A matrix LSQR iterative method to solve matrix equation AXB=C

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group(X¯1,X¯2,…,X¯q)in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

In this paper, an iterative method is presented to solve the following constrained minimum Frobenius norm residual problem: where is a linear operator from R m×n onto , , is a linear self-conjugate involution operator. By this method, for any initial matrix , a solution can be obtained in finite iteration steps in the absence of roundoff errors. The least norm solution can be derived when an appropriate initial matrix is chosen. In addition, the optimal approximation solution in the solution set of the above problem to a given matrix can also be derived by this method. Several numerical examples are given to show the efficiency of the proposed iterative method.

Let ${\mathrm R_{m,n}}$ be the set of $m\times n$ matrices over a Bezout domain $\mathrm R $ with identity $e\not= 0$ and let $0_{m,k}$ be the zero $m\times k$ matrix. Further, let $d_i(A)\in \mathrm{R}$ be an ideal generated by the $i$-th order minors of the matrix $A\in \mathrm{R}_{m,n},$ $i = 1, 2, \dots, \min\{m, n\}.$ In this article, we investigate a structure of solutions of a matrix equation $AX=B$, where $A\in {\mathrm R}_{m,n}$ and $B \in {\mathrm R}_{m,k}$ are known matrices and $X$ is unknown matrix over ${\mathrm R}$. It is known that matrix equation $AX=B$ is solvable over a Bezout domain $\mathrm{R}$ if and only if $ {\rm rank } A = {\rm rank }A_B=r$ and $d_i(A) = d_i(A_B)$ for all $i = 1, 2, \dots , r,$ where $A_B= \begin{bmatrix} A & B \end{bmatrix}.$ On the other hand, $AX=B$ is solvable over $\mathrm{R}$ if and only if matrices $\begin{bmatrix} A & 0_{m,k} \end{bmatrix}$ and $A_B$ are right-equivalent, that is, the Hermitian normal forms of these matrices coincide. In this article, we give alternative necessary and sufficient conditions for the solvability of equation $AX=B$ over a Bezout domain $\mathrm R .$ If a solution of this equation exists, we also give an algorithm for its construction. We prove also that the matrix equation $AX=B$ over $\mathrm{R}$ has a symmetric solution if and only if $AX=B$ has a solution over $\mathrm{R}$ and the matrix $AB^T$ is symmetric. If symmetric solution exists, we propose the method for its construction.

The bisymmetric solutions of the matrix equation [formula omitted] and its optimal approximation

The general coupled matrix equations over generalized bisymmetric matrices

The usual solution methods of discretised partial differential equations are based exclusively on matrix-vector multiplications as basis operation. On the one hand, this is supported by the use of sparse matrices (cf. §1.3.2.5); on the other hand, one tries to apply fast iterative methods (e.g., the multigrid method [124], [119, §12]) whose basic steps are matrix-vector multiplications. Krylov methods are based on the same concept. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation1 and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. The $$\mathcal{H}$$ -matrix arithmetic allows the solution of these matrix equations efficiently. Here, the use of hierarchical matrix operations and matrix-valued functions is only one part of the solution method. Another essential fact is that the solution $$ X \in \mathbb{R}^{I \times I}(n==\!\!\!\!\!/ \!\!\!\!/I)$$ can be replaced by an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ . If one considers the equation $$f(X)\;=\;0$$ as a system of n 2 equation for the n 2 components of X, even an optimal solution method has complexity $$\mathcal{O}(n^2)$$ , since this is linear complexity in a number of unknowns (cf. Remark 1.1). Using traditional techniques, the solution of large-scale matrix equations is not feasible. Only an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ with $$\mathcal{O}(n\; \mathrm{log}^*n)$$ data instead of n 2 admits a solution with a cost almost linear with respect to n. Section 15.1 introduces Lyapunov and Sylvester equations and discusses their solution. In Section 15.2 we consider quadratic Riccati equation. An interesting approach uses the matrix version of the sign function from §14.1.1. General nonlinear matrix equations may be solved iteratively by Newton’s method or related methods (cf. Section 15.3). As an example, computing the square root of a positive definite matrix is described in §15.3.1. The influence of the truncation error introduced by H-matrix arithmetic is investigated in Section 15.3.

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 ∈ I={X |X= U(X)}, where Ai is a linear operator from Cmxn onto Cpixqi, Ei ∈ Cpixqi, 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 X0 ∈ I, 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. Mathematical subject classification: 15A24, 65D99,65F30.

This survey provides a comprehensive overview of the solutions to the matrix equation AXB=C over real numbers, complex numbers, quaternions, dual quaternions, dual split quaternions, and dual generalized commutative quaternions, including various special solutions. Additionally, we summarize the numerical algorithms for these special solutions. This matrix equation plays an important role in solving linear systems and control theory. We specifically explore the application of this matrix equation in color image processing, highlighting its unique value in this field. Taking the dual quaternion matrix equation AXB=C as an example, we design a scheme for simultaneously encrypting and decrypting two color images. The experimental results demonstrate that this scheme is highly feasible.

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

An efficient algorithm for solving general coupled matrix equations and its application

An attempt is made to establish the existence of solutions of nonlinear matrix equations of the form X(z)A(z)X(z)=X(z), where all matrices X(z), A(z) are holomorphic matrices. Such matrix equations arise in nonlinear control theory of nonlinear dynamical systems where A(z) is a given holomorphic matrix. Numerical methods of solutions of such nonlinear matrix equations are given. Use is made of parallel processing and supercomputers to carry out the numerical calculations of such problems. Examples of methods used for problems of the above type are given. >

In this paper, first, we use the Rich type iterative method for numerical solution of nonlinear fuzzy Fredholm integral equation. The main structure of this method is based on the Richardson iterative method and trapezoidal quadrature formula for Numerical solution of nonlinear fuzzy integral equations. Convergence, error analysis and numerical stability of the proposed method are also discussed.

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB − CYD, EXF − GYH) = (M, N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1, Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round‐off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair to a given matrix pair [X0, Y0] in Frobenius norm can be derived by finding the least‐norm generalized reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equation pair , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

In this paper, we extend the well-known Jacobi and Gauss-Seidel iterations to present a large family of iterative methods. The proposed methods are applied to develop iterative solutions to the matrix equation AXB = F and the generalized Sylvester matrix equation AXB + CXD = F by means of a hierarchical identification principle. We prove that the iterative solutions converge to the exact solutions for any initial values. The algorithms proposed require less storage capacity than the existing numerical ones. The iterative methods can be applied to system parameter identification problems.