Modified Richardson iteration

Modified Richardson iteration is an <a href="/facts/Iterative_method/uKMAJhEh">iterative method</a> for solving a <a href="/facts/System_of_linear_equations/fk9CFdgp">system of linear equations</a>. Richardson iteration was proposed by <a href="/facts/Lewis_Fry_Richardson/kKQR6lFu">Lewis Fry Richardson</a> in his work dated 1910. It is similar to the <a href="/facts/Jacobi_method/aNf3WE63">Jacobi</a> and <a href="/facts/Gauss%25E2%2580%2593Seidel_method/eK94QeFt">Gauss–Seidel method</a>.
We seek the solution to a set of linear equations, expressed in matrix terms as

A
        x
        =
        b
        .
        
      
    
    {\displaystyle Ax=b.\,}

The Richardson iteration is

x
          
            (
            k
            +
            1
            )
          
        
        =
        
          x
          
            (
            k
            )
          
        
        +
        ω
        
          (
          
            b
            −
            A
            
              x
              
                (
                k
                )
              
            
          
          )
        
        ,
      
    
    {\displaystyle x^{(k+1)}=x^{(k)}+\omega \left(b-Ax^{(k)}\right),}

where 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 is a scalar parameter that has to be chosen such that the sequence 
 
 
 
 
 x
 
 (
 k
 )
 
 
 
 
 {\displaystyle x^{(k)}}
 
 converges.
It is easy to see that the method has the correct <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a>, because if it converges, then 
 
 
 
 
 x
 
 (
 k
 +
 1
 )
 
 
 ≈
 
 x
 
 (
 k
 )
 
 
 
 
 {\displaystyle x^{(k+1)}\approx x^{(k)}}
 
 and 
 
 
 
 
 x
 
 (
 k
 )
 
 
 
 
 {\displaystyle x^{(k)}}
 
 has to approximate a solution of 
 
 
 
 A
 x
 =
 b
 
 
 {\displaystyle Ax=b}
 
.

Modified Richardson iteration open-in-new

Modified Richardson iteration