Numerical Computing :: Project Seven

Description

Consider the parameterized family of functions,

fθ(x) =	1		, x ∈ [−5, 5].
	1 + exp( θx)
		−

The parameter θ controls how smooth f_θ is near x = 0, as shown:

	1
	0.9
	0.8
	0.7
	0.6
( )X	0.5
θ
F
	0.4
	0.3
	0.2
			θ=1
	0.1		θ=2
			θ=10
	0
	-5	0	5
		X

To start this homework, let θ = 1.

1. Generate training data: Create a vector with n = 7 evenly spaced points in the interval [−5, 5]. For each point x_i in this vector, compute y_i = f_θ(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 coefficients of the unique degree-6 interpolating polynomialp₆(x). Make a nice table of the 7 coefficients. And make a plot showing both f_θ(x) and p₆(x) 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 x^′_i, compute y_i^′ = f_θ(x^′_i).

4. Compute the testing error: Compute and report the the absolute testing error:

error = errorθ=1,n=7	= maximum	| yi′ − p6(xi′ ) |
			|yi′ |
				i

If you’re wondering how to compute p₆(x^′_i), look up (np.polyval and use the coefficients you computed in Step 2. You’re evaluating the polynomial model’s prediction of f_θ(x^′_i).

5. Repeat steps 1-4 with θ = 10. How does the error change? What does that tell you about the quality of the polynomial approximation for the two functions?

2