Brahmagupta interpolation formula

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In trigonometry, the Brahmagupta interpolation formula is a special case of the Newton-Stirling interpolation formula to the second-order, which Brahmagupta used in 665 to interpolate new values of the sine function from other values already tabulated. The formula gives an estimate for the value of a function $f$ at a value a + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when its value is already known at a − h, a and a + h.

The formula for the estimate is:

$f( a + x h ) \approx f(a) + x \left(\frac{\Delta f(a) + \Delta f(a-h)}{2}\right) + \frac{x^2 \Delta^2 f(a-h)}{2!}.$

where Δ is the first-order forward-difference operator, i.e.

$\Delta f(a) \ \stackrel{\mathrm{def}}{=}\ f(a+h) - f(a).$

[edit] References

George G. Joseph (2000). The Crest of the Peacock, p. 285–286. Princeton University Press. ISBN 0691006598.

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