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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^[1]

{\begin{aligned}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{aligned}}

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 $p(x)$ .

Notes

^ Fateman & Kahan 2000

References

Fateman, R. J.; Kahan, W. (2000). Improving exact integrals from symbolic algebra systems (PDF) (Report). PAM. University of California, Berkeley: Center for Pure and Applied Mathematics.

Misc

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This is the English translation of Bessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen". Astronomische Nachrichten. 4 (16): 241–254. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601.

Bessel, F. W. (1825). "Ueber die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen" [The calculation of longitude and latitude from geodesic measurements]. Astronomische Nachrichten (in German). 4 (16): 241–253. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601.
Karney, C. F. F. (2007). "Quaternions in molecular modeling". Journal of Molecular Graphics and Modelling. 25 (5): 595–604. arXiv:physics/0506177. doi:10.1016/j.jmgm.2006.04.002. PMID 16777449.
Karney, C. F. F. (2011). "Transverse Mercator with an accuracy of a few nanometers". Journal of Geodesy. 85 (8): 475–476. arXiv:1002.1417. Bibcode:2011JGeod..85..475K. doi:10.1007/s00190-011-0445-3.

[Fateman_&_Kahan-1] Fateman & Kahan 2000

[1]