二次PE方法的研究

【作者】 王自然；

【导师】 张凯院；

【作者基本信息】 西北工业大学 ， 计算数学， 2001， 硕士

【摘要】 在工程领域，如网络，控制等方面，许多实际问题都归结为解一个系数矩阵A为大型分块三对角矩阵的线性方程组 AX＝f解这类方程组通常用迭代法来求解；本文对这种类型系数矩阵的方程给出了一种迭代解法，主要讨论了如下几个问题： （1）在对系数矩阵实行不完全LU分解的基础上，导出了二次 PE方法，对系数矩阵A为正定矩阵，对角优势L-矩阵， 证明了二次PE方法的收敛性，并且在最后通过实例计算 说明了二次PE方法是一种收敛性比较好的方法。 （2）在二次PE方法的基础上，通过引入参数k，导出了二次 PE_k方法，并且对系数矩阵A为Hermite矩阵，M-矩阵及 H-矩阵证明了二次PE_k方法的收敛性；最后通过实例计算 说明了对适当的参数k，二次PE_k方法比PE方法及二次PE 方法收敛性都好。更多还原

【Abstract】 In the networks and controlling fields of engineering, many problenis can be described as solving a system of linear algebraic equationsAX=fWhose matrix of coefficients is large tridiagonal blocked matrix The usual method of solving this kind of equations is the iterative method Iii this paper, a new iterative method will be given. It is to be discussec as follows:(1)The quadratic PE method is derived on the base of incompletely LU triangular decomposition. Based on the results, the convergence of the quadratic PE method in positive definite matrix and diagomdlv dominant L matrix is proved, and the availability of the method is ~iIso illustrated .ln the end, some examples are given to illustrate the method(2)On the results of quadratic PE method, the parameter k is introduced and the convergence of the quadratic PE method is impro\ ed. The convergence of the quadratic PE k method is proved about the Hermite matrix, M-matrix and H-matrix. In chapter four, some examples are given to illustrate that the convergence of quadratic PEk method is better than the quadratic PE method to proper parameter k.更多还原