ExerciseInterpolationadvanced

Exercise: Hermite with the Bessel function

Full construction of the degree-5 Hermite polynomial with three nodes to approximate J0(0.75)J_0(0.75), step by step.

Statement and data

The Bessel functions of the first kind satisfy J0′(x)=−J1(x)J_0'(x)=-J_1(x), so the table gives both value and derivative of J0J_0 at the nodes 00, 0.50.5 and 11. With n=2n=2 the Hermite polynomial has degree 5.

xxJ0J_0 (value)J0′=−J1J_0'=-J_1 (derivative)
01.00000
0.50.9385−0.2423
10.7652−0.4401

Building the polynomial

Example

Follow the five steps: LiL_i, Li′L_i', HiH_i, H^i\hat H_i and the final sum.

  1. Lagrange functions and their derivatives:

    L0=2x2−3x+1,L0′=4x−3L1=−4x2+4x,L1′=−8x+4L2=2x2−x,L2′=4x−1\begin{aligned} L_0&=2x^2-3x+1, & L_0'&=4x-3\\ L_1&=-4x^2+4x, & L_1'&=-8x+4\\ L_2&=2x^2-x, & L_2'&=4x-1 \end{aligned}

  2. Blocks HiH_i and H^i\hat H_i (degree 5):

    H0=24x5−68x4+66x3−23x2+1H1=16x4−32x3+16x2H2=−24x5+52x4−34x3+7x2\begin{aligned} H_0&=24x^5-68x^4+66x^3-23x^2+1\\ H_1&=16x^4-32x^3+16x^2\\ H_2&=-24x^5+52x^4-34x^3+7x^2 \end{aligned}

  3. And the H^i\hat H_i:

    H^0=4x5−12x4+13x3−6x2+xH^1=16x5−40x4+32x3−8x2H^2=4x5−8x4+5x3−x2\begin{aligned} \hat H_0&=4x^5-12x^4+13x^3-6x^2+x\\ \hat H_1&=16x^5-40x^4+32x^3-8x^2\\ \hat H_2&=4x^5-8x^4+5x^3-x^2 \end{aligned}

  4. Summing f(xi)Hi+f′(xi)H^if(x_i)H_i+f'(x_i)\hat H_i gives:

    H5(x)=−0.002x5+0.0192x4−0.0029x3−0.2491x2+1H_5(x)=-0.002x^5+0.0192x^4-0.0029x^3-0.2491x^2+1

Evaluating at 0.750.75 approximates J0(0.75)J_0(0.75):

J0(0.75)≈H5(0.75)=0.86426J_0(0.75)\approx H_5(0.75)=0.86426