r-置换因子循环线性系统求解的快速算法

Fast Algorithm for Solution of r-Permutation Factor Circulant Linear Systems

在置换因子循环矩阵的基础上给出了r-置换因子循环矩阵的概念,得到以这类矩阵为系数的线性方程组AX=b有解的判定条件和快速算法。当r-置换因子循环矩阵非奇异时,该快速算法求出线性方程组的唯一解,即存在唯一的r-置换因子循环矩阵C∈PRCMn,使AX=b的唯一解是C第一列;当r-置换因子循环矩阵奇异时,该快速算法求出线性方程组的特解与通解,即存在唯一的r-置换因子循环矩阵H∈PRCMn及C∈PRCMn,使得C的第一列X1是AX=b的一个特解,而且X=X1+(I-H)Z是AX=b的通解,这里Z是任意的n维列向量。

In this paper,r-permutation factor circulant matrix is defined based on the permutation factor circulant matrix,and a fast algorithm for conditions of solution and solution of r-permutation factor circulant matrix linear equations AX = b are presented.When r-permutation factor circulant matrix are nonsingular,this algorithm computes the single solution of r-permutation factor circulant matrix linear equations,that is,there exists a unique r-permutation factor circulant matrix C∈PRCMn,which the only solution of AX = b is the first column of C;When r-permutation factor circulant matrix are singular,it computes the special solution and general of r-permutation factor circulant matrix linear equations,which there is a unique r-permutation factor circulant matrix H∈PRCMn and C∈PRCMn makes the first column X1 of C is a special solution of AX = b,but also the X = X1 ＋ （I-H） Z is the general solution of AX = b,here Z is an n arbitrary-dimensional column vectors.