Implementation of Chebyshev Sine Approximation in Octave

John Bryan

• Equations.
  
  Chebyshev polynomial approximation, \begin{equation} f(x) \approx P_n(x) = \sum_{j=0}^n c_jT_j(2\frac{x-a}{b-a}-1) \tag{eq. 1} \end{equation} for x $\in$ [a,b].
  
  Polynomial node locations, \begin{equation} x_k=\frac{b+a}{2}+\frac{b-a}{2}\cos(\frac{2k+1}{2n+2}\pi) \tag{eq. 2} \end{equation} for k=0,1,2,...,n.
  
  Coefficients, \begin{equation} c_0=\frac{1}{n+1}\sum_{k=0}^nf(x_k). \tag{eq. 3} \end{equation} \begin{equation} c_j = \frac{2}{n+1}\sum_{k=0}^nf(x_k)\cos(j\frac{2k+1}{2n+2}\pi) \tag{eq. 4} \end{equation} for j=1,2,...n.
  
  Chebyshev polynomials,
  \begin{equation} T_i(z) = 2xT_{i-1}(z)-T_{i-2}(z) \tag{eq. 5} \end{equation} for i=2,3,...n, with $ T_0(z)=1 $ and $T_1(z)=z$.
  
  
  Approximation error upper bound, \begin{equation} |f(x)-P_n(x)| \le \left(\frac{b-a}{2}\right)^{n+1}\frac{B_{n+1}}{2^n(n+1)!}, \tag{eq. 6} \end{equation} with $B_{n+1}=\text{max}_{x\in[a,b]}|f^{(n+1)}(x)|$.
  
  
• Octave implementation.
  
  
• Results.
  $$ f(x)=\sin(x), x \in [0,\pi/2] $$ $$ \begin{array}{c|lcr} \text{n=7 theoretical approximation error upper bound} & 2.80 \times 10^{-8} \\ \text{n=7 Simulation approximation maximum error} & 2.11 \times 10^{-8} \\ \text{n=11 theoretical approximation error upper bound} & 5.61 \times 10^{-14} \\ \text{n=11 Simulation approximation maximum error} & 4.19 \times 10^{-14} \\ \end{array} $$