Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.

Description of Students Thinking on Warshall-Floyd Algorithm

<em>The shortest path continues to be a trend until now that is always discussed and developed. This study focuses on the construction process and description of the students' understanding in deciding the shortest route based on the matrix iteration according to the Floyd-Warshall algorithm. The research approach used is descriptive qualitative research and the data collection technique is a test technique. The research data included problem-solving by four students with different abilities and the result data were analyzed inductively. The results showed that the matrix iteration with the formula for determining the entry of the iteration matrix x_n = minimum(d_ijk-1, d_ikk-1+ d_kjk-1) was followed without any constraints by all students until the 7th iteration. It was found that the inaccuracy of taking entries in the 1st iteration by students with low ability caused calculation errors and failure in finding the shortest route.</em>

Constructing a High-Order Globally Convergent Iterative Method for Calculating the Matrix Sign Function

This work is concerned with the construction of a new matrix iteration in the form of an iterative method which is globally convergent for finding the sign of a square matrix having no eigenvalues on the axis of imaginary. Toward this goal, a new method is built via an application of a new four-step nonlinear equation solver on a particulate matrix equation. It is discussed that the proposed scheme has global convergence with eighth order of convergence. To illustrate the effectiveness of the theoretical results, several computational experiments are worked out.

Numerical method to a class of boundary value problems

A class of boundary value problems can be transformed uniformly to a least square problem with Toeplitz constraint. Conjugate gradient least square, a matrix iteration method, is adopted to solve this problem, and the solution process is elucidated step by step so that the example can be used as a paradigm for other applications.

Solving a class of boundary value problems by LSQR

Boundary value problems arising in fluid mechanics and thermal science can be transformed uniformly to a set of linear equations, whose coefficient matrix is circulant. This paper adopts a matrix iteration LSQR to solve the inverse of coefficient matrix. The solution process is elucidated step by step, and the numerical results reveal the effectiveness and feasibility of the presented method.