Iterative refinement

<p>
Iterative refinement is an <a href="/facts/Iterative_method/uKMAJhEh">iterative method</a> proposed by <a href="/facts/James_H._Wilkinson/4aKUNcrg">James H. Wilkinson</a> to improve the accuracy of numerical solutions to <a href="/facts/Systems_of_linear_equations/fk9CFdgp">systems of linear equations</a>.
</p><p>When solving a linear system 
  
    
      
        A
        
          x
        
        =
        
          b
        
        
        ,
      
    
    {\displaystyle A\mathbf {x} =\mathbf {b} \,,}
  
 due to the compounded accumulation of <a href="/facts/Round-off_error/aF9N8X4I">rounding errors</a>, the computed solution 
  
    
      
        
          
            
              
                x
              
              ^
            
          
        
      
    
    {\displaystyle {\hat {\mathbf {x} }}}
  
 may sometimes deviate from the exact solution 
  
    
      
        
          
            x
          
          
            ⋆
          
        
        
        .
      
    
    {\displaystyle \mathbf {x} _{\star }\,.}
  
 Starting with 
  
    
      
        
          
            x
          
          
            1
          
        
        =
        
          
            
              
                x
              
              ^
            
          
        
        
        ,
      
    
    {\displaystyle \mathbf {x} _{1}={\hat {\mathbf {x} }}\,,}
  
 iterative refinement computes a sequence 
  
    
      
        {
        
          
            x
          
          
            1
          
        
        ,
        
        
          
            x
          
          
            2
          
        
        ,
        
        
          
            x
          
          
            3
          
        
        ,
        …
        }
      
    
    {\displaystyle \{\mathbf {x} _{1},\,\mathbf {x} _{2},\,\mathbf {x} _{3},\dots \}}
  
 which converges to 
  
    
      
        
          
            x
          
          
            ⋆
          
        
        
        ,
      
    
    {\displaystyle \mathbf {x} _{\star }\,,}
  
 when certain assumptions are met.
</p>

Iterative refinement open-in-new

Iterative refinement