Turn in your own writeup that includes your code. List any resources you used including collaborating with others. Submit a PDF on Canvas by Friday, Oct. 22 at 5pm.
Consider the parameterized family of functions,
f (x) =
1
; x 2 [ 5; 5]:
1 + exp( x)
The parameter controls how smooth f is near x = 0, as shown:
To start this homework, let = 1.
1. Generate training data: Create a vector with n = 7 evenly spaced points in the interval [ 5; 5]. (Matlab/Numpy: (np.)linspace.) For each point xi in this vector, compute yi = f (xi). You should now have 7 pairs (xi; yi). Make a nice table with the seven input/output pairs.
2. Train the model: Construct the Vandermonde system and solve for the coe cients of the unique degree-6 interpolating polynomial p6(x). Make a nice table of the 7 coe cients. And make a plot showing both f and pp over the domain [ 5; 5]. Does this look like a good approximation? Explain your assessment.
3. Generate testing data: Create a new vector with 101 evenly spaced points in [ 5; 5]. For each point x0i, compute yi0 = f (x0i). Report the mean ((np.)mean) and standard deviation ((np.)std) from the set of points y10 ; : : : ; y1010 .
4. Compute the testing error: Compute and report the the absolute testing error:
error = error =1;n=7
= maximum
j yi0
p6(xi0 ) j
jyi0 j
i
1
If you’re wondering how to compute p6(x0i), look up (np.)polyval and use the coe cients you computed in Step 2. You’re evaluating the polynomial model’s prediction of f (x0i).
5. Repeat steps 1-4 with = 10. How does the error change? What does that tell you about the quality of the polynomial approximation for the two functions?
6. EXTRA CREDIT (15pts): Repeat steps 1-4 with both = 1 and = 10 for n = 8; 9; : : : ; 15. Plot error versus n on a semilog scale. Describe the convergence (that is, at what rate the error goes to zero) as n increases.
