Problem: Hermite's Interpolation, lnx, [2,3], x=2.7, N=2 [ Calculator, Method and examples ]
Solution:
Your problem `->` Hermite's Interpolation, lnx, [2,3], x=2.7, N=2
Equation is `f(x)=ln(x)`.
`h=(b-a)/N`
`h=(3-2)/2=0.5`
The value of table for `x`, `f(x)` and `f'(x)`
| x | y=f(x) | y'=f'(x) | | 2 | 0.6931 | 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'`
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