Question

Gregory newton backward difference interpolation formula

Answer 1

Answer:

Newton-Gregory Backward Difference Interpolation polynomial:

If the data size is big then the divided difference table will be too long. Suppose the desired intermediate value $ (\widetilde{x})$ at which one needs to estimate the function $ (i.e.f(\widetilde{x}))$ falls towards the end or say in the second half of the data set then it may be better to start the estimation process from the last data set point. For this we need to use backward-differences and backward difference table.  

Let us first define backward differences and generate backward difference table, say for the data set $ (x_{i},f_{i}),i=0,1,2,3,4.$