Take a simple equation Sin(x)-0.5=0. Now, In the range (0-pi) find out at 5 equally spaced points. Using these 5 points generate a polynomial of order 4 using interpolation.
Now find the difference of output at multiple points order than the nodes. You’ll notice that an error is present which also varies. Since interpolation takes into consideration only a finite number of points this error exists. If you keep on increasing the dataset then the error produced will decrease but the calculation increase.
From the multiple interpolation methods available such as newton divided difference, newton forwards and backward divided difference, you can use any to approximate your results.
Make sure that the newton divided difference can work with uneven step size but newton forward or backward difference does not work with uneven step size, you need to have a constant step size for it.
When considering a constant step size for second-order polynomial it can be shown that there is a limit to the step size which can be used. One important application of interpolation is using it to find the integration of complicated functions.
If you know the output value at a few points then you can limit your approximation under a desired order of error. Numerical analysis is also majorly used where the integration of the function does not exist.
For example consider the function xx, integration of this function cannot express in the form of elementary functions. Use numerical analysis here to find the value of the antiderivative.
The only limitation is that you can only find definite antiderivatives of function and you need to know the value of those nodes.