Absence Of Round-off Errors Research Articles

Generalized conjugate direction algorithm for solving general coupled Sylvester matrix equations

In this paper, a generalized conjugate direction algorithm (GCD) is proposed for solving general coupled Sylvester matrix equations. The GCD algorithm is an improved gradient algorithm, which can realize gradient descent by introducing matrices Pj(k) and Tj(k) to construct parameters α(k) and β(k). The matrix Pj(k) and Tj(k) are iterated in a cross way to accelerate the convergence rate. In addition, it is further proved that the algorithm converges to the exact solution in finite iteration steps in the absence of round-off errors if the system is consistent. Also, the sufficient conditions for least squares solutions and the minimum F-norm solutions are obtained. Finally, numerical examples are given to demonstrate the effectiveness of the GCD algorithm.

Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equations

This paper focuses on constructing a conjugate gradient-based (CGB) method to solve the generalized periodic coupled Sylvester matrix equations in complex space. The presented method is developed from a point of conjugate gradient methods. It is proved that the presented method can find the solution of the considered matrix equations within finite iteration steps in the absence of round-off errors by theoretical derivation. Some numerical examples are provided to verify the convergence performance of the presented method, which is superior to some existing numerical algorithms both in iteration steps and computation time.

A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations

The purpose of this paper is to present an iterative algorithm for solving the general discrete-time periodic Sylvester matrix equations. It is proved by theoretical analysis that this algorithm can get the exact solutions of the periodic Sylvester matrix equations in a finite number of steps in the absence of round-off errors. Furthermore, when the discrete-time periodic Sylvester matrix equations are consistent, we can obtain its unique minimal Frobenius norm solution by choosing appropriate initial periodic matrices. Finally, we use some numerical examples to illustrate the effectiveness of the proposed algorithm.

An iterative algorithm for the generalized centro-symmetric solution of the generalized coupled Sylvester matrix equations AV+BW=EVF+C, MV+NW=GVH+D

In this paper, an iterative algorithm for solving of the generalized coupled Sylvester matrix equations AV+BW=EVF+C, MV+NW=GVH+D over the generalized centro-symmetric matrices (V,W ) is proposed. For any initial generalized centro-symmetric matrices V0 and W0, a generalized centro-symmetric solution (V,W ) is obtained within a finite number of iterations in the absence of round-off errors. Two numerical examples are presented to support the theoretical results where the efficiency and accuracy of the suggested algorithm are shown.

Developing iterative algorithms to solve Sylvester tensor equations

This paper is concerned with solving high order Sylvester tensor equation arising in control theory. We propose the tensor forms of the bi-conjugate gradient and bi-conjugate residual methods for solving the tensor equation. To improve their performance, two preconditioned iterative algorithms based on the nearest Kronecker product are developed for finding its solution. We also prove that the proposed algorithms are convergent to an exact solution within finite iteration steps for any initial tensor in the absence of round-off errors. At last, some numerical examples are provided to illustrate the feasibility and validity of the algorithms proposed.

An iterative algorithm for generalized periodic multiple coupled Sylvester matrix equations

The paper is indicated to constructing a modified conjugate gradient iterative (MCG) algorithm to solve the generalized periodic multiple coupled Sylvester matrix equations. It can be proved that the proposed approach can find the solution within finite iteration steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm solution of the system. Some numerical examples are illustrated to show the performance of the proposed approach and its superiority over the existing method CG.

Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

As is well known, linear operator equations have wide applications in many areas of engineering and applied mathematics. In the present paper, we are interested in solving the linear operator equation $$\begin{aligned} {\mathscr {F}}(J)+{\mathscr {G}}(K)+{\mathscr {H}}(L)=N, \end{aligned}$$ where J, K and L should be partially bisymmetric under a prescribed submatrix constraint. Three conjugate gradient-like algorithms are derived for solving this constrained operator equation including the Lyapunov, Stein and Sylvester matrix equations and the quadratic inverse eigenvalue problem as special cases. The algorithms converge to the solutions of the linear operator equation within a finite number of iterations in the absence of round-off errors. At the end, the accuracy and efficiency of the introduced algorithms are demonstrated numerically with three examples.

Conjugate gradient-like methods for solving general tensor equation with Einstein product

Tensors have a wide application in data mining, chemistry, information sciences, documents analysis and medical engineering. In this work, we study the general tensor equation ∑i=1lFi*PX*QGi=H with Einstein product where Fi,Gi,H, for i=1,2,…,l, are known tensors and X is an unknown tensor to be determined. The main motivation for this study is the investigation of conjugate gradient-like methods for solving this tensor equation. We show that the conjugate gradient-like methods converge to tensor solutions in a finite number of steps in the absence of round-off errors. Numerical examples confirm ’oretical results and demonstrate the accuracy and computational efficiency of the methods.

Modified CGLS iterative algorithm for solving the generalized Sylvester-conjugate matrix equation

By introducing the real inner product, this paper offers an modified conjugate gradient least squares iterative algorithm (MCGLS)for solving the generalized Sylvester-conjugate matrix equation. The properties of this algorithm are discussed and the finite convergence of this algorithm is proven. This new iterative method can obtain the symmetric least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.

Minimum norm partial quadratic eigenvalue assignment for vibrating structures using receptance method

Abstract In this paper, we consider the minimum norm partial quadratic eigenvalue assignment problem (MNPQEAP) for multi-input vibration control systems. Our work is based on the recent paper proposed by Bai et al. (2018) who solve this problem by using receptances and system matrices. By using receptance method, we transform this problem into finding the minimum norm solutions of the matrix equations and propose a modified conjugate gradient (MCG) method for solving this problem. Our method can solve the MNPQEAP with finite iteration steps in the absence of roundoff errors. The numerical examples show the efficiency of our method.

Three types of biconjugate residual method for general periodic matrix equations over generalized bisymmetric periodic matrices

As is well known, periodic matrix equations have wide applications in many areas of control and system theory. This paper is devoted to a study of the numerical solutions of a general type of periodic matrix equations. We present three types of biconjugate residual (BCR) method to find the generalized bisymmetric periodic solutions [Formula: see text] of general periodic matrix equations [Formula: see text] The main theorems of this paper show that the presented methods can compute the generalized bisymmetric periodic solutions in a finite number of steps in the absence of round-off errors. We give two numerical examples to illustrate and interpret the theoretical results.

An iterative algorithm to solve the generalized Sylvester tensor equations

ABSTRACT This paper is concerned with the conjugate gradient least squares algorithm to solve a class of tensor equations via the Einstein product. The proposed algorithm uses tensor computations with no matricizations involved. We prove that the solution (or the least squares solution) of the tensor equation can be obtained within a finite number of iterative steps in the absence of round-off errors. By selecting the appropriate initial tensor, the least Frobenius norm solution (or the least Frobenius norm least squares solution) of the tensor equation can be obtained. Numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper.

The conjugate gradient methods for solving the generalized periodic Sylvester matrix equations

This work is devoted to designing two conjugate gradient methods for the least Frobenius norm solution of the generalized periodic Sylvester matrix equations. When the studied problem is consistent, the first conjugate gradient method can find its solution within finite iterative steps in the absence of round-off errors. Furthermore, its least Frobenius norm solution can be obtained with some special kind of initial matrices. When the studied problem is inconsistent, the second conjugate gradient method with some special kind of initial matrices can find its least squares solution with the least Frobenius norm within finite iterative steps in the absence of round-off errors. Finally, several numerical examples are tested to validate the performance of the proposed methods.

Finite iterative Hamiltonian solutions of the generalized coupled Sylvester – conjugate matrix equations

In this paper, we present an iterative algorithm to solve a generalized coupled Sylvester – conjugate matrix equations over Hamiltonian matrices. When the considered systems of matrix equations are consistent, it is proven that the solution can be obtained within finite iterative steps for any arbitrary initial generalized Hamiltonian matrices in the absence of round off errors. Two numerical examples are given to illustrate the effectiveness of the proposed method.

Solving constrained quadratic inverse eigenvalue problem via conjugate direction method

Inverse eigenvalue problems are among the most important problems in numerical linear algebra. This paper is concerned with the problem of designing an iterative method for a quadratic inverse eigenvalue problem of the form MXΛ2+GXΛ+KX=0 where M, G and K should be partially doubly symmetric under a prescribed submatrix constraint. This kind of quadratic inverse eigenvalue problem includes the classical and generalized inverse eigenvalue problems as special cases. We propose the conjugate direction (CD) method for computing the least Frobenius norm solutions of this constrained quadratic inverse eigenvalue problem in a finite number of steps in the absence of round-off errors. The CD method can automatically determine the solvability of this problem. Finally, the overall performance and efficiency of the CD method for computing approximate solutions of the problem are discussed.

An efficient algorithm based on Lanczos type of BCR to solve constrained quadratic inverse eigenvalue problems

This paper proposes the Lanczos type of biconjugate residual (BCR) algorithm for solving the quadratic inverse eigenvalue problem L 2 X Λ 2 + L 1 X Λ + L 0 X = 0 where L 2 , L 1 and L 0 should be partially bisymmetric under a prescribed submatrix constraint. An analysis reveals that the algorithm obtains the solutions of the constrained quadratic inverse eigenvalue problem in finitely many steps in the absence of round-off errors. Finally numerical results are performed to confirm the analysis and to illustrate the efficiency of the proposed algorithm.

The least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint

In this paper, we present an iterative method for finding the least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint. We prove that if the constrained matrix equations are consistent, the solution can be obtained within finite iterative steps in the absence of round-off errors; if constrained matrix equations are inconsistent, the least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Finally, numerical examples are provided to illustrate the efficiency of the proposed method and testify the conclusions suggested in this paper.

An iterative method for obtaining the Least squares solutions of quadratic inverse eigenvalue problems over generalized Hamiltonian matrix with submatrix constraints

In this paper, we consider a class of constrained matrix quadratic inverse eigenvalue problem and its optimal approximation problem. It is proved that the proposed algorithm always converge to the generalized Hamiltonian solutions with a submatrix constraint of Problem 1.1 within finite iterative steps in the absence of roundoff error. In addition, by choosing a special kind of initial matrices, it is shown that the minimum norm solution of Problem 1.1 can be obtained consequently. At last, for a given matrix group in the solution set of Problem 1.1, it is proved that the unique optimal approximation solution of Problem 1.2 can be also obtained. Some numerical results are reported to demonstrate the efficiency of our algorithm.

Finite iterative algorithm for the symmetric periodic least squares solutions of a class of periodic Sylvester matrix equations

The present work proposes a finite iterative algorithm to find the least squares solutions of periodic matrix equations over symmetric ξ-periodic matrices. By this algorithm, for any initial symmetric ξ-periodic matrices, the solution group can be obtained in finite iterative steps in the absence of round-off errors, and the solution group with least Frobenius norm can be obtained by choosing a special kind of initial matrices. Furthermore, in the solution set of the above problem, the unique optimal approximation solution group to a given matrix group in the Frobenius norm can be derived by finding the least Frobenius norm symmetric ξ-periodic solution of a new corresponding minimum Frobenius norm problem. Finally, numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper.

Computing symmetric solutions of general Sylvester matrix equations via Lanczos version of biconjugate residual algorithm

Many problems in control theory can be studied by obtaining the symmetric solution of linear matrix equations. In this investigation, we deal with the symmetric solutions X, Y and Z of the general Sylvester matrix equations A1XB1+C1YD1+E1ZF1=G1,A2XB2+C2YD2+E2ZF2=G2,⋮⋮⋮⋮AtXBt+CtYDt+EtZFt=Gt.The Lanczos version of biconjugate residual (BCR) algorithm is generalized to compute the symmetric solutions of the general Sylvester matrix equations. The convergence properties of this algorithm are discussed and it is proven that it smoothly converges to the symmetric solutions of the general Sylvester matrix equations in a finite number of iterations in the absence of round-off errors. Finally, the comparative numerical results are carried out to support that the current algorithm may be more efficient than other iterative algorithms.