基于QR分解求解带顶点三对角带状线性方程组

Solution of Tridiagonal Banded Linear Equation Set with Diagonal Points Based on QR Decomposition

带顶点三对角带状线性方程组在实际问题的求解过程中经常遇到 ,一般情况下此类方程组没有实用有效的求解方法 .与现有一般基于LU分解的或其他一些迭代方法不同 ,基于实际很少采用的矩阵QR分解方法 ,利用其对各类矩阵普遍适用的优点 ,给合此类带状线性方程组的特点 ,提出并探讨了将QR分解应用于该类方程组的求解过程 ,既利用了QR分解保证足够的精度 ,又避免了一般QR分解过大的计算量 .分析和实际计算均表明 ,该方法在计算精度及计算量方面均满足实际应用的要求 .

A characteristic linear equation set is a tridiagonal banded linear equation set with two diagonal points. Generally, it is very difficult to solve the equation set perfectly with methods of linear algebra. Different from the algebra based on LU decomposition and iterative algebra, a solution of the equation set through QR decomposition has been put forward and discussed. QR decomposition is ubiquitous but used few, because of its vast amount of computation. Tridiagonal banded linear equation set with two diagonal points has its characteristic, so QR decomposition can be used to solve the linear equation set with satisfactory precision and economic amount of computation. Analysis and example show that the proposed algebra meets the requirements on precision and computation amount in application.