User:Cffk/sandbox

Extension to compute the divided difference of a polynomial
Horner's method can be modified to compute the dividing difference, $$(p(x) - p(y))/(x - y)$$. Given the polynomial (as before)


 * $$p(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n,$$

proceed as follows


 * $$\begin{align}

b_n & = a_n,                &\quad d_n &= b_n, \\ b_{n-1} & = a_{n-1} + b_n x, &\quad d_{n-1} &= b_{n-1} + d_n y, \\ & {}\ \  \vdots             &\quad &  {}\ \ \vdots\\ b_1 & = a_1 + b_2 x,        &\quad d_1 &= b_1 + d_2 y,\\ b_0 & = a_0 + b_1 x. \end{align}$$

At completion, we have $$b_0 = p(x)$$ and $$d_1 = (p(x) - p(y))/(x - y)$$. This computation of the divided difference is subject to much less round-off error than evaluating $$p(x)$$ and $$p(y)$$ separately, particularly when $$ x \simeq y$$. Substituting $$y = x$$ in this method gives $$d_1 = p'(x)$$, the derivative of $$p(x)$$.

Misc
/Theoretical Gravity

See


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