The basic idea in linear operator approximation is to apply the exact operator to an approximation of the input. These exercises explore and reinforce this concept.

1Track 1

1. Use Taylor series expansions to verify the approximations

   ${\displaystyle f^{\text{'}}\left(t\right)\cong \frac{1}{12h}[-f(t+2h)+8f(t+h)-8f(t-h)+f(t-2h)],}$ ${\displaystyle f^{\text{'}}\left(t\right)\cong \frac{1}{2h}[-3f\left(t\right)+4f(t+h)-f(t+2h)].}$

   Determine the error term. Construct the polynomial approximant $p_n\left(t\right)\cong f\left(t\right)$ whose derivative leads to the above formula. Conduct a convergence study as $h\to 0$ for $f\left(t\right)\in \{\sin (\pi t/4),e^{10t},e^{-10t}\}$ at $t_0=1$, and compare the observed order of convergence with the theoretical estimate.

2. As above for

   ${\displaystyle f^{\text{'}\text{'}}\left(t\right)\cong \frac{1}{12h^2}[-f(t+2h)+16f(t+h)-30f\left(t\right)+16f(t-h)-f(t-2h)],}$ ${\displaystyle f^{\text{'}\text{'}\text{'}}\left(t\right)\cong \frac{1}{h^3}[f(t+3h)-3f(t+2h)+3f(t+h)-f\left(t\right)].}$

3. Construct a recursive function RecInt(a,b,err,f,Q) that has arguments scalars $a,b,\mathrm{err}$ and functions $f,Q$ and approximates

   ${\displaystyle I\left(f\right)=\underset{a}{\overset{b}{\int }}f\left(t\right)dt}$

   through repeated application of quadrature rule $Q(f,a,b)$ according to the algorithm

   Algorithm

   Recursive quadrature

   RecInt(a,b,err,f,Q)

   $c=a+(b-a)/2$

   $Q_{\mathrm{ab}}=Q(f,a,b);Q_{\mathrm{ac}}=Q(f,a,c);Q_{\mathrm{cb}}=Q(f,c,b)$

   $e=|Q_{\mathrm{ac}}+Q_{\mathrm{cb}}-Q_{\mathrm{ab}}|/|Q_{\mathrm{ac}}+Q_{\mathrm{cb}}|$

   if $e<\mathrm{err}$

   return $Q_{\mathrm{ac}}+Q_{\mathrm{cb}}$

   else

   return RecInt(a,c,err,f,Q)+RecInt(c,b,err,f,Q)

   Test the recursive integration procedure with trapezoid, Simpson, and Gauss-Legendre rules of orders 2,3 on the integral

   ${\displaystyle \underset{-1}{\overset{1}{\int }}\cos \left(\frac{1}{t}\right).}$

   For each case, present plots of the integrand and the evaluation points used in the recursive quadrature algorithm. Construct convergence plots by executing the algorithm for various error thresholds ${\epsilon }_k$ and recording the number of evaluation points $n_k$. Plot $(\log n_k,\log {\epsilon }_k)$ and comment on whether the observed order of convergence is that predicted by theoretical quadrature error estimates.

2Track 2

1. Use the finite difference expressions of the derivative

   ${\displaystyle \frac{d}{dt}=\frac{1}{h}(\Delta -\frac{1}{2}{\Delta }^2+\dots )=\frac{1}{h}(\nabla +\frac{1}{2}{\nabla }^2+\dots )=\frac{1}{h}(\delta -\frac{{\delta }^3}{24}+\cdots )}$

   to obtain the approximations.

   ${\displaystyle f^{\text{'}}\left(t\right)\cong \frac{1}{12h}[-f(t+2h)+8f(t+h)-8f(t-h)+f(t-2h)],}$ ${\displaystyle f^{\text{'}}\left(t\right)\cong \frac{1}{2h}[-3f\left(t\right)+4f(t+h)-f(t+2h)].}$

   Conduct a convergence study as $h\to 0$ for $f\left(t\right)\in \{\sin (\pi t/4),e^{10t},e^{-10t}\}$ at $t_0=1$, and compare the observed order of convergence with theoretical estimates. How do the three functions differ, and what effect does this have on derivative approximation?

2. As above for

   ${\displaystyle f^{\text{'}\text{'}}\left(t\right)\cong \frac{1}{12h^2}[-f(t+2h)+16f(t+h)-30f\left(t\right)+16f(t-h)-f(t-2h)],}$ ${\displaystyle f^{\text{'}\text{'}\text{'}}\left(t\right)\cong \frac{1}{h^3}[f(t+3h)-3f(t+2h)+3f(t+h)-f\left(t\right)].}$

   Use the series products

   ${\displaystyle \frac{d^2}{d{t}^2}=\frac{d}{dt}\frac{d}{dt}=\frac{1}{h^2}(\Delta -\frac{1}{2}{\Delta }^2+\dots )(\Delta -\frac{1}{2}{\Delta }^2+\dots ).}$

3. Romberg integration is a combination of trapezoid quadrature over decreasing subintervals and Aitken extrapolation. Implement Romberg integration and test on

   ${\displaystyle \underset{0}{\overset{1}{\int }}{e}^t\cos (\pi t)dt.}$

   Present a convergence sutdy. What is the observed order of convergence?