关键词: 共轭向量基, 共轭方向法, 最速下降法, 病态线性方程组

Abstract:

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.

Key words: conjugate direction method

conjugate vector base