Polynomial and Lagrange Interpolation

Initially, we want to find the velocity for an arbitrary point between i and i + 1, using only the two adjacent points. Here, the highest polynomial is a first order polynomial or a 

Solving the linear algebraic system (eqn. (7.5)) results in a linear interpolation equation given by 

Following the same procedure for a point x that is between i — 1 and i, the desired relation will be 

Similarly, we fit a second order polynomial using three points 

After solving for the constants we obtain the following second order interpolation function, 

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