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: $$\begin{array}{rl}{f}^{\prime }(x_0)& =\frac{1}{2h}[-3f(x_0)+4f(x_1)-f(x_2)]+\frac{h^2}{3}{f}^{(3)}({\xi }_0)\\ f^{\prime }(x_1)& =\frac{1}{2h}[f(x_2)-f(x_0)]-\frac{h^2}{6}{f}^{(3)}({\xi }_1)\\ f^{\prime }(x_2)& =\frac{1}{2h}[f(x_0)-4f(x_1)+3f(x_2)]+\frac{h^2}{3}{f}^{(3)}({\xi }_2)\end{array}$$
2. Write a Python function called ThreePointDiff that takes as input an array $y=(f(x_0),f(x_1),\dots ,$ $f(x_n))$ and a step-size $h$, and returns estimates for the derivatives $y_j^{\prime }=f^{\prime }(x_j)$ using 3-point formula estimates. For the end-point derivatives $y_0^{\prime }$ and $y_n^{\prime }$ 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 $f^{\prime }(x)$ 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 $x=(x_0,x_1,\dots ,x_n)$ and the second column are the values $y=(y_0,y_1,\dots ,y_n)$. 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 ${\hat{y}}_j^{\prime }$ for the derivatives $y_j^{\prime }=f^{\prime }(x_j)$ for $j=0,1,\dots ,n$. 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 $(x_j,{\hat{y}}_j^{\prime }$ 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 $\{(x_j,y_j,{\hat{y}}_j^{\prime }){\}}_{j=0}^n$ on a set of grid points $u=(u_0,u_1,\dots ,u_N)$ where $u_0=x_0$ and $u_N=x_n$. Pick a value of $N$ so that $N>n$.