求解对称正定线性方程组的正交基变换方法

Orthogonal Basis Transformation Method for Solving System of Symmetric Positive Definite Linear Equations

求解对称正定线性方程组是线性代数和数值分析一项重要内容.通过证明对称正定线性方程组与函数逼近理论中正规方程组一一对应,将对称正定线性方程组类比为函数逼近理论中正规方程组,利用施密特正交化方法将对称正定线性方程组转化为对角方程组进行求解,提出并推导了求解对称正定线性方程组的正交基变换方法.数值算例表明该算法有效、可靠,且计算量小于平方根法.为求解对称正定线性方程组提供了新方法.

Solving system of symmetric positive definite linear equations is one of important content of NumericalAnalysis and Linear Algebra. The theorem that there are one-one corresponding relations in the symmetric positivedefinite linear equation and the normal equations in function approximation theory is proved. Orthogonal basistransformation method and its derivation are proposed in this paper. Symmetric positive definite linear equations arecast into diagonal linear system by using Gram-Schmidt method. Numerical examples show that the method iseffective and reliable;its amount of computations is less than Cholesky decomposition. A new method for solvingsystem of symmetric positive definite linear equations is provided in this paper.