Absence Of Round-off Errors Research Articles

Iterative algorithms for solving some tensor equations

ABSTRACTWe propose a class of iterative algorithms to solve some tensor equations via Einstein product. These algorithms use tensor computations with no matricizations involved. For any (special) initial tensor, a solution (the minimal Frobenius norm solution) of related problems can be obtained within finite iteration steps in the absence of roundoff errors. Numerical examples are provided to confirm the theoretical results, which demonstrate that this kind of iterative methods are effective and feasible for solving some tensor equations.

Biconjugate residual algorithm for solving general Sylvester-transpose matrix equations

The present paper is concerned with the solution of the coupled generalized Sylvester-transpose matrix equations {A1XB1 + C1XD1 + E1XTF1 = M1, A2XB2 + C2XD2 + E2XTF2 = M2, including the well-known Lyapunov and Sylvester matrix equations. Based on a variant of biconjugate residual (BCR) algorithm, we construct and analyze an efficient algorithm to find the (least Frobenius norm) solution of the general Sylvester-transpose matrix equations within a finite number of iterations in the absence of round-off errors. Two numerical examples are given to examine the performance of the constructed algorithm.

An iterative algorithm for solving the generalized Sylvester-conjugate matrix equation

This paper aims to extend the conjugate gradient least squares method to solve the least squares problem of the generalized Sylvester-conjugate matrix equation. For any initial values, the proposed iterative method can obtain the least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are given to show the efficiency of the presented iterative method. And it’s also proved that our proposed iterative algorithm is better than the existing LSQR iterative algorithm.

Convergence properties of BCR method for generalized Sylvester matrix equation over generalized reflexive and anti-reflexive matrices

ABSTRACTSolving the well-known Lyapunov and Sylvester matrix equations appears in a wide range of applications such as in control theory and signal processing. This article establishes the matrix form of the biconjugate residual (BCR) algorithm for computing the generalized reflexive solution X and the generalized anti-reflexive solution Y of the generalized Sylvester matrix equation It is proven that the proposed BCR algorithm converges within a finite number of iterations in the absence of round-off errors. At the end, various numerical implementations illustrating the effectiveness and accuracy of the proposed BCR algorithm are presented and discussed.

An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations

The iterative method of the generalized coupled Sylvester-conjugate matrix equations $\sum\limits _{j=1}^{l}\left (A_{ij}X_{j}B_{ij}+C_{ij}\overline {X}_{j}D_{ij}\right )=E_{i} (i=1,2,\cdots ,s)$ over Hermitian and generalized skew Hamiltonian solution is presented. When these systems of matrix equations are consistent, for arbitrary initial Hermitian and generalized skew Hamiltonian matrices X j (1), j = 1,2,⋯ , l, the exact solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm Hermitian and generalized skew Hamiltonian solution of the problem. Finally, numerical examples are presented to demonstrate the proposed algorithm is efficient.

An efficient iterative updating method for hysteretic damping models

Finite element model updating techniques are used to update the finite element model of a structure in order to improve its correlation with the experimental dynamic test data. This paper presents an efficient iterative method for finite element matrix updating problem in a hysteretic damping model based on a few of complex measured vibration modal data. 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 matrix triple. Some theorems are stated and proved, 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.

Generalized conjugate direction method for solving a class of generalized coupled Sylvester-conjugate transpose matrix equations over generalized Hamiltonian matrices

In this paper, a generalized conjugate direction (GCD) method for finding the generalized Hamiltonian solutions of a class of generalized coupled Sylvester-conjugate transpose matrix equations is proposed. Furthermore, it is proved that the algorithm can compute the least Frobenius norm generalized Hamiltonian solution group of the problem by choosing a special initial matrix group within a finite number of iterations in the absence of round-off errors. Numerical examples are also presented to illustrate the efficiency of the algorithm.

BCR method for solving generalized coupled Sylvester equations over centrosymmetric or anti-centrosymmetric matrix

This paper introduces another version of biconjugate residual method (BCR) for solving the generalized coupled Sylvester matrix equations over centrosymmetric or anti-centrosymmetric matrix. We prove this version of BCR algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round-off errors. Furthermore, a method is provided for choosing the initial matrices to obtain the least norm solution of the problem. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

On the convergence of conjugate direction algorithm for solving coupled Sylvester matrix equations

In this work, we obtain the matrix form of the conjugate direction (MCD) method for solving the coupled Sylvester matrix equations (CSMEs)\(\left\{ \begin{array}{ll} \sum _{i=1}^s A_iXB_i=C, &{} \hbox {} \\ \sum _{j=1}^t D_jXE_j=F, &{} \hbox {} \end{array}\right. \)which are defined in the domain of real numbers. We prove that the MCD method converges to the solution of the CSMEs for any initial guess within a finite number of iterations in the absence of round-off errors. Also we show that the MCD method can find the least Frobenius norm solution of the CSMEs with special initial guess. Finally three numerical examples show that the MCD method is efficient to solve some matrix equations.

Symmetric least squares solution of a class of Sylvester matrix equations via MINIRES algorithm

The purpose of this paper is deriving the minimal residual (MINIRES) algorithm for finding the symmetric least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the symmetric least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares symmetric solution of the problem. Finally, we give some numerical examples to illustrate the performance of MINIRES algorithm.

Convergence of HS version of BCR algorithm to solve the generalized Sylvester matrix equation over generalized reflexive matrices

In recent years, several linear matrix equations such as Lyapunov and Sylvester matrix equations have received considerable attention due to their important applications in engineering and applied mathematics. In this work, an iterative technique based on the Hestenes–Stiefel (HS) version of biconjugate residual (BCR) algorithm is introduced to solve the generalized Sylvester matrix equation∑i=1f(AiXBi)+∑j=1g(CjYDj)=E,over the generalized reflexive matrices X and Y. We show that the proposed iterative technique converges to the generalized reflexive solutions within a finite number of iterations in the absence of round-off errors. Numerical examples confirm the efficiency and accuracy of the proposed iterative technique and also comparisons with other algorithms are made.

Convergence analysis of the MCGNR algorithm for the least squares solution group of discrete-time periodic coupled matrix equations

The conjugate gradient normal equations residual minimizing (CGNR) algorithm is a popular tool for solving large nonsymmetric linear systems. In this study, we propose the matrix form of the CGNR (MCGNR) algorithm to find the least squares solution group of the discrete-time periodic coupled matrix equations [Formula: see text] with periodic matrices. We prove that the MCGNR algorithm converges in a finite number of steps in the absence of round-off errors, that is, this algorithm has the finite termination property. Also we show that the norms of the residual matrices of the MCGNR algorithm decrease monotonically during its iteration, that is, this algorithm has the residual reducing property. To show the efficiency of the MCGNR algorithm, two numerical examples are presented.

Lanczos version of BCR algorithm for solving the generalised second‐order Sylvester matrix equation

It is well known that the biconjugate residual (BCR) algorithm and its variants are powerful procedures to find the solution of large sparse non-symmetric systems equation A x = b . In this study, the authors develop the Lanczos version of BCR algorithm for computing the solution pair ( V , W ) of the generalised second-order Sylvester matrix equation E V F + G V H + B V C = D W E + M , which includes the second-order Sylvester, Lyapunov and Stein matrix equations as special cases. The convergence results show that the algorithm with any initial matrices converges to the solutions within a finite number of iterations in the absence of round-off errors. Finally, two numerical examples are provided to support the theoretical findings and to testify the effectiveness and usefulness of the algorithm.

Finite iterative algorithms for the generalized reflexive and anti-reflexive solutions of the linear matrix equation AXB = C

In this paper, an iterative method is presented to solve the linear matrix equation AXB = C over the generalized reflexive (or anti-reflexive) matrix X (A ? Rpxn, B ? Rmxq, C ? Rpxq, X ? Rnxm). By the iterative method, the solvability of the equation AXB = C over the generalized reflexive (or anti-reflexive) matrix can be determined automatically. When the equation AXB = C is consistent over the generalized reflexive (or anti-reflexive) matrix X, for any generalized reflexive (or anti-reflexive) initial iterative matrix X1, the generalized reflexive (anti-reflexive) solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized reflexive (or anti-reflexive) iterative solution of AXB = C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB = C is inconsistent is given. Furthermore, the optimal approximate solution of AXB = C for a given matrix X0 can be derived by finding the least-norm generalized reflexive (or anti-reflexive) solution of a new corresponding matrix equation AXB = C. Finally, several numerical examples are given to support the theoretical results of this paper.

Finding solutions for periodic discrete-time generalized coupled Sylvester matrix equations via the generalized BCR method

The periodic discrete-time matrix equations have wide applications in stability theory, control theory and perturbation analysis. In this work, the biconjugate residual algorithm is generalized to construct a matrix iterative method to solve the periodic discrete-time generalized coupled Sylvester matrix equations [Formula: see text] The constructed method is shown to be convergent in a finite number of iterations in the absence of round-off errors. By comparing with other similar methods in practical computation, we give numerical results to demonstrate the accuracy and the numerical superiority of the constructed method.

Iterative solutions of generalized inverse eigenvalue problem for partially bisymmetric matrices

In this paper, for given , , an iterative algorithm is constructed to find the solutions of generalized inverse eigenvalue problem , where A and B should be partially bisymmetric under a prescribed submatrix constraint. For any initial constrained matrices, a solution pair can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least norm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained by this iterative algorithm. Numerical examples are given to illustrate efficiency of the proposed algorithm.

Symmetric solutions of the coupled generalized Sylvester matrix equations via BCR algorithm

The symmetric solutions of linear matrix equations are extensively required in mathematics and engineering problems. The purpose of this paper is on deriving the biconjugate residual (BCR) algorithm for finding the least Frobenius norm symmetric solution pair (X,Y) of the coupled generalized Sylvester matrix equations{A1XB1+C1XD1+E1YF1=G1,A2XB2+C2XD2+E2YF2=G2.The convergence analysis shows that the BCR algorithm can compute the least Frobenius norm symmetric solution pair of the coupled generalized Sylvester matrix equations within a finite number of iterations in the absence of round-off errors. Finally, we give three numerical examples to illustrate the performance of the BCR algorithm.

A finite iterative method for solving the generalized Hamiltonian solutions of coupled Sylvester matrix equations with conjugate transpose

ABSTRACTFor given skew-Hermitian unitary matrix J, i.e. , , a matrix is termed generalized Hamiltonian matrix if . In this paper, an iterative method is constructed to solve the generalized Hamiltonian solutions of the coupled Sylvester matrix equations with conjugate transpose. It is proved that the iterative method is unconditionally convergent for any initial generalized Hamiltonian matrices. With it, the generalized Hamiltonian solutions can be obtained within finite iteration steps in the absence of roundoff errors. Finally, numerical examples are presented to illustrate the efficiency and applicability of the method.

Generalized conjugate direction algorithm for solving the general coupled matrix equations over symmetric matrices

Symmetric solutions of the linear matrix equations have wide applications in both mechanical and electrical engineering. In this work, an analytic study of the generalized conjugate direction (CD) algorithm for finding the symmetric solution group (X1,X2,...,Xm) of the general coupled matrix equations źj=1mAijXjBij=Ci,i=1,2,...,n,$$\sum\limits_{j=1}^{m}A_{ij}X_{j}B_{ij}=C_{i},\qquad i=1,2,...,n, $$ is performed. We show that the generalized CD algorithm can compute the (least Frobenius norm) symmetric solution group of the general coupled matrix equations for any (special) initial symmetric matrix group within a finite number of iterations in the absence of round-off errors. In order to illustrate the effectiveness of the generalized CD algorithm, two numerical examples are finally given.

Least-Squares Solutions of Generalized Sylvester Equation with Xi Satisfies Different Linear Constraint

In this paper, an iterative method is constructed to find the least-squares solutions of generalized Sylvester equation , where is real matrices group, and satisfies different linear constraint. By this iterative method, for any initial matrix group within a special constrained matrix set, a least squares solution group with satisfying different linear constraint can be obtained within finite iteration steps in the absence of round off errors, and the unique least norm least-squares solution can be obtained by choosing a special kind of initial matrix group. In addition, a minimization property of this iterative method is characterized. Finally, numerical experiments are reported to show the efficiency of the proposed method.