Question

For this problem, let ke = max{2,the largest even number in your student number}, ko = max{3,the largest odd number in your student number} So, e.g., if your student number is 5135731, then ke = 2 and ko = 7. The data in Table 2 represents points (xi,yi) sampled from an experimentally generated triangular wave function with period 2π.
(a) Use the least-squares technique we developed in class to estimate the coefficients a, b and c for the optimal least-squares fit of the data points to the function yf(x) = acos(x) + bcos(kox) + ccos(kex)
(b) The repeating triangular wave function can be expressed as y(x) =(2x−π π 0 ≤ x ≤ π3 π−2x π π ≤ x ≤ 2π Using the inner product hf(x),g(x)i = 1 πR2π 0 f(x)g(x) dx, compare the least-squares coefficients you have obtained with the inner productshy(x),cos(nx)i, for suitable choices of n. How do they compare? What can you conclude?
x y 0.00000 -0.96468 0.50000 -0.64637 1.00000 -0.32806 1.50000 -0.00975 2.00000 0.30856 2.50000 0.62687 3.00000 0.94518 3.50000 0.80715 4.00000 0.48884 4.50000 0.17053 5.00000 -0.14778 5.50000 -0.46609 6.00000 -0.78440

Answer 1

Given that

we have to Use the Least-squares technique we developed in class to estimate the coefficients a,b and c for the optimal least -squares fit of the data points to the function

y_f(x) = a cos(x) + b cos(k_ox) + c cos(k_ex)