Question

3. Determine (analytically) the spacing h in a table of evenly spaced values of the function f(x)vx between 1 and 2, so that interpolation with a) degree polynomial b) 2"d degree polynomial in this table will yield a desired accuracy of at least N places after zero. Choose your own N.

Answer 1

For 1st degree poly n = 1 x_0 = 0 x_1 = 1 y_0 = 0 y_1 = squareroot 1 = 1 here n = 1 The approximation is f(x) = f(x_0) + 4 delta F(x_0) delta f(x_0) = y_1 - y_0 = 1, u = x - x_0/h = x - 0/1 = x/1 f(x) = 0 + x1 = x. which is the linear approximation. After decimal it will not meet any kind of accuracy. For n = 2, x_2 = 2 y_2 = squareroot 2 The approximation is f(x) = f(x_0) + u delta f(x_0) + 4(4 - 1)/2! Delta^2 f(x_0) delta^2 f(x_0) = delta } f(x_1) - f(x_0) = {f(x_2) - f(x_1)} - } f(x_1) - f(x_0)} = f(x_2) - 2f (x_1) + f(x_0) = squareroot 2 - 2 times 1 = squareroot 2 - 2 4 = x - 0/1 = x f(x) = 0 + x + x(x - 1)/2 (squareroot 2 - 2) = x(1 - 1/2 (squareroot 2 - 2)) + x^2/2 (squareroot 2 - 2)