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This paper presents three tensor randomized extended Kaczmarz methods for solving a tensor inconsistent linear system of equations under the t-product between tensors. The randomized extended Kaczmarz, the randomized extended block Kaczmarz, and the randomized extended greedy block Kaczmarz methods in tensor form are proposed to solve inconsistent tensor linear system of equations, respectively. The convergence of each method is proved. Several numerical examples are given to show the efficiency and effectiveness of our methods. [ABSTRACT FROM AUTHOR]

The parameterized block splitting (PBS) is a convergent and efficient iterative method to solve the large complex symmetric linear systems. In this paper, by using PBS iterative technique, the Newton equation is approximately solved, then we establish the modified Newton-PBS iterative method to solve the complex nonlinear systems whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, the local convergence analysis are explored under appropriate conditions. Ultimately, we apply the new method and several known methods to experimental numerical examples, and experimental results verify the superiority and efficiency of our new method. Especially, in terms of CPU time and iteration steps, our method is obviously better. [ABSTRACT FROM AUTHOR]

We study a two-dimensional coordinate descent method to solve large linear least squares problems expanding on the method presented by Leventhal and Lewis. For an overdetermined system with full column rank, they proved its convergence in expectation, regardless of its consistency. In our work, we present a block version of the same. We also present an update on the extension done by Ma et al. to address non-full rank linear systems or underdetermined linear systems. Convergence is analyzed for the stated methods, and numerical experiments are provided to demonstrate their efficiency. [ABSTRACT FROM AUTHOR]

In this work, we propose a simple yet generic preconditioned Krylov subspace method for a large class of nonsymmetric block Toeplitz all-at-once systems arising from discretizing evolutionary partial differential equations. Namely, our main result is to propose two novel symmetric positive definite preconditioners, which can be efficiently diagonalized by the discrete sine transform matrix. More specifically, our approach is to first permute the original linear system to obtain a symmetric one and subsequently develop desired preconditioners based on the spectral symbol of the modified matrix. Then, we show that the eigenvalues of the preconditioned matrix sequences are clustered around ± 1 , which entails rapid convergence when the minimal residual method is devised. Alternatively, when the conjugate gradient method on the normal equations is used, we show that our preconditioner is effective in the sense that the eigenvalues of the preconditioned matrix sequence are clustered around unity. An extension of our proposed preconditioned method is given for high-order backward difference time discretization schemes, which can be applied on a wide range of time-dependent equations. Numerical examples are given, also in the variable-coefficient setting, to demonstrate the effectiveness of our proposed preconditioners, which consistently outperforms an existing block circulant preconditioner discussed in the relevant literature. [ABSTRACT FROM AUTHOR]

We develop structure-preserving incomplete LU type factorizations for preconditioning centrosymmetric matrices and use them to numerically solve centrosymmetric and nearly centrosymmetric linear systems arising from spectral methods for partial differential equations. Our algorithm builds in part on direct solution techniques previously developed for this type of linear systems, featuring double-cone factorizations. We illustrate our findings on discretizations of model problems involving the Poisson, diffusion, Helmholtz, and biharmonic equations in one, two, and three dimensions. [ABSTRACT FROM AUTHOR]

The method of Kerzman and Trummer (J. Comp. Appl. Math. 14, 111–123, 1986) for computing the Riemann mapping function of a smooth domain is extended to include the case of simply connected convex regions with corners, in particular convex polygons. The connection between the Szegö kernel and the Riemann mapping function is classical. The integral equation in Kerzman and Trummer (J. Comp. Appl. Math. 14, 111–123, 1986) that determines the Szegö kernel is no longer defined in the presence of corners. We modify the equation by using new oblique projections. This approach is equivalent to employing preliminary mappings. [ABSTRACT FROM AUTHOR]