NUMERICAL COMPUTATION Homework 5

Description

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:

	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 = 1.

1. Generate training data: Create a vector with n = 7 evenly spaced points in the interval [ 5; 5]. (In Matlab, you can do this with 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). Provide a printout of the 7 pairs (e.g., in a table).

2. Train the model: Construct the Vandermonde system and solve for the coe cients of the unique degree 6 interpolating polynomial p₆(x). Provide a printout of the 6 coe cients.

3. 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 (mean in Matlab) and standard deviation (std in Matlab) from the set of points y₁⁰; : : : ; y₁₀₁⁰.

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

error = error	= maximum				j	y0	p	(x0)	j
=1;n=7	1	i		101		i	6	i

Note that the error depends on the value of and the number of training points / the degree of the polynomial.

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 = 10 and n = 8; 9; : : : ; 15. Plot error _=10;n versus n on a semilog scale. How does the polynomial approximation converge as n increases?

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