求解病态线性方程组的共轭向量基算法

结合最速下降法计算量小和共轭方向法收敛速度快的特点,提出了一种求解病态方程组的共轭向量基的方法。线性方程组的精确解能够由共轭向量基线性表示,利用迭代的方式给出了构造共轭向量基以及对应系数的方法,证明了算法所构造的向量基的共轭性。同时给出了一个改进算法以适合不同精度要求,加快迭代的收敛速度。通过对5000阶的Hilbert方程组进行求解,结果的相对误差小于0.45％,并与当前普遍使用有效的方法进行了比较,数值实验结果表明,该算法适合求解大型病态线性方程组,且具有快速收敛,精度较高的特性。

Conjugate vector base algorithm for solving ill-conditioned linear equations

The characteristics of the steepest descent method’s small amount of computations and conjugate direction method’s fast convergence combined, a conjugate vector base method for solving ill-conditioned linear equations was proposed. The accurate solution of linear equations could be expressed linearly by the conjugate vector base, and the iterative strategy was used to construct the conjugate vector base groups and the corresponding coefficient. The constructed conjugate vector base groups were proved to be conjugated. Meanwhile, an improved algorithm fitting different required precisions was also given, which can accelerate the iterative convergence. 5000-order Hilbert ill-conditioned linear equations were solved, and the relative error was less than 0.45％. Numerical experiments verified that the method was efficient compared with the efficient methods used commonly nowadays. Numerical experiments results showed that the conjugate vector base method was suited for solving large-scale ill-conditioned linear equations with fast convergence and high precision.