In recent years, quaternion subspace methods have been widely used to solve non-Hermitian quaternion linear systems. However, due to the complexity associated with high-dimensional data computations, these methods are not suitable for large-scale linear systems. In this paper, we propose a novel structure-preserving randomized quaternion full orthogonalization (RQFOM) method, which is based on the randomized quaternion Arnoldi procedure. By utilizing random sketching, the method projects high-dimensional data into a low-dimensional space, significantly reducing the computational cost. Theoretical results for the proposed method are provided, and numerical experiments demonstrate their feasibility and efficiency. Additionally, we apply the proposed RQFOM to the encryption and decryption of color images. The resulting encryption scheme exhibits robust performance and security, effectively resisting various attacks.

A parameter-free method, namely the generalization of the Gauss-Seidel (GGS) method, is developed to solve generalized absolute value equations (GAVE). Some results in the recent work of Edalatpour et al. [A generalization of the Gauss-Seidel iteration method for solving absolute value equations. Appl Math Comput. 2017;293:156-167] are extended. For solving linear complementarity problem, the GGS method can be seen as a special case of the accelerated modulus-based matrix splitting iteration method [Zheng and Yin. Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem. Numer Algorithms. 2013;64:245-262], which is characterized by solving a GAVE subproblem in each iteration. Moreover, new convergence results of the GGS method are given. Numerical results demonstrate the effectiveness and efficiency of the GGS method.

Dual quaternions are pivotal for modeling rigid-body motions in robotics, aerospace, and computer graphics, with solving generalized Sylvester dual quaternion matrix equations central to kinematic analysis and trajectory planning. Existing methods often have slow convergence or limited applicability to coupled forms, restricting use. This paper addresses a class of such equations. We establish a necessary and sufficient condition for solution existence via real representation of quaternion matrices, vectorization, and Kronecker products. Two novel algorithms are proposed: dual quaternion generalized conjugate direction and conjugate gradient least squares algorithms, both converging in finite iterations under no rounding errors for theoretical efficiency. Numerical simulations demonstrate the algorithms outperform existing methods in convergence speed and apply to practical engineering like robotic kinematics. Furthermore, experiments on simultaneous restoration of two hand tremor-induced mixed-direction blurred color images validate their effectiveness in handling complex real-world degradation, expanding their application scope to image processing and highlighting strong practical potential.

For large and sparse linear systems,it is effective to reduce the original problem into a lower dimensional linear system and solve the derived equation instead. The main contribution of this paper is that a novel adaptive parameter iteration algorithm is constructed from the perspective of numerical optimization for a class of two-by-two linear systems. The new algorithm adopts a prediction-correction two-step iteration, which uses delayed information to define the iterations. Global convergence results are established, and the algorithm enjoys at least a Q-linear convergence rate under suitable conditions. Numerical experiments demonstrate the efficiency and effectiveness of the new algorithm in applications to elliptic PDE-constrained optimization problems,complex symmetric systems, and saddle point problems, in comparison with existing similar algorithms.

In this work, we establish a generalized MASSOR (GMASSOR) method for solving saddle point problems. The proposed method can be used to both nonsingular and singular cases. In addition, we deduce the convergence and semi-convergence of the GMASSOR method under the appropriate constraints on the iteration parameters. Numerical results are given to verify the effectiveness of the proposed method.

A new Newton-based matrix splitting iterative method is proposed for solving generalized absolute value equation with nonlinear term. We give the global convergence of this method. Further some new convergence conditions are proposed when A = M - N is an H-compatible splitting. Numerical results indicate that the new Newton-based matrix splitting iterative method for solving generalized absolute value equation with nonlinear term is effective.

A family of neural networks is proposed to solve linear complementarity problems (LCP). The neural networks are constructed from the novel equivalent model of LCP, which is reformulated by utilizing the modulus and smoothing technologies. Some important properties of the proposed novel equivalent model are summarized. In addition, the stability properties of the proposed steepest descent-based neural networks for LCP are analyzed. In order to illustrate the theoretical results, we provide some numerical simulations and compare the proposed neural networks with existing neural networks based on the NCP-functions. Numerical results indicate that the performance of the proposed neural networks is effective and robust.

Building on the classical Uzawa approach and the shift-splitting technique, we develop a class of Uzawa-type shift-splitting methods and preconditioners for solving block problems of three-by-three with double saddle point matrices. Theoretical analysis confirms that the iteration methods converge under appropriate conditions, and the proposed preconditioners have computational advantages to implement within Krylov subspace acceleration. Moreover, we propose a relaxed variant for the preconditioner and analyze the spectral properties of the corresponding preconditioned matrices. Finally, numerical experiments demonstrate the practicality and efficiency of the proposed preconditioners.

Based on the triangular splitting technique, we introduce a free-parameter alternating triangular splitting (FPATS) method for solving block two-by-two linear systems with applications to time-harmonic parabolic models. In addition, we demonstrate that the FPATS method is unconditionally convergent and outperforms other methods. Numerical results are provided to show the practicality and efficiency of our method.

In this paper, we use the scaled alternating direction multiplier method (SADMM) and its variant form to obtain the least squares solution of the generalized Sylvester matrix equation. Unlike currently ADMM algorithms for solving matrix equations, we consider the effect of the penalty coefficients of the augmented Lagrangian function on the convergence of the SADMM algorithm and give a way to adjust the values of the penalty coefficients. Optimality conditions and termination criteria for the SADMM algorithm are given. Two types of convergence analyses under different assumptions are also given and it is shown that the SADMM algorithm converges for any initial matrix under satisfying the assumptions. The simplicity and effectiveness of the new method are shown by the reported numerical results.

Recently,inspired by a modified generalized shift-splitting iteration method for complex symmetric linear systems,we propose two variants of the modified generalized shift-splitting iteration(MGSS)methods for solving com-plex symmetric linear systems.One is a parameterized MGSS iteration method and the other is a modified parameterized MGSS iteration method.We prove that the proposed methods are convergent under appropriate constraints on the parameters.In addition,we also give the eigenvalue distributions of differ-ent preconditioned matrices to verify the effectiveness of the preconditioners proposed in this paper.

It is well known that Levenberg-Marquardt (LM) method is widely used for solving nonlinear equations. In this paper, we give an extension of LM method and propose a nonmonotone LM method with correction which produces the LM parameter according to the new nonmonotone strategy of Grippo, Lampariello and Lucidi. Moreover, not only an LM step but also a correction step are computed at every iteration in our proposed nonmonotone LM method with correction. The cubic convergence of the proposed method is proved under the local error bound condition which is weaker than nonsingularity. Some numerical results confirm the feasibility and effectiveness of the proposed algorithm.

A new equivalent reformulation of the absolute value equations associated with second-order cone (SOCAVEs) is emphasised, from which two dynamical models based on projection operator for solving SOCAVEs are constructed. Under suitable conditions, it is proved that the equilibrium points of the dynamical systems exist and could be (globally) asymptotically stable. The effectiveness of the proposed methods are illustrated by some numerical simulations.

The system of generalized absolute value equations (GAVE) has attracted more and more attention in the optimization community. In this paper, by introducing a smoothing function, we develop a smoothing Newton algorithm with non-monotone line search to solve the GAVE. We show that the non-monotone algorithm is globally and locally quadratically convergent under a weaker assumption than those given in most existing algorithms for solving the GAVE. Numerical results are given to demonstrate the viability and efficiency of the approach.

A high-performance sparse model is very important for processing high-dimensional data. Therefore, based on the quadratic approximate greed pursuit (QAGP) method, we can make full use of the information of the quadratic lower bound of its approximate function to get the relaxation quadratic approximate greed pursuit (RQAGP) method. The calculation process of the RQAGP method is to construct two inexact quadratic approximation functions by using the m-strongly convex and L-smooth characteristics of the objective function and then solve the approximation function iteratively by using the Iterative Hard Thresholding (IHT) method to get the solution of the problem. The convergence analysis is given, and the performance of the method in the sparse logistic regression model is verified on synthetic data and real data sets. The results show that the RQAGP method is effective.

Recently, Yu et al. presented a modified fixed point iterative (MFPI) method for solving large sparse absolute value equation (AVE).In this paper, we consider using accelerated overrelaxation (AOR) splitting to develop the modified fixed point iteration (denoted by MFPI-JS and MFPI-GSS) methods for solving AVE.Furthermore, the convergence analysis of the MFPI-JS and MFPI-GSS methods for AVE are also studied under suitable restrictions on the iteration parameters, and the functional equation between the parameter τ and matrix Q.Finally, numerical examples show that the MFPI-JS and MFPI-GSS are efficient iteration methods.

In this article, based on the real representation and Kronecker product, Cramer’s rule for a class of coupled Sylvester commutative quaternion matrix equations is studied and its expression is obtained. The proposed algorithm is very simple and convenient because it only involves real operations. Some numerical examples are provided to illustrate the feasibility of the proposed algorithm.

Inspired by MSOR-like iteration method proposed by Guo et al. [25] and NMSSOR iteration method proposed by Najafi et al. [32], parameter acceleration technique and preconditioning matrix Q are used, we establish a parameterized modified SOR-like (PMSOR-like) method for non-singular and singular saddle point problems. Theoretical analysis show that the convergence and semi-convergence properties of the PMSOR-like method can be guaranteed under suitable conditions. For nonsingular cases, we give the local theoretical optimal parameters of PMSOR-like method and the corresponding convergence factor. In addition, numerical examples are given to verify the efficiency of the proposed method.

In this paper, a modified conjugate gradient (MCG) iterative algorithm is proposed for solving the general Sylvester matrix equation. Some numerical examples are given to compare the accuracy and efficiency of the MCG iterative method with CG method in the literature. The numerical results suggest that the MCG method is efficient, especially for some problems that can’t be solved using CG method.