Please do parts a and b In ab below you will obtain the quad

In (a)-(b) below, you will obtain the quadratic Lagrange interpolation polynomial for f(x) = 1/2x^2+x+1 in the interval [1.00, 3.00] using the data: (a) Write the basic Lagrange interpolation polynomials L_1(x), L_2(x), L_3(x), where the subscript f in Ij(x) indicates the interpolation point j in the table. (nc MATLAB codes, just mathematical formulas) (b) Give the expression for the Lagrange interpolation polynomial P_2(x) in terms of these basic Lagrange interpolation polynomials L_1(x), L_2(x), L_3(x) and the data values f_1 = f(1.0), f_2 = f(2.0), f = f(3, 0). No need to expand and simplify the polynomials.

Solution

solution;

1)here lagranges polynomial for any point x is given by

y=y1*L1+y2*L2+y3*L3

where

y1=1/4

y2=1/11

y3=1/22

now lagranges polynomials given by

L(x1)=(x-x2)(x-x3)/(x1-x2)(x1-x3)=(x-2)(x-3)/(1-2)(1-3)

on putting value we get that

L(x1)=(x-2)(x-3)/2

L(x2)=(x-x1)(x-x3)/(x2-x1)(x2-x3)=(x-1)(x-3)/(2-1)(2-3)

on putting value we get that

L(x2)=-(x-1)(x-3)/1

L(x3)=(x-x1)(x-x2)/(x3-x1)(x3-x2)=(x-1)(x-2)/(3-1)(3-2)

on putting value we get that

L(x3)=(x-1)(x-2)/2

in this way lagranges polynomial are

L(x1)=(x-2)(x-3)/2

L(x2)=-(x-1)(x-3)/1

L(x3)=(x-1)(x-2)/2

where interpoltion function is given by

y=f(x)=y1*L(x1)+y2*L(x2)+y3*L(x3)

on putting value we get that

y=(1/4)[x^2-5x+6]-(1/11)[x^2-4x+3]+(1/44)[x^2-4x+3]

on simplfying we get that interpolation function as below

y=(1/3872)[220x^2-1364x+2112]