Computer Engineering and Applications ›› 2015, Vol. 51 ›› Issue (5): 65-70.

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Iterative algorithm for optimal approximation reflexive solutions of matrix equations AXB+CYD=E

YANG Jiawen1, SUN Heming2   

  1. 1.Chuzhou Vocational and Technical College, Chuzhou, Anhui 239000, China
    2.College of Science, Hohai University, Nanjing 211100, China
  • Online:2015-03-01 Published:2015-04-08



  1. 1.滁州职业技术学院,安徽 滁州 239000
    2.河海大学 理学院,南京 211100

Abstract: The iterative algorithm can be used to calculate the optimal approximation reflexive solutions of the Sylvester matrix equations[AXB+CYD=E]by using the hybrid steepest descent method. But the convergent speed of the algorithm is very slow. So it presents an iterative algorithm by using the conjugate direction method. Whatever matrix equations [AXB+CYD=E]are consistent or not, for arbitrary initial reflexive matrix[X1]and[Y1], the optimal approximation reflexive solutions can be obtained within finite iteration steps by using the given algorithm. Two numerical examples show that the proposed algorithm is efficient, and the convergent speed is faster.

Key words: Sylvester matrix equations, Kronecker product, optimal approximation, reflexive matrix, conjugate direction

摘要: 利用复合最速下降法的迭代算法能够求出矩阵方程[AXB+CYD=E]的最佳逼近自反解,但其收敛速度很慢。针对这一问题,提出一种利用共轭方向法的迭代算法。对于任给初始自反矩阵[X1]和[Y1],无论矩阵方程[AXB+CYD=E]是否相容,该算法都可以经过有限次迭代计算出其最佳逼近自反解。两个数值例子表明该算法是可行的,且收敛速度更快。

关键词: Sylvester矩阵方程, Kronecker积, 最佳逼近, 自反矩阵, 共轭方向