Find Solution using Hermite's formula
| x | f(x) |
| 2 | 0.69315 |
| 2.5 | 0.91629 |
| 3 | 1.09861 |
x = 2.7
Finding f(2)
Solution:
The value of table for `x`, `f(x)` and `f'(x)`
| x | y=f(x) | y'=f'(x) |
| 2 | 0.6932 | 0.5 |
| 2.5 | 0.9163 | 0.4 |
| 3 | 1.0986 | 0.3333 |
The Polynomials `I_i(x)` are
`I_0(x)=((x - x_1)(x - x_2))/((x_0 - x_1)(x_0 - x_2))=((x -2.5)(x -3))/((2 -2.5)(2 -3))=((x -2.5)(x -3))/((-0.5)(-1))=(x^2-5.5x+7.5)/(0.5)=2x^2-11x+15`
`I_1(x)=((x - x_0)(x - x_2))/((x_1 - x_0)(x_1 - x_2))=((x -2)(x -3))/((2.5 -2)(2.5 -3))=((x -2)(x -3))/((0.5)(-0.5))=(x^2-5x+6)/(-0.25)=-4x^2+20x-24`
`I_2(x)=((x - x_0)(x - x_1))/((x_2 - x_0)(x_2 - x_1))=((x -2)(x -2.5))/((3 -2)(3 -2.5))=((x -2)(x -2.5))/((1)(0.5))=(x^2-4.5x+5)/(0.5)=2x^2-9x+10`
`I_0'(x)=4x-11`
`I_1'(x)=-8x+20`
`I_2'(x)=4x-9`
`I_0'(x_0)=I_0'(2)=4 xx 2-11=-3`
`I_1'(x_1)=I_1'(2.5)=-8 xx 2.5+20=0`
`I_2'(x_2)=I_2'(3)=4 xx 3-9=3`
Hermite Interpolation Formula is
`H(x)=sum u_i(x)*y_i + sum v_i(x)*y_i'`
where `u_i(x)=[1-2(x-x_i) I_i'(x_i)][I_i(x)]^2` and `v_i(x)=(x-x_i)[I_i(x)]^2`
`u_0(x)=[1-2(x-x_0) I_0'(x_0)][I_0(x)]^2`
`=>u_0(x)=[1-2(x-2) I_0'(2)][I_0(x)]^2`
`=>u_0(x)=[1-2(x-2) * (-3)][I_0(x)]^2`
`=>u_0(x)=[1 +6x-12][I_0(x)]^2`
`=>u_0(x)=(6x-11)(2x^2-11x+15)^2`
`v_0(x)=(x-x_0)[I_i(x)]^2`
`=>v_0(x)=(x-2)(2x^2-11x+15)^2`
`u_1(x)=[1-2(x-x_1) I_1'(x_1)][I_1(x)]^2`
`=>u_1(x)=[1-2(x-2.5) I_1'(2.5)][I_1(x)]^2`
`=>u_1(x)=[1-2(x-2.5) * (0)][I_1(x)]^2`
`=>u_1(x)=[1 +0][I_1(x)]^2`
`=>u_1(x)=(1)(-4x^2+20x-24)^2`
`v_1(x)=(x-x_1)[I_i(x)]^2`
`=>v_1(x)=(x-2.5)(-4x^2+20x-24)^2`
`u_2(x)=[1-2(x-x_2) I_2'(x_2)][I_2(x)]^2`
`=>u_2(x)=[1-2(x-3) I_2'(3)][I_2(x)]^2`
`=>u_2(x)=[1-2(x-3) * (3)][I_2(x)]^2`
`=>u_2(x)=[1 -6x+18][I_2(x)]^2`
`=>u_2(x)=(-6x+19)(2x^2-9x+10)^2`
`v_2(x)=(x-x_2)[I_i(x)]^2`
`=>v_2(x)=(x-3)(2x^2-9x+10)^2`
Hermite Interpolation formula is
`H(x)=u_0(x)*y_0+v_0(x)*y_0'+u_1(x)*y_1+v_1(x)*y_1'+u_2(x)*y_2+v_2(x)*y_2'`
`H(x)=(6x-11)(2x^2-11x+15)^2 * (0.6932) + (x-2)(2x^2-11x+15)^2 * (0.5)+(1)(-4x^2+20x-24)^2 * (0.9163) + (x-2.5)(-4x^2+20x-24)^2 * (0.4)+(-6x+19)(2x^2-9x+10)^2 * (1.0986) + (x-3)(2x^2-9x+10)^2 * (0.3333)`
Putting x=2.7 and simplifying, we obtain
`H(2.7)=0.9933`
This material is intended as a summary. Use your textbook for detail explanation.
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