Consider y = f(x) = x3 over the range 1≤x≤3. Usethethreenodesx0 =1,x1 =2 and x3 = 3 to construct a quadratic interpola- tion polynomial P2(x) using the Lagragian in- terpolation. P2 (x) =
To construct a quadratic interpolation polynomial using Lagrange interpolation, we need to find the Lagrange basis polynomials L0(x), L1(x), and L2(x) as well as the coefficients c0, c1, and c2.
The Lagrange basis polynomials are given by:
L0(x) = (x - x1)(x - x2) / (x0 - x1)(x0 - x2)
L1(x) = (x - x0)(x - x2) / (x1 - x0)(x1 - x2)
L2(x) = (x - x0)(x - x1) / (x2 - x0)(x2 - x1)
where x0, x1, x2 are the given nodes, which in this case are 1, 2, and 3.
Let's calculate the Lagrange basis polynomials:
L0(x) = (x - 2)(x - 3) / (1 - 2)(1 - 3)
= (x - 2)(x - 3) / 2
L1(x) = (x - 1)(x - 3) / (2 - 1)(2 - 3)
= (x - 1)(x - 3) / -1
L2(x) = (x - 1)(x - 2) / (3 - 1)(3 - 2)
= (x - 1)(x - 2) / 2
Now we can calculate the coefficients c0, c1, and c2 by evaluating f(x) at the given nodes:
c0 = f(x0) / L0(x0)
= f(1) / L0(1)
= (1^3) / ((1 - 2)(1 - 3)/2)
= 1 / (1/2)
= 2
c1 = f(x1) / L1(x1)
= f(2) / L1(2)
= (2^3) / ((2 - 1)(2 - 3)/-1)
= 8 / (-1/-1)
= 8
c2 = f(x2) / L2(x2)
= f(3) / L2(3)
= (3^3) / ((3 - 1)(3 - 2)/2)
= 27 / (2/2)
= 27
Finally, we can write the quadratic interpolation polynomial P2(x) as:
P2(x) = c0 * L0(x) + c1 * L1(x) + c2 * L2(x)
= 2 * ((x - 2)(x - 3) / 2) + 8 * ((x - 1)(x - 3) / -1) + 27 * ((x - 1)(x - 2) / 2)
= (x - 2)(x - 3) - 8(x - 1)(x - 3) + 13.5(x - 1)(x - 2)
Therefore, P2(x) = (x - 2)(x - 3) - 8(x - 1)(x - 3) + 13.5(x - 1)(x - 2).