Interpolate the value of the function corresponding to x=4 , using Lagrange's interpolation formula where (x,f(x)) has the following values (2,10), (3,15), (5,25), (8,40), (12,60)

Question

The question belongs to numerical methods concept of engineering first year

Answer 1

We have 5 points (x[n],y[n]=f(x[n])) where n goes from 1 to 5.

If the Lagrange polynomial is P(x) then:

P(x)=((x-x2)(x-x3)(x-x4)(x-x5)/((x1-x2)(x1-x3)(x1-x4)(x1-x5)))y1+...

P(x)=10(x-3)(x-5)(x-8)(x-12)/((2-3)(2-5)(2-8)(2-12))+

15(x-2)(x-5)(x-8)(x-12)/((3-2)(3-5)(3-8)(3-12))+

25(x-2)(x-3)(x-8)(x-12)/((5-2)(5-3)(5-8)(5-12))+

40(x-2)(x-3)(x-5)(x-12)/((8-2)(8-3)(8-5)(8-12))+

60(x-2)(x-3)(x-5)(x-8)/((12-2)(12-3)(12-5)(12-8)).

P(x)=(x-3)(x-5)(x-8)(x-12)/18-

(x-2)(x-5)(x-8)(x-12)/6+

25(x-2)(x-3)(x-8)(x-12)/126-

(x-2)(x-3)(x-5)(x-12)/9+

(x-2)(x-3)(x-5)(x-8)/42

All that remains is to put x=4 to find an interpolated value for f(4):

f(4)=-16/9-(-32/3)+800/63-16/9+4/21=20, which does in fact lie between 15 and 25.