Implementation of Adaptive Fejer First Quadrature Rule in Octave

John Bryan

• Setup
  
  
  • Adaptive Fejer First Quadrature Rule implemented.
    
    
  • Chebyshev polynomial zeros used as nodes.
    
    
  • a=lower limit of integration.
    
    
  • b=upper limit of integration.
    
    
  • n=number of nodes.
    
    
  • $x_{k,n}$ = kth node of the n nodes \begin{equation} x_{k,n} = \cos \Bigl( \frac{(2k+1) \pi}{2n}\Bigr) \end{equation}
    
    
  • $w_{k,n}$= kth weight of the n weights.
    
    \begin{equation} w_{k,n} = \int_{-1}^{1} p_{k}(x) dx \end{equation}
    
    \begin{equation} p_{k}(x) = \frac{\displaystyle \prod_{\substack{j=1 \\ j \neq k}}^{n}(x-x_{j})}{\displaystyle \prod_{\substack{j=1 \\ j \neq k}}^{n}(x_{k}-x_{j})} \end{equation}
    
    
  • c=(b-a)/2
    
    
  • d=(b+a)/2
    
    
  • Quadrature approximation, \begin{equation} \int_a^b f(x) dx \approx c\sum_{k=1}^n w_{k,n}f(cx_{k,n}+d) \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.
    
    
  • Continue to calculate integration approximations over smaller regions until the absolute value of the difference between two successive level approximations is smaller than a chosen tolerance.
    
    
• Octave implementation.
  
  
• Results.
  
  

  $\int_0^{50} sin(x) dx$
  Exact	3.503397150788667e-02
  Adaptive Fejer First Quadrature Rule	3.503397150788645e-02
  Adaptive Fejer First Quadrature Rule difference from exact	2.220446049250313e-16
  eps	2.220446049250313e-16