Approximating of Higher Order Derivatives¶

It also possible to use Taylor series to approximate higher order derivatives (e.g., $f''(x_j), f'''(x_j)$, etc.). For example, taking the Taylor series around $a = x_j$ and then computing it at $x = x_{j-1}$ and $x_{j+1}$ gives

\[ f(x_{j-1}) = f(x_j) - hf^{\prime}(x_j) + \frac{h^2f''(x_j)}{2} - \frac{h^3f'''(x_j)}{6} + \cdots\]

and

\[f(x_{j+1}) = f(x_j) + hf^{\prime}(x_j) + \frac{h^2f''(x_j)}{2} + \frac{h^3f'''(x_j)}{6} + \cdots.\]

If we add these two equations together, we get

\[f(x_{j-1}) + f(x_{j+1}) = 2f(x_j) + h^2f''(x_j) + \frac{h^4f''''(x_j)}{24} + \cdots,\]

and with some rearrangement gives the approximation $$f''(x_j) \approx \frac{f(x_{j+1}) - 2f(x_j) + f(x_{j-1})}{h^2},$$ and is $O(h^2)$.

< 20.2 Finite Difference Approximating Derivatives | Contents | 20.4 Numerical Differentiation with Noise >

Python Numerical Methods

Approximating of Higher Order Derivatives¶