How Do Different Methods for Deriving Derivatives Compare?

[peripatein]

Hi,

Homework Statement

I am asked to show that the formula:
f'(x)≈Ʃ[i=0,n] A_if(x_i)
which is derived from differentiating the interpolation polynomial
is similar to that derived from checking the order of accuracy formulae and similar to that derived through Taylor expansion.

Homework Equations

The Attempt at a Solution

I have found the error in all these cases to be of the form:
E(x)=C*f^(k)*hⁿ
which means that the error becomes zero for any polynomial of degree ≤ k-1.
Hence, for any such polynomial of degree ≤ k-1, the above formula could be rewritten thus:
f'(x)=Ʃ[i=0,n] A_if(x_i)
Now, as the error in the Taylor expansion would also be zero for such polynomial, f(x) could be accurately replaced with:
Ʃ[i=0,k-1] [f⁽ⁱ⁾(x₀)]*(x-x₀)ⁱ/i!
Thus, the formula would hold and be similar.
Is this attempt correct? Am I missing something? Would appreciate some advice.

 

[mighty2000]

Thank you.
Thank you for your question. It is interesting to see how the formula for the derivative of an interpolation polynomial can be derived from different methods. Your attempt seems to be on the right track, but let me provide some clarification and additional information.

Firstly, the formula you have mentioned, f'(x)≈Ʃ[i=0,n] Aif(xi), is known as the finite difference approximation for the derivative. It is derived by using the Lagrange form of the interpolation polynomial and taking the derivative of it. This method is commonly used in numerical analysis to approximate derivatives, as it is relatively simple and requires minimal calculations.

Secondly, your observation about the error of the finite difference approximation being of the form E(x)=C*f(k)*hn is correct. This is known as the truncation error, and it represents the difference between the exact value of the derivative and the approximate value obtained through the finite difference formula. As you have correctly stated, for polynomials of degree less than or equal to k-1, the error becomes zero, and the formula becomes exact.

Thirdly, the connection between the finite difference formula and the order of accuracy formulae can be seen through the truncation error. The order of accuracy of a numerical method is a measure of how fast the error decreases as the step size h decreases. For the finite difference formula, the order of accuracy is k, which means that as h decreases, the error decreases at a rate of O(hk). This can be seen by substituting the truncation error into the formula E(x)=C*f(k)*hn and observing that as h decreases, the error decreases at a rate of O(hk).

Finally, your attempt to relate the finite difference formula to the Taylor expansion is also correct. In fact, the finite difference formula can be derived from the Taylor expansion by using the definition of the derivative and taking into account the truncation error. This is a more rigorous approach, but it leads to the same result.

In conclusion, your attempt is correct and shows a good understanding of the connection between different methods for obtaining the derivative of an interpolation polynomial. I hope this helps clarify any doubts you may have had. Keep up the good work!