鳞状因子循环矩阵方程解的条件与求解的快速算法

利用多项式快速算法,给出了鳞状因子循环矩阵方程AX=b可解的条件与求解的快速算法.当鳞状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;当鳞状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.该快速算法仅用到鳞状因子循环矩阵的第一行元素及对角矩阵中的对角上的常数进行计算.在计算机上实现时只有舍入误差.特别地,在有理数域上用计算机求得的结果是精确的.

The Fast Algorithms for Judging the Existence of Solution and the Solution of the Scaled Factor Circulant Matrix Equation

Fast algorithms for judging the existence of solution and the solution of scaled factor circulant matrices AX = b are presented by utilizing the fast algorithm for computing polynomials. When scaled factor circulant matrices are nonsingular, we can find the single solution of the scaled factor circulant matrix equation; When scaled factor circulant matrices are singular, we can find the special as well as the general solution of the scaled factor circulant matrix equation; There is only an error of approximation when the fast algorithm is implemented on computers, and only the elements in the first row of the scaled factor circulant matrix and the constants in the diagonal matrix are needed by the fast algorithm. Especially, the neumerical result would be accurate over the rational number field.