Lyapunov矩阵方程的一种数值解法及部分并行处理

A Numerical Solution of Lyapunov Matrix Equation and a Partly Parallel Treatment

本文设计了求解Lyapunov矩阵方程的一种新方法。所考虑的矩阵方程是 AX—XB=C(1)其中A,B,C分别是m×m,n×n和m×n的已知矩阵。 该方法首先是将系数矩阵A,B初等相似约化为三对角矩阵,即存在可逆矩阵U,V,使U^(-1)AU=A,V^(-1)BV=B,其中A,B为三对角矩阵。然后设计了矩阵方程AY—YB=C的公式解法,分三步: 1)求f(λ)=det(λI—A)的λ各次幂的系数a_0,…,a_m; 2)计算sum from i=1 to m (A_(m-i)-CB^(m-i)),f(B); 3)求解Y。解方程AY—YB=C的方法称为THR算法。 最后经逆变换获得原矩阵方程(1)的解X。 求解矩阵方程(1)的方法称为R—THR算法。该方法的计算量约为m^3+4/3n^3+7m^2n+5nm^2+m^2。 本文给出了R—THR的串行计算的数值例子,并给出了THR算法的并行计算格式。最后通过几种数值方法的比较,表明该方法是可行的,也是有效的。

This paper discusses the new method of solving Lyapunov matrix equation. The matrix equation isAX - XB = C (1)where A,B,C respectively refers to m × m, n × n, m × n as the known matrices.Firstly, the coefficient matrices A, B are reduced to the tridiagonal forms by elementary similarity transformation. That is, inverted matrices U, V satisfy U-1 AU = A, V-1 BV = B , in which A, B are tridiagonal matrices.Secondly, the new method provides formula solution of matrix equation AY - YB = C which falls into three steps:1)to find coefficients a0,..., am of λs each power in f(λ) =det (λ- A) .2)to calculate;3)to solve Y . The method of solving AY - YB = C is called the THR algorithm.Finally, the solution of original matrix equation (1) is obtained through inverse transformation.The method of solving matrix equation (1) is called the R-THR algorithm. Thenumber of multiplications is approximately given by m3 + 4/3n3 + 7m2n + 5nm2 + m2 .This paper presents several numerical examples of R-THR's sequential algorithm and parallel computing formula of THR. In comparison with several numerical mthods, this method proves feasible and efficient.