Numerical Analysis I

Problems

1. Use the general 3-point formula derived in class to show that when $x_0, x_1, x_n$ are equally spaced nodes we obtain the following: \begin{align*} f'(x_0) &= \frac{1}{2h}\left[-3f(x_0) + 4f(x_1) - f(x_2)\right] + \frac{h^2}{3} f^{(3)}(\xi_0)\\[2ex] f'(x_1) &= \frac{1}{2h}\left[ f(x_2) - f(x_0)\right] - \frac{h^2}{6} f^{(3)}(\xi_1) \\[2ex] f'(x_2) &= \frac{1}{2h}\left[f(x_0)-4f(x_1)+3f(x_2)\right] + \frac{h^2}{3} f^{(3)}(\xi_2) \end{align*}
2. Write a Python function called ThreePointDiff that takes as input an array $y=(f(x_0), f(x_1), \ldots,$ $f(x_n))$ and a step-size $h$, and returns estimates for the derivatives $y'_j=f'(x_j)$ using 3-point formula estimates. For the end-point derivatives $y'_0$ and $y'_n$ use the appropriate end-point formula, and for the remaining $n-2$ derivatives use the 3-point mid-point formula.

3. Consider the following data:

   $x$	$f(x)$
   −1.5	1.0
   −1.0	0.0
   −0.5	−1.0
   0.0	2.0
   0.5	4.0

   Use the 3-point formulas to estimate $f'(x)$ for each $x$. For the end-point derivatives use the appropriate end-point formula, and for the remaining derivatives use the 3-point mid-point formula. Do this by hand for practice, and then use your ThreePointDiff function to check your answers.

4. Download the file xy-data.txt. The file is a text file containing two columns where the first column are the values \(x=(x_0, x_1, \ldots, x_n)\) and the second column are the values \(y=(y_0, y_1, \ldots, y_n)\). Download the file, save it in an appropriate directory, and then in Python navigate to this directory. Load the contents of the data file and save it to the variable z as follows:

   z = np.loadtxt( 'xy-data.txt' )

   Use your ThreePointDiff function to compute estimates \(\hat{y}'_j\) for the derivatives \(y'_j = f'(x_j)\) for \(j=0,1,\ldots, n\). The value of \(h\) can be determined from the \(x\) data; notice that the nodes are equally spaced.

   1. Plot the \((x_j,y_j)\) data and the data \((x_j, \hat{y}'_j\) on the same axis. Using your knowledge from calculus, comment on how well the derivatives are computed numerically.
   2. Plot the Hermite polynomial \(H(x)\) interpolating the data set \(\{(x_j, y_j, \hat{y}'_j)\}_{j=0}^n\) on a set of grid points \(u=(u_0,u_1,\ldots,u_N)\) where \(u_0=x_0\) and \(u_N = x_n\). Pick a value of \(N\) so that \(N > n\).