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.
The AGM is defined as the limit of the interdependent sequences and :
The arithmetic–geometric mean can be extended to complex numbers and when the branches of the square root are allowed to be taken inconsistently, it is, in general, a multivalued function.[1]
Example edit
To find the arithmetic–geometric mean of a0 = 24 and g0 = 6, iterate as follows:
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 an and gn 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 edit
The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were further analyzed by Gauss.[1]
Properties edit
The geometric mean of two positive numbers is never greater than the arithmetic mean.[3] So (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ≤ M(x, y) ≤ an. These are strict inequalities if x ≠ y.
M(x, y) is thus a number between the geometric and arithmetic mean of x and y; it is also between 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]
The arithmetic–geometric mean is connected to the Jacobi theta function by[6]
Related concepts edit
The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is Gauss's constant.
In 1941, (and hence ) was proven transcendental by Theodor Schneider.[note 2][7][8] The set is algebraically independent over ,[9][10] but the set (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over . In fact,[11]
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 edit
The inequality of arithmetic and geometric means implies that
Proof of the integral-form expression edit
This proof is given by Gauss.[1] Let
Changing the variable of integration to , where
This yields
gives
Thus, we have
Finally, we obtain the desired result
Applications edit
The number π edit
According to the Gauss–Legendre algorithm,[15]
where
with and , which can be computed without loss of precision using
Complete elliptic integral K(sinα) edit
Taking and yields the AGM
where K(k) is a complete elliptic integral of the first kind:
That is to say that this quarter period may be efficiently computed through the AGM,
Other applications edit
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 (ex, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms.[18]
See also edit
References edit
Notes edit
- ^ By 1799, Gauss had two proofs of the theorem, but neither of them was rigorous from the modern point of view.
- ^ In particular, he proved that the beta function is transcendental for all such that . The fact that is transcendental follows from
Citations edit
- ^ a b c d Cox, David (January 1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330.
- ^ agm(24, 6) at Wolfram Alpha
- ^ 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.
- ^ 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..
- ^ Dimopoulos, Hercules G. (2011). Analog Electronic Filters: Theory, Design and Synthesis. Springer. pp. 147–155. ISBN 978-94-007-2189-0.
- ^ 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
- ^ 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.
- ^ Todd, John (1975). "The Lemniscate Constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. S2CID 85873.
- ^ G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
- ^ G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 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. p. 45
- ^ 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.
- ^ 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.
- ^ King, Louis V. (1924). On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge University Press.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM. New York: Wiley. ISBN 0-471-83138-7. MR 0877728.
Sources edit
- Daróczy, Zoltán; Páles, Zsolt (2002). "Gauss-composition of means and the solution of the Matkowski–Suto problem". Publicationes Mathematicae Debrecen. 61 (1–2): 157–218. doi:10.5486/PMD.2002.2713.
- "Arithmetic–geometric mean process". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
- Weisstein, Eric W. "Arithmetic–geometric mean". MathWorld.