NUMERICAL COMPUTATION Homework 7

Description

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 x_i in this vector, compute y_i = f (x_i). You should now have 7 pairs (x_i; y_i). 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 p_p over the domain [ 5; 5]. Does this look like a good approximation? Explain your assessment.

2. Generate testing data: Create a new vector with 101 evenly spaced points in [ 5; 5]. For each point x⁰_i, compute y_i⁰ = f (x⁰_i). Report the mean ((np.)mean) and standard deviation ((np.)std) from the set of points y₁⁰ ; : : : ; y₁₀₁⁰ .

2. 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

If you’re wondering how to compute p₆(x⁰_i), look up (np.)polyval and use the coe cients you computed in Step 2. You’re evaluating the polynomial model’s prediction of f (x⁰_i).

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?

5. 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.

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