Arithmetic–geometric mean

This article is about the particular type of mean. For the similarly named inequality, see Inequality of arithmetic and geometric means.

In mathematics, the arithmetic–geometric mean (AGM or agM^[1]) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some mathematical constants, in particular, computing π.

Plot of the arithmetic–geometric mean ${\displaystyle \operatorname {agm} (1,x)}$ among several generalized means.

The AGM is defined as the limit of the interdependent sequences ${\displaystyle a_{i}}$ and ${\displaystyle g_{i}}$. Assuming ${\displaystyle x\geq y\geq 0}$, we write:$${\displaystyle {\begin{aligned}a_{0}&=x,\\g_{0}&=y\\a_{n+1}&={\tfrac {1}{2}}(a_{n}+g_{n}),\\g_{n+1}&={\sqrt {a_{n}g_{n}}}\,.\end{aligned}}}$$These two sequences converge to the same number, the arithmetic–geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y) or AGM(x, y).

The arithmetic–geometric mean can be extended to complex numbers and, when the branches of the square root are allowed to be taken inconsistently, generally it is a multivalued function.^[1]

Example

To find the arithmetic–geometric mean of a₀ = 24 and g₀ = 6, iterate as follows:$${\displaystyle {\begin{array}{rcccl}a_{1}&=&{\tfrac {1}{2}}(24+6)&=&15\\g_{1}&=&{\sqrt {24\cdot 6}}&=&12\\a_{2}&=&{\tfrac {1}{2}}(15+12)&=&13.5\\g_{2}&=&{\sqrt {15\cdot 12}}&=&13.416\ 407\ 8649\dots \\&&\vdots &&\end{array}}}$$The first five iterations give the following values:

n	an	gn
0	24	6
1	15	12
2	13.5	13.416 407 864 998 738 178 455 042...
3	13.458 203 932 499 369 089 227 521...	13.458 139 030 990 984 877 207 090...
4	13.458 171 481 745 176 983 217 305...	13.458 171 481 706 053 858 316 334...
5	13.458 171 481 725 615 420 766 820...	13.458 171 481 725 615 420 766 806...

The number of digits in which a_n and g_n agree (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.^[2]

History

The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were further analyzed by Gauss.^[1]

Properties

Both the geometric mean and arithmetic mean of two positive numbers x and y are between the two numbers. (They are strictly between when x ≠ y.) The geometric mean of two positive numbers is never greater than the arithmetic mean.^[3] So the geometric means are an increasing sequence g₀ ≤ g₁ ≤ g₂ ≤ ...; the arithmetic means are a decreasing sequence a₀ ≥ a₁ ≥ a₂ ≥ ...; and g_n ≤ M(x, y) ≤ a_n for any n. These are strict inequalities if x ≠ y.

M(x, y) is thus a number between x and y; it is also between the geometric and arithmetic mean of x and y.

If r ≥ 0 then M(rx, ry) = r M(x, y).

There is an integral-form expression for M(x, y):^[4]$${\displaystyle {\begin{aligned}M(x,y)&={\frac {\pi }{2}}\left(\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {x^{2}\cos ^{2}\theta +y^{2}\sin ^{2}\theta }}}\right)^{-1}\\&=\pi \left(\int _{0}^{\infty }{\frac {dt}{\sqrt {t(t+x^{2})(t+y^{2})}}}\right)^{-1}\\&={\frac {\pi }{4}}\cdot {\frac {x+y}{K\left({\frac {x-y}{x+y}}\right)}}\end{aligned}}}$$where K(k) is the complete elliptic integral of the first kind:$${\displaystyle K(k)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}}$$Since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in elliptic filter design.^[5]

The arithmetic–geometric mean is connected to the Jacobi theta function ${\displaystyle \theta _{3}}$ by^[6]$${\displaystyle M(1,x)=\theta _{3}^{-2}\left(\exp \left(-\pi {\frac {M(1,x)}{M\left(1,{\sqrt {1-x^{2}}}\right)}}\right)\right)=\left(\sum _{n\in \mathbb {Z} }\exp \left(-n^{2}\pi {\frac {M(1,x)}{M\left(1,{\sqrt {1-x^{2}}}\right)}}\right)\right)^{-2},}$$which upon setting ${\displaystyle x=1/{\sqrt {2}}}$ gives$${\displaystyle M(1,1/{\sqrt {2}})=\left(\sum _{n\in \mathbb {Z} }e^{-n^{2}\pi }\right)^{-2}.}$$

Related concepts

The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is Gauss's constant.$${\displaystyle {\frac {1}{M(1,{\sqrt {2}})}}=G=0.8346268\dots }$$In 1799, Gauss proved^{[note 1]} that$${\displaystyle M(1,{\sqrt {2}})={\frac {\pi }{\varpi }}}$$where ${\displaystyle \varpi }$ is the lemniscate constant.

In 1941, ${\displaystyle M(1,{\sqrt {2}})}$ (and hence ${\displaystyle G}$) was proved transcendental by Theodor Schneider.^{[note 2][7][8]} The set ${\displaystyle \{\pi ,M(1,1/{\sqrt {2}})\}}$ is algebraically independent over ${\displaystyle \mathbb {Q} }$,^[9][10] but the set ${\displaystyle \{\pi ,M(1,1/{\sqrt {2}}),M'(1,1/{\sqrt {2}})\}}$ (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over ${\displaystyle \mathbb {Q} }$. In fact,^[11]$${\displaystyle \pi =2{\sqrt {2}}{\frac {M^{3}(1,1/{\sqrt {2}})}{M'(1,1/{\sqrt {2}})}}.}$$The geometric–harmonic mean GH can be calculated using analogous sequences of geometric and harmonic means, and in fact GH(x, y) = 1/M(1/x, 1/y) = xy/M(x, y).^[12] The arithmetic–harmonic mean is equivalent to the geometric mean.

The arithmetic–geometric mean can be used to compute – among others – logarithms, complete and incomplete elliptic integrals of the first and second kind,^[13] and Jacobi elliptic functions.^[14]

Proof of existence

The inequality of arithmetic and geometric means implies that$${\displaystyle g_{n}\leq a_{n}}$$and thus$${\displaystyle g_{n+1}={\sqrt {g_{n}\cdot a_{n}}}\geq {\sqrt {g_{n}\cdot g_{n}}}=g_{n}}$$that is, the sequence g_n is nondecreasing and bounded above by the larger of x and y. By the monotone convergence theorem, the sequence is convergent, so there exists a g such that:$${\displaystyle \lim _{n\to \infty }g_{n}=g}$$However, we can also see that:$${\displaystyle a_{n}={\frac {g_{n+1}^{2}}{g_{n}}}}$$ and so: $${\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }{\frac {g_{n+1}^{2}}{g_{n}}}={\frac {g^{2}}{g}}=g}$$

Q.E.D.

Proof of the integral-form expression

This proof is given by Gauss.^[1] Let

$${\displaystyle I(x,y)=\int _{0}^{\pi /2}{\frac {d\theta }{\sqrt {x^{2}\cos ^{2}\theta +y^{2}\sin ^{2}\theta }}},}$$

Changing the variable of integration to ${\displaystyle \theta '}$, where

$${\displaystyle \sin \theta ={\frac {2x\sin \theta '}{(x+y)+(x-y)\sin ^{2}\theta '}}\Rightarrow d(\sin \theta )=d\left({\frac {2x\sin \theta '}{(x+y)+(x-y)\sin ^{2}\theta '}}\right)\Rightarrow \cos \theta \ d\theta =2x{\frac {(x+y)-(x-y)\sin ^{2}\theta '}{((x+y)+(x-y)\sin ^{2}\theta ')^{2}}}\ \cos \theta 'd\theta '}$$

$${\displaystyle \cos \theta ={\frac {\sqrt {(x+y)^{2}-2(x^{2}+y^{2})\sin ^{2}\theta '+(x-y)^{2}\sin ^{4}\theta '}}{(x+y)+(x-y)\sin ^{2}\theta '}}={\frac {\cos \theta '{\sqrt {(x-y)^{2}\cos ^{2}\theta '+4xy}}}{(x+y)+(x-y)\sin ^{2}\theta '}}={\frac {\cos \theta '{\sqrt {(x+y)^{2}\cos ^{2}\theta '+4xy\sin ^{2}\theta '}}}{(x+y)+(x-y)\sin ^{2}\theta '}},}$$

$${\displaystyle \Rightarrow \cos \theta \ d\theta ={\frac {\cos \theta '{\sqrt {(x+y)^{2}\cos ^{2}\theta '+4xy\sin ^{2}\theta '}}}{(x+y)+(x-y)\sin ^{2}\theta '}}\ d\theta =2x{\frac {(x+y)-(x-y)\sin ^{2}\theta '}{((x+y)+(x-y)\sin ^{2}\theta ')^{2}}}\ \cos \theta 'd\theta ',}$$

$${\displaystyle \Rightarrow d\theta ={\frac {x((x+y)-(x-y)\sin ^{2}\theta ')}{((x+y)+(x-y)\sin ^{2}\theta ')}}{\frac {2d\theta '}{\sqrt {(x+y)^{2}\cos ^{2}\theta '+4xy\sin ^{2}\theta '}}}\ ,}$$ $${\displaystyle {\sqrt {x^{2}\cos ^{2}\theta +y^{2}\sin ^{2}\theta }}={\frac {\sqrt {x^{2}((x+y)^{2}-2(x^{2}+y^{2})\sin ^{2}\theta '+(x-y)^{2}\sin ^{4}\theta ')+4x^{2}y^{2}\sin ^{2}\theta '}}{((x+y)+(x-y)\sin ^{2}\theta ')}}={\frac {x((x+y)-(x-y)\sin ^{2}\theta ')}{((x+y)+(x-y)\sin ^{2}\theta ')}}}$$

This yields $${\displaystyle {\frac {d\theta }{\sqrt {x^{2}\cos ^{2}\theta +y^{2}\sin ^{2}\theta }}}={\frac {2d\theta '}{\sqrt {(x+y)^{2}\cos ^{2}\theta '+4xy\sin ^{2}\theta '}}}={\frac {d\theta '}{\sqrt {(({\frac {x+y}{2}})^{2}\cos ^{2}\theta '+({\sqrt {xy}})^{2}\sin ^{2}\theta '}}},}$$

gives

$${\displaystyle {\begin{aligned}I(x,y)&=\int _{0}^{\pi /2}{\frac {d\theta '}{\sqrt {(({\frac {x+y}{2}})^{2}\cos ^{2}\theta '+({\sqrt {xy}})^{2}\sin ^{2}\theta '}}}\\&=I{\bigl (}{\tfrac {x+y}{2}},{\sqrt {xy}}{\bigr )}.\end{aligned}}}$$

Thus, we have

$${\displaystyle {\begin{aligned}I(x,y)&=I(a_{1},g_{1})=I(a_{2},g_{2})=\cdots \\&=I{\bigl (}M(x,y),M(x,y){\bigr )}=\pi /{\bigr (}2M(x,y){\bigl )}.\end{aligned}}}$$ The last equality comes from observing that ${\displaystyle I(z,z)=\pi /(2z)}$.

Finally, we obtain the desired result

$${\displaystyle M(x,y)=\pi /{\bigl (}2I(x,y){\bigr )}.}$$

Applications

The number π

According to the Gauss–Legendre algorithm,^[15]

$${\displaystyle \pi ={\frac {4\,M(1,1/{\sqrt {2}})^{2}}{1-\displaystyle \sum _{j=1}^{\infty }2^{j+1}c_{j}^{2}}},}$$

where

$${\displaystyle c_{j}={\frac {1}{2}}\left(a_{j-1}-g_{j-1}\right),}$$

with ${\displaystyle a_{0}=1}$ and ${\displaystyle g_{0}=1/{\sqrt {2}}}$, which can be computed without loss of precision using

$${\displaystyle c_{j}={\frac {c_{j-1}^{2}}{4a_{j}}}.}$$

Complete elliptic integral K(sinα)

Taking ${\displaystyle a_{0}=1}$ and ${\displaystyle g_{0}=\cos \alpha }$ yields the AGM

$${\displaystyle M(1,\cos \alpha )={\frac {\pi }{2K(\sin \alpha )}},}$$

where K(k) is a complete elliptic integral of the first kind:

$${\displaystyle K(k)=\int _{0}^{\pi /2}(1-k^{2}\sin ^{2}\theta )^{-1/2}\,d\theta .}$$

That is to say that this quarter period may be efficiently computed through the AGM, $${\displaystyle K(k)={\frac {\pi }{2M(1,{\sqrt {1-k^{2}}})}}.}$$

Other applications

Using this property of the AGM along with the ascending transformations of John Landen,^[16] Richard P. Brent^[17] suggested the first AGM algorithms for the fast evaluation of elementary transcendental functions (e^x, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms.^[18]

See also

• Landen's transformation
• Gauss–Legendre algorithm
• Generalized mean

References

Notes

1. By 1799, Gauss had two proofs of the theorem, but neither of them was rigorous from the modern point of view.
2. In particular, he proved that the beta function ${\displaystyle \mathrm {B} (a,b)}$ is transcendental for all ${\displaystyle a,b\in \mathbb {Q} \setminus \mathbb {Z} }$ such that ${\displaystyle a+b\notin \mathbb {Z} _{0}^{-}}$. The fact that ${\displaystyle M(1,{\sqrt {2}})}$ is transcendental follows from ${\displaystyle M(1,{\sqrt {2}})={\tfrac {1}{2}}\mathrm {B} \left({\tfrac {1}{2}},{\tfrac {3}{4}}\right).}$

Citations

1. Cox, David (January 1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330.
2. agm(24, 6) at Wolfram Alpha
3. Bullen, P. S. (2003). "The Arithmetic, Geometric and Harmonic Means". Handbook of Means and Their Inequalities. Dordrecht: Springer Netherlands. pp. 60–174. doi:10.1007/978-94-017-0399-4_2. ISBN 978-90-481-6383-0. Retrieved 2023-12-11.
4. Carson, B. C. (2010). "Elliptic Integrals". In Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.). NIST Handbook of Mathematical Functions. Cambridge University Press. ISBN 978-0-521-19225-5. MR 2723248..
5. Dimopoulos, Hercules G. (2011). Analog Electronic Filters: Theory, Design and Synthesis. Springer. pp. 147–155. ISBN 978-94-007-2189-0.
6. Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. pages 35, 40
7. Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik. 183 (19): 110–128. doi:10.1515/crll.1941.183.110. S2CID 118624331.
8. Todd, John (1975). "The Lemniscate Constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. S2CID 85873.
9. G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
10. G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
11. Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 45
12. Newman, D. J. (1985). "A simplified version of the fast algorithms of Brent and Salamin". Mathematics of Computation. 44 (169): 207–210. doi:10.2307/2007804. JSTOR 2007804.
13. Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 17". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 598–599. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
14. King, Louis V. (1924). On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge University Press.
15. Salamin, Eugene (1976). "Computation of π using arithmetic–geometric mean". Mathematics of Computation. 30 (135): 565–570. doi:10.2307/2005327. JSTOR 2005327. MR 0404124.
16. Landen, John (1775). "An investigation of a general theorem for finding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom". Philosophical Transactions of the Royal Society. 65: 283–289. doi:10.1098/rstl.1775.0028. S2CID 186208828.
17. Brent, Richard P. (1976). "Fast Multiple-Precision Evaluation of Elementary Functions". Journal of the ACM. 23 (2): 242–251. CiteSeerX 10.1.1.98.4721. doi:10.1145/321941.321944. MR 0395314. S2CID 6761843.
18. Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM. New York: Wiley. ISBN 0-471-83138-7. MR 0877728.

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