Gauss–Legendre algorithm: Difference between revisions

Revision as of 16:26, 7 July 2019

Revision as of 17:43, 9 July 2019

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where <math>E(k)</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]:

:<math>E(k) = \int_0^{\pi/2}\sqrt {1-k^2 \sin^2\theta}\,; d\theta</math> and <math>K(k) = \int_0^{\pi/2}\frac{d\theta}{\sqrt {1-k^2 \sin^2\theta}}.</math>

Gauss knew of both of these results.<ref name="brent">{{Citation

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=== Legendre’s identity ===

For <math>\varphi</math> and <math>\theta</math> such that <math display="inline">\varphi+\theta={1 \over 2}\pi</math> Legendre proved the identity:

:<math>K(\sin \varphi) E(\sin \theta ) + K(\sin \theta ) E(\sin \varphi) - K(\sin \varphi) K(\sin \theta) = {1 \over 2}\pi.</math><ref name="brent" />

:Equivalently,

:<math>\forall \varphi: K(\sin\varphi)[E(\cos\varphi)-K(\cos\varphi)] + K(\cos\varphi)E(\sin\varphi) = \frac{\pi}{2}</math>

=== Gauss–Euler method ===