Posted: 11/17/21

Due: 11/24/21, 11:55PM

1Track 1

1. Review the construction of Fig.01 and Fig. 02 in L02. Construct convergence plots for the approximation of $f^{\text{'}}(\pi /4)$, $f\left(x\right)=\sin \left(\cos \left(x\right)\right)+\cos \left(\sin \left(x\right)\right)$ using truncations to orders $𝒪\left(h^p\right)$, $p=1,2,3,4$ using the forward finite difference operator $\Delta f\left(x\right)=f(x+h)-f\left(x\right)$

   ${\displaystyle \frac{d}{dt}=\frac{1}{h}(\Delta -\frac{1}{2}{\Delta }^2+\frac{1}{3}{\Delta }^3+\frac{1}{4}{\Delta }^4-\dots ).}$

   Solution.

   $\circ$Define finite difference operators

   In[1]:=

   $Assumptions = h > 0;

   ${\displaystyle \text{Null}}$

   In[2]:=

2. Use the polynomial interpolation error formula to find the error

   ${\displaystyle e=|I\left(f\right)-Q\left(f\right)|=\frac{{(b-a)}^3}{12}{f}^{\text{'}\text{'}}\left(\xi \right),a<\xi <b,}$

   of the trapezoid quadrature rule

   ${\displaystyle I\left(f\right)=\underset{a}{\overset{b}{\int }}f\left(t\right)dt\cong Q\left(f\right)=\frac{b-a}{2}(f\left(a\right)+f\left(b\right)).}$

   Construct a convergence plot of the error

   ${\displaystyle e\left(h\right)=\sum _{k=1}^n|\underset{x_{k-1}}{\overset{x_k}{\int }}f\left(t\right)dt-\frac{h}{2}(f_{k-1}+f_k)|,}$

   with $h=1/n$, $f_k=f\left(x_k\right)$, $x_k=kh$ in approximating the integral

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

3. Repeat the above for the Simpson rule

   ${\displaystyle Q\left(f\right)=\frac{b-a}{6}(f\left(a\right)+4f\left(\frac{a+b}{2}\right)+f\left(b\right)).}$

4. Extra credit (3 points). Repeat the above for Gauss-Legendre quadrature of orders 2 and 3. Note that each subinterval over which you apply a Gauss quadrature rule has to be mapped to [-1,1].

2Track 2

1. The expansion of the differentiation operator in terms of the forward finite difference operator arise from

   ${\displaystyle \frac{d}{dt}=\frac{1}{h}\ln (E+\Delta )}$

   and leads to the series

   ${\displaystyle \frac{d}{dt}=\frac{1}{h}(\Delta -\frac{1}{2}{\Delta }^2+\frac{1}{3}{\Delta }^3+\frac{1}{4}{\Delta }^4-\dots ).}$

   Deduce the analogous series that arises from

   ${\displaystyle \frac{d}{dt}=\frac{2}{h}\mathrm{arcsinh}\left(\frac{\delta }{2}\right),}$

   with $\delta$ the central finite difference operator

   ${\displaystyle \delta f\left(x\right)=\frac{f(x+h/2)-f(x-h/2)}{2}.}$

   Construct convergence plots for the approximation of $f^{\text{'}}(\pi /4)$, $f\left(x\right)=\sin \left(\cos \left(x\right)\right)+\cos \left(\sin \left(x\right)\right)$ using $p=1,2,3$ terms from both above series. Compare with the behavior of Fig. 02, L02, and comment.

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

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

   What is the observed order of convergence?

3. Review Lesson 37, “From Lanczos to Gauss Quadrature” of Trefethen & Bau. Compute the roots of the first 10 Legendre polynomials by finding the eigenvalues of the associated Jacobi matrices.

4. Extra credit (3points). 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 2,3,4 on the integral

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