Numerical Analysis I

Problems

1. Consider the following data $(x_j,y_j)$, $j=0,1,2,3$:

   $x$	$y$
   -1	-1
   0	0
   3	-1
   5	-3

   1. Compute the divided-difference table associated to the given data.

   2. Write out the interpolating polynomial $P(x)$ of the given data in the divided-difference representation (but do not expand).

   3. Compute $P(2.0)$.

   4. Now suppose that we add the new data point $(x_4,y_4)=(4,-2)$. What is the divided-difference table associated to the extended data points $(x_j,y_j)$ for $j=0,1,2,3,4$?

   5. Compute $P(2.0)$, where $P$ is the interpolating polynomial for the extended data points.

2. Fill in the unknown entries in the divided-difference table given below:

   	0	1	2	3
   $x$	$f[\cdot ]$	$f[\cdot ,\cdot ]$	$f[\cdot ,\cdot ,\cdot ]$	$f[\cdot ,\cdot ,\cdot ,\cdot ]$
   $2$	$f[x_0]$	$\star$	$\star$	$\star$
   $3$	$f[x_1]$	$f[x_0,x_1]$	$\star$	$\star$
   $4$	$-7$	$4$	$2$	$\star$
   $5$	$1$	$f[x_2,x_3]$	$f[x_1,x_2,x_3]$	$f[x_0,x_1,x_2,x_3]$

   Using your results, write out the interpolating polynomial in expanded form.

3. Write a Python function called DividedDiff whose inputs are Numpy arrays $x=(x_0,x_1,\dots ,x_n)$ and $y=(y_0,y_1,\dots ,y_n)$, and returns the divided-difference coefficients $a=(a_0,a_1,\dots ,a_n)$ as a Numpy array. To test your function, try it with the data from Problem 1. Only include your code and not the results of your tests.

4. Write a Python function called DividedDiffInterp whose inputs are Numpy arrays $x=(x_0,x_1,\dots ,x_n)$, $y=(y_0,y_1,\dots ,y_n)$, and $u=(u_0,u_1,\dots ,u_N)$, and returns the array $v=(v_0,v_1,\dots ,v_N)$, where $v_i=P(u_i)$ and $P$ is the interpolating polynomial for the $(x_j,y_j)$ data points. The polynomial $P$ is to be evaluated using the divided-difference representation. Your function should first call DividedDiff to compute the divided-difference coefficients $a$ and then evaluate $P$ using the divided-difference representation. To test your function, compare the results using your LagrangeInterp function on any given set of $x,y$, and $u$ data points (e.g., you could use the data from Homework 4 Problems 6 or 7, or the data from Problem 1 above). Only include your code and not the results of your tests.

5. Consider the following data $$\begin{array}{rl}x& =(0,1,2,3)\\ y& =(1,-1,-5,-17)\end{array}$$ and let $P(x)$ be the interpolating polynomial.

   1. Compute the divided-difference representation of $P(x)$ and then expand the polynomial in the form $$P(x)=c_3{x}^3+c_2{x}^2+c_{1}x+c_0.$$

   2. Compute the Lagrange form of $P(x)$ and then expand the polynomial in the same form as in part (a).

   3. Comment on what you obtained in parts (a) and (b).

6. Let $f(x)=\cos (2\pi x)$ for $x\in [0,1]$ and let $x_0,x_1,\dots ,x_n$ be the Chebyshev nodes in $[0,1]$. Let $u=(u_0,u_1,u_2,\dots ,u_N)$ be equally spaced points inside the interval $[0,1]$ where $N=200$. Evaluate the interpolating polynomial $P(x)$ for the data $(x_j,f(x_j))$ at the points $u$ in two different ways:

   1. Using your LagrangeInterp function.

   2. Using your newly created DividedDiffInterp function.

   Then do the following:

   1. Plot the error function $|f(u)-P(u)|$ for each case (1) and (2) on the same plot. Do this for $n=10$, $n=40$, and $n=90$ (or experiment with other values of $n$) but include only your plot for $n=90$.

   2. What happens to the error function for each case (1) and (2) as $n$ increases?