**Suppose we are numerically approximating the second-order derivative **

**A common way is to use finite difference scheme which uses Taylor expansion.**

1. Perform Taylor expansion for , , .etc

2. Assemble the Taylor series and cancel the high-order terms to construct the desired-order scheme, i.e. finding a set of coefficients which makes the summation of high-order terms to be zero.

For example, the second-order central difference scheme:

**An alternative way to approximate the derivative.**

1. Express as

**We can see , so the objective is to find the coefficient .**

2. Use the current data to fit this second-order polynomial (a parabola, actually a curve fitting problem)

3. Solve this fitting problem

Since we have three unknown coefficients , , and , we only need three data points. However, we can choose more data points to fit this curve, which is actually least square fitting. Now, let us write the above equations in a general vector form:

where

This is an over-determined system if , which we can approximate it with least square fitting.

See https://en.wikipedia.org/wiki/Overdetermined_system

4. Solution

The approximated/fitted solution is: , where the second-order derivative