解线性方程组的广义共轭梯度法的一种推广

解线性方程的广义共轭梯度法可以看成是一种Ｋｒｙｌｏｖ子空间的方法。本文从这点出发给出了ＧＣＧ法的一种推广。新方法所求得的近解能使得残量范数在相应的Ｋｒｙｌｏｖ子空间上取得最小值。在处理对称正定问题时，它等价于共轭残量法。但由于迭代过程中不再产生和存储Ａ－共轭向量，方法的实现更为简单。

A Variant of GCG for Solving Linear Systems

The generalized conjugate gradient (GCG) method for solving linear systems of algebraic equations can be viewed as a Krylov subspace method of the Galerkin type. With this view point, a variant of GCG is proposed, which can be viewed as a Krylov subspace method of the least squares type. It is equivalent to the conjugate residual method (CR) in the case of symmetric positive definite. The new method guarantees the approximate solution minimizes the residual norm over all candidate vectors in the associated Krylov subspace, however, there is no A conjugate vectors to be constructed and stored, so it may be easily performed with less cost.