Implementation of Adaptive Gauss-Kronrod Quadrature in Octave and Python

John Bryan

• Setup
  
  
  • Adaptive GK(15,7) quadrature implemented.
    
    
  • The implementation uses
    
    
    • K15 interlacing zeros of $E_{8}$ and $L_{7}$ as nodes.
      
      
    • G7 using zeros of $L_{7}$ as nodes.
      
      
    • $\lvert K15-G7 \rvert$ used as error estimate.
      
      
  • a=lower limit of integration.
    
    
  • b=upper limit of integration.
    
    
  • n=number of nodes.
    
    
  • $x_{k,n}$ = kth node of the n nodes.
    
    
  • $w_{k,n}$= kth weight of the n weights.
    
    
  • c=(b-a)/2
    
    
  • d=(b+a)/2
    
    
  \begin{equation} \int_a^b f(x) dx \approx c\sum_{k=1}^n w_{k,n}f(cx_{k,n}+d) \end{equation}
• Octave implementation.
  
  
• Python implementation.
  
  
• Results.
  
  

  $\int_0^{50} sin(x) dx$
  Exact	3.503397150788667e-02
  Adaptive GK(15,7)	3.503397150788645e-02
  Adaptive GK(15,7) difference from exact	2.220446049250313e-16
  eps	2.220446049250313e-16