¯ F 2 + G 2 is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian R -conjugate solution matrices V 1 , W 1 . Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations ∑ j = 1 q A i j X j B i j = F i , i = 1 , 2 , … , p . We prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors. Also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum-norm solution. [ABSTRACT FROM AUTHOR]

This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over generalized centro-symmetric matrices. When the considered matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial generalized centro-symmetric matrix group is chosen. In addition, for a given generalized centro-symmetric matrix group, the optimal approximation generalized centro-symmetric solution group can be obtained by finding the least Frobenius norm generalized centro-symmetric solution group of new coupled Sylvester-transpose matrix equations. Finally, a numerical example is given to demonstrate the efficiency of the introduced iterative algorithm. [ABSTRACT FROM AUTHOR]

We improve the usual relative error bound for the computation of x through iterated multiplications by x in binary floating-point arithmetic. The obtained error bound is only slightly better than the usual one, but it is simpler. We also discuss the more general problem of computing the product of n terms. [ABSTRACT FROM AUTHOR]

In the present paper, we propose an iterative algorithm for solving the generalized (P, Q)-reexive solution of the quaternion matrix equation Σul=1 AlXBl + Σvs=1 CsX̃Ds = F. By this iterative algorithm, the solvability of the problem can be determined automatically. When the matrix equation is consistent over a generalized (P, Q)-reexive matrix X, a generalized (P, Q)-reexive solution can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm generalized (P, Q)-reexive solution can be obtained by choosing an appropriate initial iterative matrix. Furthermore, the optimal approximate generalized (P, Q)-reexive solution to a given matrix X0 can be derived by finding the least Frobenius norm generalized (P, Q)-reexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: We present a roundoff error analysis of the method for solving the linear least squares problem with full column rank matrix A, using only factors Σ and V from the SVD decomposition of . This method (called here) is an analogue of the method of seminormal equations ( ), where the solution is computed from using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both and are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the method. We show that, under the condition , this method (called ) produces a forward stable solution, where denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only it is generally not forward stable, and has similar numerical properties to the corresponding method. Our forward error bounds for the are slightly better than for the since the terms are not present. We illustrate our analysis by numerical experiments. [Copyright &y& Elsevier]

An efficient iterative algorithm is presented to solve a system of linear matrix equations A1X1B1 + A2X2B2 = E, C1X1D1 + C2X2D2 = F with real matrices X1 and X2. By this iterative algorithm, the solvability of the system can be determined automatically. When the systemis consistent, for any initial matrices X01 and X02, a solution can be obtained in the absence of roundoff errors, and the least normsolution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions X1 and X2 to the given matrices X1 and X2 in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations A1X1B1 + A2X2B2 = Ē, C1X1D1 + C2X2D2 = F, where Ē = E - A1X1B1 - A2X2B2, F = F - C1X1D1 - C2X2D2. The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices A1, A2, B1, B2, C1, C2, D1, D2 are large, our algorithm is efficient as well. [ABSTRACT FROM AUTHOR]

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analyzed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps, the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps, the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a modified fixed point iterative algorithm to solve the fourth-order PDE model for image restoration problem. Compared with the standard fixed point algorithm, the proposed algorithm neednʼt to compute inverse matrices so that it can speed up the convergence and reduce the roundoff error. Furthermore, we prove the convergence of the proposed algorithm and give some experimental results to illustrate its effectiveness by comparing with the standard fixed point algorithm, the time marching algorithm and the split Bregman algorithm. [Copyright &y& Elsevier]

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: ... min over generalized reflexive matrix X. For any initial generalized reflexive matrix X1, by the iterative algorithm, the generalized reflexive solution X* can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution X* can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution ... to a given matrix X0 in Frobenius norm can be derived by finding the least-norm generalized reflexive solution ...* of a new corresponding minimum Frobenius norm residual problem: min ... with ... Finally, several numerical examples are given to illustrate that our iterative algorithm is effective [ABSTRACT FROM AUTHOR]

Abstract: An efficient iterative method for updating the mass, gyroscopic and stiffness matrices simultaneously using a few of complex measured modal data is developed. By using the proposed iterative method, the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices. Numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters. [Copyright &y& Elsevier]

Abstract: In this paper, two iterative algorithms are proposed to solve the linear matrix equations . When the matrix equations are consistent, by the first algorithm, a solution can be obtained within finite iterative steps in the absence of roundoff-error for any initial value, furthermore, the minimum-norm solution can be got by choosing a special kind of initial matrix. Additionally, the unique optimal approximation solution to a given matrix can be derived by finding the minimum-norm solution of a new matrix equations . When the matrix equations are inconsistent, we present the second algorithm to find the least-squares solution with the minimum-norm. Finally, two numerical examples are tested by MATLAB, the results show that these iterative algorithms are efficient. [Copyright &y& Elsevier]

We consider a new adaptive finite element (AFEM) algorithm for self-adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite-dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we present a novel improvement of Wilkinson's iterative refinement for the solution of linear system by using stability results of numerical solution for a dynamic system associated with the linear system. The convergence analysis is shown and roundoff error analysis is considered for this new refinement. Numerical experiments are accompanied to support our analysis. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this note, a technical error is pointed out in the proof of a lemma in the above paper. A correct proof of this lemma is given. In addition, a further result on the algorithm in the above paper is also given. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, two efficient iterative methods are presented to solve the symmetric and skew symmetric solutions of a linear matrix equation , respectively, with real pair matrices and . By these two iterative methods, the solvability of the symmetric and skew symmetric solutions for the matrix equation can be determined automatically. When the matrix equation has symmetric and skew symmetric solutions, then, for any initial pair matrices and , symmetric and skew symmetric solutions can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum norm of the symmetric and skew symmetric solutions can be obtained by choosing a special kind of initial pair matrices. In addition, the unique optimal approximation pair solution and to the given matrices and in Frobenius norm can be obtained by finding the minimum norm solution of a new matrix equation , where . The given numerical examples demonstrate that the iterative methods are quite efficient. [Copyright &y& Elsevier]

In this paper, an efficient iterative method is presented to solve the linear matrix equation [image omitted] (X) = E with real matrix X. By this iterative method, the solvability of the linear matrix equation can be determined automatically. When the matrix equation is consistent, then, for any initial matrix X0, a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. We also propose an iterative algorithm to obtain the solution or the least norm solution of the consistent matrix system. The given numerical examples demonstrate the efficiency of these two algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method is constructed to solve the following constrained linear matrix equations system: [A1(X),A2(X), ⋯ ,Ar (X)] = [E1, E2, ⋯ , Er ], X ϵ L = {X∣X = U (X)}, where Ai is a linear operator from Cm×n onto C pi×qi , Ei ϵ C pi×qi , i = 1, 2, ⋯ , r, and U is a linear self-conjugate involution operator. When the above constrained matrix equations system is consistent, for any initial matrix X0ϵ L, a solution can be obtained by the proposed iterative method in finite iteration steps in the absence of roundoff errors, and the least Frobenius norm solution can be derived when a special kind of initial matrix is chosen. Furthermore, the optimal approximation solution to a given matrix can be derived. Several numerical examples are given to show the efficiency of the presented iterative method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an oblique projection iterative method is presented to compute matrix equation of a square matrix with . By this iterative method, when taken the initial matrix , the group inverse can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation , a group inverse of matrix is got, then we use the formulae , the Drazin inverse can be obtained. [Copyright &y& Elsevier]

Abstract: A matrix is called a generalized reflection matrix if and . An matrix is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix if . In this paper, three iterative algorithms are proposed for solving the linear matrix equation over reflexive (anti-reflexive) matrices and . When this matrix equation is consistent over reflexive (anti-reflexive) matrices, for any reflexive (anti-reflexive) initial iterative matrices, the reflexive (anti-reflexive) solutions can be obtained within finite iterative steps in the absence of roundoff errors. By the proposed iterative algorithms, the least Frobenius norm reflexive (anti-reflexive) solutions can be derived when spacial initial reflexive (anti-reflexive) matrices are chosen. Furthermore, we also obtain the optimal approximation reflexive (anti-reflexive) solutions to the given reflexive (anti-reflexive) matrices in the solution set of the matrix equation. Finally, some numerical examples are presented to support the theoretical results of this paper. [Copyright &y& Elsevier]

Methods based on iterative integration as well as generalized averaging and special derivatives were proposed to smooth and identify the experimental trends. Examples of constructing the mathematical trend models were presented. [ABSTRACT FROM AUTHOR]

We consider a 4n2-algorithm for computing the Gohberg­Semencul representation of the inverse of a nonsymmetric Toeplitz matrix and for obtaining the solution of a Toeplitz system. The original algorithm is unstable when one or more of the leading principal submatrices are ill-conditioned. In our stabilized version we slightly perturb the matrix when we encounter a division by a small number. As a consequence, the solution is also perturbed, and its accuracy can be improved by taking a small number of iterative refinement steps. We present a roundoff error analysis of the new algorithm as well as numerical results that support our analysis. [ABSTRACT FROM AUTHOR]

We present a roundoff error analysis of a null space method for solving quadratic programming minimization problems. This method combines the use of a direct QR factorization of the constraints with an iterative solver on the corresponding null space. Numerical experiments are presented which give evidence of the good performances of the algorithm on sparse matrices. [ABSTRACT FROM AUTHOR]