‎In this paper, we delve into frame theory to create an innovative iterative method for resolving the operator equation $ Lu=f $. In this case, $ L:H\rightarrow H $, a bounded, invertible, and self-adjoint linear operator, operates within a separable Hilbert space denoted by $H$. Our methodology, which is based on the GMRES projective method, introduces an alternate search space, which brings another dimension to the problem-solving process. Our investigation continues with the assessment of convergence, where we look at the corresponding convergence rate. This rate is intricately influenced by the frame bounds, shedding light on the effectiveness of our approach. Furthermore, we investigate the ideal scenario in which the equation finds an exact solution, providing useful insights into the practical implications of our work.

Ill-conditioned linear systems may frequently arise in discretization of integral equations and many other real world applications. Solving such systems by classical METHODS might fail or result to solutions that are meaningless from practical point view. Moreover a slight perturbation in the right hand side vector might also lead to an enormous change of the solution vector due to ill-conditioned ness. To find meaningful solutions of such systems, the Tikhonov regularization is an effective technique that has been widely used. In this paper we use it to solve ill-conditioned linear systems and also to find closest feasible linear systems to nearly feasible linear systems by smallest changes in problem data. Throughout the paper numerical results are reported to demonstrate the practical efficiency of the presented algorithms.

The global generalized minimum residual (Gl-GMRES) method is examined for solving the generalized Sylvester matrix equation Sqi=1 AiXBi=C. Some new theoretical results are elaborated for the proposed method by employing the Schur complement. These results can be exploited to establish new convergence properties of the Gl-GMRES method for solving general (coupled) linear matrix equations. In addition, the Gl-GMRES method for solving the generalized Sylvester-transpose matrix equation is briey studied. Finally, some numerical experiments are presented to illustrate the efficiently of the Gl-GMRES method for solving the general linear matrix equations.

This paper aims to extend a Krylov subspace technique based on an in-complete orthogonalization of Krylov tensors (as a multidimensional exten-sion of the common Krylov vectors) to solve generalized Sylvester tensor equations via the Einstein product. First, we obtain the tensor form of the quasi-GMRES method, and then we lead to the direct variant of the proposed algorithm. This approach has the great advantage that it uses previous data in each iteration and has a low computational cost. More-over, an upper bound for the residual norm of the approximate solution is found. Finally, several experimental problems are given to show the acceptable accuracy and efficiency of the presented method.

Voltage collapse is an event that causes major concern to the power system utility nowadays. The effect can be catastrophic to the power system where it can cause total collapse to the operation of the system. The study of the voltage collapse phenomenon can provide a way to prevent this event from happening. Static analysis provides the best way to study this phenomenon. There have been many METHODS developed to study the criteria of this phenomenon. Conventional Newton Raphson method has the singularity problem on its Jacobian matrix and thus could not give the solution. To overcome this problem, many METHODS had been proposed that can avoid the problem and one of them is the Continuation Power Flow (CPF) method. CPF method is a very powerful method that can provide the solution without having the singularity problem. The CPF method is improved using the new CPF-GMRES method. Although NRS (Newton - Raphson - Seydel) is an old method, it is fast and accurate. This paper expands NRS method and proves that Expanded NRS is more reliable and faster than CPF-GMRES. The idea is demonstrated on 350 bus network in IRAN (Khorasan region).

Due to improvements in computational resources, interest has recently increased in using implicit scheme for solving flow equations on 3D unstructured grids. However, most of the implicit schemes produce greater numerical diffusion error than their corresponding explicit schemes. This stems from the fact that in linearizing implicit fluxes, it is conventional to replace the Jacobian matrix in the dissipation term by its constant spectral radius. The objective of the present study is to develop a modified implicit solver based on Roe scheme so that its numerical dissipation is as much as the explicit one. In the proposed scheme, the Krylov subspace method with a LU decomposition preconditioner (GMRES+LU-SGS) is used to solve the linear systems. The efficiency of this method is shown by presenting some examples at the end.

Increasing precision and stability in the online estimation of a spacecraft's model, due to the uncertainty and noise is one of the challenges in the attitude control of the space systems. The least squares error method in combination with the Kalman filter is one of the effective METHODS for estimating these types of dynamic models. Based on the development of the GMRES (generalized minimal residual) METHODS, in order to increase the performance of the estimation method, a newonline meta-heuristic algorithm is proposed. The algorithm is an iterative-based method that uses previous step information based on user experience, or a new online meta-heuristic method which determines the number of steps to solve the matrix equations system in the Krylov subspace and improves overall convergence to the response. In order to evaluate the accuracy of this estimation method, the GMRES, Bi-CG (Bi Conjugate Gradient), CGS (Conjugate Gradients Squared), BI-CGSTAB (Bi Conjugate Gradient Stabilized) METHODS are compared that the online meta-heuristic GMRES method shows the highest accuracy and stability in the estimation.