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PLAGUE
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- TL;DR
- How to prove newtons forward difference interpolation formula using induction?
Say, $$y_n (x) = a_0 + a_1(x -x_0) + a_2(x-x_1)(x - x_0) + ... +a_n(x-x_0)(x-x_1)...(x-x_{n-1})$$
Now, $$y_0(x_0) = a_0$$
$$y_1(x_1) = a_0 + a_1(x_1 - x_0)$$
or, $$a_1 = \frac{\Delta y_0}{h}$$
Here, $$h = \frac{x_i - x_0}{i}$$
Similarly, $$a_n = \frac{(\Delta)^n y_0}{h^n n!}$$
Next substituting the values of a, we get the Newton's Forward Interpolation Formula.
It is not difficult to see that ##a_n = \frac{(\Delta)^n y_0}{h^n n!}##. But how do I prove this by induction method? Or any other rigorous way?
Now, $$y_0(x_0) = a_0$$
$$y_1(x_1) = a_0 + a_1(x_1 - x_0)$$
or, $$a_1 = \frac{\Delta y_0}{h}$$
Here, $$h = \frac{x_i - x_0}{i}$$
Similarly, $$a_n = \frac{(\Delta)^n y_0}{h^n n!}$$
Next substituting the values of a, we get the Newton's Forward Interpolation Formula.
It is not difficult to see that ##a_n = \frac{(\Delta)^n y_0}{h^n n!}##. But how do I prove this by induction method? Or any other rigorous way?
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