Question

For this problem take f(x) = cos(2x 2 ), and consider n − 1-degree polynomial interpolation. Also recall that the Chebyshev interpolation nodes over the interval [−1, 1] are given by the formula xj = cos ((2j−1)π /2n ) for j = 1, 2, . . . n. Find an upper bound for the error for x ∈ [−1, 1] when the Chebyshev nodes are used with n = 6.

Answer 1

The student is asking for the upper bound of the error when interpolating the function cos(2x^2) using Chebyshev nodes with n=6. While the interpolation error formula for Chebyshev nodes is provided, the sixth derivative of the given function is required to calculate the upper bound. The error is dependent on both the nth derivative of the function and the number of nodes used.

Step-by-step explanation:

The question is asking for an upper bound for the error when using Chebyshev nodes for polynomial interpolation of the function f(x) = cos(2x2) with n = 6. The Chebyshev nodes are given by xj = cos((2j-1)π/2n) for j = 1, 2 … n. The general form of the interpolation error when using Chebyshev nodes is En(x) ≤ (1/(2n-1 * n!)) * max|f(n)(θ)| over θ ∈ [-1, 1], where f(n)(x) is the n-th derivative of the function being interpolated.

To find the upper bound for the error, we must firstly determine the max absolute value of the n-th derivative of f(x) within the interval [-1, 1]. Since the question does not provide a formula for the nth derivative of f(x), a specific error bound cannot be calculated without this information. However, the approach would involve taking the sixth derivative of f(x), evaluating its maximum absolute value within the interval, and then applying the formula mentioned above with n = 6.