Residual (numerical analysis)

<h2 id="residual-of-the-approximation-of-a-function">Residual of the approximation of a function</h2>
<p>Similar terminology is used dealing with <a href="/facts/Differential_equation/9MGbE3Ca">differential</a>, <a href="/facts/Integral_equation/LCu4Imfq">integral</a> and <a href="/facts/Functional_equation/4ovU4CAt">functional equations</a>. For the approximation 
  
    
      
        
          f
          
            a
          
        
      
    
    {\displaystyle f_{\text{a}}}
  
 of the solution 
  
    
      
        f
      
    
    {\displaystyle f}
  
 of the equation
</p>

T
        (
        f
        )
        (
        x
        )
        =
        g
        (
        x
        )
        
        ,
      
    
    {\displaystyle T(f)(x)=g(x)\,,}

<p>the residual can either be the function
</p>

g
        (
        x
        )
         
        −
         
        T
        (
        
          f
          
            a
          
        
        )
        (
        x
        )
      
    
    {\displaystyle ~g(x)~-~T(f_{\text{a}})(x)}
  
,
<p>or can be said to be the maximum of the norm of this difference
</p>

max
          
            x
            ∈
            
              
                X
              
            
          
        
        
          |
        
        g
        (
        x
        )
        −
        T
        (
        
          f
          
            a
          
        
        )
        (
        x
        )
        
          |
        
      
    
    {\displaystyle \max _{x\in {\mathcal {X}}}|g(x)-T(f_{\text{a}})(x)|}

<p>over the domain 
  
    
      
        
          
            X
          
        
      
    
    {\displaystyle {\mathcal {X}}}
  
, where the function 
  
    
      
        
          f
          
            a
          
        
      
    
    {\displaystyle f_{\text{a}}}
  
 is expected to approximate the solution 
  
    
      
        f
      
    
    {\displaystyle f}
  
,
</p><p>or some integral of a function of the difference, for example:
</p>

∫
          
            
              X
            
          
        
        
          |
        
        g
        (
        x
        )
        −
        T
        (
        
          f
          
            a
          
        
        )
        (
        x
        )
        
          
            |
          
          
            2
          
        
         
        
          d
        
        x
        .
      
    
    {\displaystyle \int _{\mathcal {X}}|g(x)-T(f_{\text{a}})(x)|^{2}~\mathrm {d} x.}

<p>In many cases, the smallness of the residual means that the approximation is close to the solution, i.e., 
</p>

|
          
            
              
                
                  f
                  
                    a
                  
                
                (
                x
                )
                −
                f
                (
                x
                )
              
              
                f
                (
                x
                )
              
            
          
          |
        
        ≪
        1.
      
    
    {\displaystyle \left|{\frac {f_{\text{a}}(x)-f(x)}{f(x)}}\right|\ll 1.}

<p>In these cases, the initial equation is considered as <a href="/facts/Well-posed/U7J4Vde8">well-posed</a>; and the residual can be considered as a measure of deviation of the approximation from the exact solution.
</p>
<h2 id="use-of-residuals">Use of residuals</h2>
<p>When one does not know the exact solution, one may look for the approximation with small residual.
</p><p>Residuals appear in many areas in mathematics, including <a href="/facts/Iterative_solver/uKMAJhEh">iterative solvers</a> such as the <a href="/facts/Generalized_minimal_residual_method/Ak7gbYzI">generalized minimal residual method</a>, which seeks solutions to equations by systematically minimizing the residual.
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Shewchuk, Jonathan Richard (1994). "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" (PDF). p. 6. <a href="https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf#page=12" target="_blank">https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf#page=12</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Residual (numerical analysis) open-in-new

Residual (numerical analysis)