Higher order derivatives of unequally spaced data
07-07-2021, 05:45 PM
Post: #2
 Albert Chan Senior Member Posts: 2,516 Joined: Jul 2018
RE: Higher order derivatives of unequally spaced data
For derivatives, we may remove 1 term of Lagrange polynomial.

(f(x))' = (f(x) - c)', for constant c

Example, with 3 points:

$$f(x) ≈ f(x_1)\left({(x-x_2)(x-x_3) \over (x_1-x_2)(x_1-x_3)}\right) + f(x_2)\left({(x-x_1)(x-x_3) \over (x_2-x_1)(x_2-x_3)}\right) + f(x_3)\left({(x-x_1)(x-x_2) \over (x_3-x_1)(x_3-x_2)}\right)$$

Let $$c = f(x_2)$$, we eliminated middle term:

$$f'(x) ≈ \Big(f(x_1) - f(x_2)\Big) \left({(x-x_2)+(x-x_3) \over (x_1-x_2)(x_1-x_3)}\right) + \Big(f(x_3) - f(x_2)\Big) \left({(x-x_1)+(x-x_2) \over (x_3-x_1)(x_3-x_2)}\right)$$

Simplify with Divided Differences Notation:

\begin{align} f'(x) &≈ {f[x_1,x_2] \over x_3-x_1} (x_2+x_3-2x) + {f[x_2,x_3] \over x_3-x_1} (2x-x_1-x_2) \\ &= {f[x_1,x_2] \over x_3-x_1} (x_2+x_3-2x) + \left(f[x_1,x_2,x_3] + {f[x_1,x_2] \over x_3-x_1}\right) (2x-x_1-x_2) \\ &= f[x_1,x_2] + f[x_1,x_2,x_3] \, (2x-x_1-x_2) \end{align}

We can confirm above, with 3-points Divided-difference formula

$$f(x) ≈ f(x_1) + (x-x_1) \Big(f[x_1,x_2] + (x-x_2)\,f[x_1,x_2,x_3] \Big)$$

Differentiate with respect to x, we get back the same f'(x)
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 Messages In This Thread Higher order derivatives of unequally spaced data - Namir - 07-06-2021, 08:48 PM RE: Higher order derivatives of unequally spaced data - Albert Chan - 07-07-2021 05:45 PM RE: Higher order derivatives of unequally spaced data - Namir - 07-10-2021, 11:05 PM RE: Higher order derivatives of unequally spaced data - Namir - 07-17-2021, 12:45 PM

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