Question

Let f(x) = ln(x 2 ) − x 4 . Let l and S be the piecewise linear and Hermite cubic spline interpolants to f on n + 1 equally spaced points 1 = x0 < x1 < · · · < xN = 2. What value of n would you have to take to ensure that

(i) max16x62 |f(x) − l(x)| 6 10−6 ?

Answer 1

Without specific error bounds or derivatives, we cannot determine the exact value of n for interpolation error under 10^-6. Fundamental concepts include logarithmic properties and interpolation error behavior.

Step-by-step explanation:

The question involves determining the minimum number of equally spaced points (defined by the variable n) required for the piecewise linear interpolant l(x) and the Hermite cubic spline interpolant S(x) to approximate the function f(x) = ln(x2) - x4 such that the maximal absolute error over the interval [1, 2] is less than or equal to 10-6. This is a typical question in numerical analysis, where one seeks to control the interpolation error. To answer this question accurately, one would need to apply the error bounds for piecewise linear and cubic Hermite splines, which depend on the derivatives of the function being interpolated as well as the distances between the interpolating points. However, since the provided information does not include specific error bound formulas or the derivatives of f, we cannot determine the exact value of n. Nevertheless, the fundamental concepts in use include the properties of logarithmic functions and the behavior of interpolation error as the number of points is increased.