I. Prove Theorem 6.3 by assuming that ∀x ∈ (a,b) the weight function ρ(x) 0.
II. Consider the Chebyshev polynomials of the firstkind.
(a) Show that they are orthogonal on [−1,1] with respect to the inner product in Theorem 6.3 with the weight function .
(b) Normalize the first three Chebyshev polynomialsto arrive at an orthonormal system.
III. Least-square approximation of a continuous function.Approximate the circular arc given by the equation√
y(x) = 1 − x2 for x ∈ [−1,1] by a quadratic polynomial with respect to the inner product in Theorem
6.3.
with Fourier expansion,
with normal equations,
IV. Discrete least square via orthonormal polynomials.Consider the example on the table of sales record in the notes.
(a) Starting from the independent list (1,x,x2), construct orthonormal polynomials by the GramSchmidt process using
N hu(t),v(t)i = Xρ(ti)u(ti)v(ti) (1)
i=1 as the inner product with N = 12 and ρ(x) = 1.
(b) Find the best approximation ˆϕ = such that for all bi ∈ R.
Verify that ˆϕ is the same as that of the example on the table of sales record in the notes.
x
0.0
0.5
1.0
1.5
2.0
2.5
3.0
y
2.9
2.7
4.8
5.3
7.1
7.6
7.7
x
3.5
4.0
4.5
5.0
5.5
6.0
6.5
y
7.6
9.4
9.0
9.6
10.0
10.2
9.7
x
7.0
7.5
8.0
8.5
9.0
9.5
10.0
y
8.3
8.4
9.0
8.3
6.6
6.7
4.1
(c) Suppose there are other tables of sales record inthe same format as that in the example . Values of N and xi’s are the same, but the values of yi’s are different. Which of the above calculations can be reused? Which cannot be reused? What advantage of orthonormal polynomials over normal equations does this reuse imply?
The first three problems weigh 6 points each while the last problem weighs 12 points.
2 C++ programming
Write a C++ function to perform discrete least square via normal equations. Your subroutine should take two arrays x and y as the input and output three coefficients a0,a1,a2 that determines a quadratic polynomial as the best fitting polynomial in the sense of least squares with the weight function ρ = 1.
.
