解线性代数方程组的二次PE方法和二次PE_k方法

Quadratic PE and PE_k Methods of Solving a System of Linear Algebraic Equations

建立了求解系数矩阵为大型分块三对角矩阵的线性代数方程组的二次 PE方法和二次 PEk方法。对系数矩阵为 Hermite正定矩阵的情形 ,通过研究迭代矩阵的拟三角分解与特征值表示 ,证明了二次 PE方法和二次 PEk

In a paper presented at AIAA 3rd Computational Fluid Dynamics, Helliwell proposed the PE(Pseudo Elimination)method of solving a system of linear algebraic equations with tridiagonal matrix discretized from certain partial differential equations . Under the condition that the system of linear algebraic equations has Hermitian positive definite matrix or diagonal dominance matrix,Hu proved convergence results about linear PE method and linear PE k method . We propose improving convergence speed by replacing linear approximation with quadratic approximation. We propose quadratic PE and PE k methods for solving a system of linear algebraic equations with large scale blocked tridiagonal matrix. Through quasi tridecomposition and eigenvalue analysis of iterative matrix, we proved the solvability and convergence of the quadratic PE and quadratic PE k methods when the coefficient matrix is Hermitian positive definite matrix. Numerical experiments show preliminarily that the convergence speed of quadratic PE method is remarkably higher than that of linear PE method. Although in each iterative step,computing time of the quadratic PE method is a little higher than that of the linear PE method,the total computing time of quadratic PE method is only about one third of that of the linear PE method.