Question

4-0 2

a) Find False Position function for this data.

b) Find the third-order interpolation function with Lagrange method

c) Find the third-order interpolation function with Newton's Divided Difference Method.

d)Find the natural spline interpolation function for the same data

e)Draw the given points in a row using the False Position function, the third order polynomial obtained by Lagrange and Newton's Divided Difference Method, and the natural spline interpolation function using MATLAB.

Answer 1

From Lagrange interpolation formula f(x) = (x - x_1)(x - x_2)(x - x_3)/(x_0 - x_1)(x_0 - x_2)(x_0 - x_3) f(x_0) + (x - x_0)(x - x_2)(x - x_3)/(x_1 - x_0)(x_1 - x_2)(x_1 - x_3) f(x_1) + (x - x_0)(x - x_1)(x - x_3)/(x_2 - x_0)(x_2 - x_1)(x_2 * x_3) f(x_2) + (x -x_0)(x - x_1)(x - x_2)/(x_3 - x_0)(x_3 - x_1)(x_3 - x_2) f(x_3) = (x - 3)(x - 4)(x - 7)/(1 - 3)(1 - 4)(1 - 7) 2 + (x - 1)(x - 4)(x - 7)/(3 - 0)(3 - 4)(3 - 7) 1 + (x - 1)(x - 3)(x - 7)/(4 - 1)(4 - 3)(4 - 7) 0 + (x - 1)(x - 3)(x - 4)/(7 - 1)(7 - 3)(7 - 4)^3 \r\n = (x - 4)[-4(x^2 - 7x + 21) + 6(x^2 - 8x + 7) + 3(x^2 - 4x + 3)]/72 (x - 4)[-4x^2 + 6x^2 + 3x^2 + x(40 - 48 - 12) - 84 + 42 + 9]/72 = 5x^3 - 20x^2 - 33x - 20x^2 + 80x + 132/72 f(x) = 5x^3 - 40x^2 + 47x + 132/72 This is the third order interpolation function with lagrange method By Newton's divided difference formula we have f(x) = 2 + (x - 1)(-1/2) + (x - 1)(x - 3)(-1/6) + (x - 1)(x - 3)(x - 4)1/9