Bartels–Stewart algorithm

<p>In <a href="/facts/Numerical_linear_algebra/jnS6Tteb">numerical linear algebra</a>, the Bartels–Stewart algorithm is used to numerically solve the <a href="/facts/Sylvester_equation/0MAnUp7F">Sylvester matrix equation</a> 
  
    
      
        A
        X
        −
        X
        B
        =
        C
      
    
    {\displaystyle AX-XB=C}
  
. Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first <a href="/facts/Numerical_stability/jRDdo8zV">numerically stable</a> method that could be systematically applied to solve such equations. The algorithm works by using the <a href="/facts/Schur_decomposition/EYNrqHmh">real Schur decompositions</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 to transform 
  
    
      
        A
        X
        −
        X
        B
        =
        C
      
    
    {\displaystyle AX-XB=C}
  
 into a triangular system that can then be solved using forward or backward substitution. In 1979, <a href="/facts/Gene_H._Golub/grRlsYzz">G. Golub</a>, <a href="/facts/Charles_F._Van_Loan/2wy1Avrl">C. Van Loan</a> and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving <a href="/facts/Sylvester_equation/0MAnUp7F"> Sylvester equations</a> when 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is of small to moderate size.
</p>

Bartels–Stewart algorithm open-in-new

Bartels–Stewart algorithm