Adaptive Simpson Octave Implementation

John Bryan

• Setup
  
  
  • a=lower limit of integration.
    
    
  • b=upper limit of integration.
    
    
  • $\text{h}_{1}=\frac{\text{b}-\text{a}}{2}$
    
    
  • $\text{h}_{2}=\frac{\text{h}_{1}}{2}$
    
    
  • \begin{equation} \text{S}_{1} = \frac{\text{h}_{1}}{3}(f(\text{a})+4f(\text{a}+\text{h}_{1})+f(\text{b})) \end{equation}
    
    \begin{equation} \text{S}_{2} = \frac{\text{h}_{2}}{3}(f(\text{a})+4f(\text{a}+\text{h}_{2})+2f(({a}+2\text{h}_{2})+4f(\text{a}+3\text{h}_{2})+f(\text{b})) \end{equation}
    
    \begin{equation} \text{Error}=\frac{|\text{S}_{2}-\text{S}_{1}|}{15} \end{equation}
    
    
  • Use recursion to successively subdivide (a,b) integration approximation into sum of (a,$\frac{b+a}{2}$) and ($\frac{b+a}{2}$,b) integration approximations. The tolerance is divided by 2 at each level.
    
    
  • Continue to calculate integration approximations over smaller regions until the error is smaller than level tolerance.
    
    
• Octave implementation.
  
  
• Results.
  
  

  $\int_0^{10} sin(x) dx$
  Exact	1.839071529076453
  Adaptive Simpson	1.839071529076453
  Difference from exact	0.000000000000000
  eps	2.220446049250313e-16