| With the development of the computers, numerical analysis concerned with the solution of linear systems of equations is always used in the practical and mathematic problems. For a linear system, we study the iterative method in this paper which is very useful to solve the large sparse coefficient matrix.The method is based on the splitting of the coefficient matrix and then we discuss the convergence of the iterative matrix to get the results. We will introduce the basic iterative methods, concluding Jacobi, Gauss-Seidel, Successive overrelaxation (SOR) and symmetric SOR (SSOR) iterative methods. Now we have many new methods, for example, accelerated overrelaxation, incomplete factorization and preconditioned methods. Those methods can be used in point and block iterative methods.In the electrical engineers, we always use a nonlinear system of ordinary differential equations (ODEs) to describe the behavior of the electrical circuit. Since how to solve the ODEs is very important to the simulation of very large scale integrated (VLSI) circuits. This paper studies the multi-splitting algorithm and the two-stage iterative methods for waveform relaxation (WR) methods, solving the initial value problems for linear systems of ordinary differential equations (ODEs) in the forms y ' (t ) +Ay(t ) =f(t ). And we will mainly discuss the coefficient matrix A and its splitting. The multi-splitting algorithm uses several splittings instead of one singular splitting, and through some algebra manipulations we will get a new better result.Two-stage iterative methods, which are also called inner/outer methods, split the matrix twice; i.e., let the splitting A = M-N2-N1. Research on the solution of those iterative methods has been done by some authors.Here, convergent theorems and comparison theorems of both cases are analyzed when the coefficient matrix is hermitian positive definite and the splitting is P-regular or hermitian P-regular. Some numerical examples are given. |
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