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istory tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia mathematical function of two positive real arguments this article is about the particular type of mean for the similarly named inequality see inequality of arithmetic and geometric means plot of the arithmetic geometric mean agm 1 x displaystyle operatorname agm 1 x among several generalized 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 the agm is defined as the limit of the interdependent sequences a i displaystyle a_ i and g i displaystyle g_ i assuming x y 0 displaystyle x geq y geq 0 we write a 0 x g 0 y a n 1 1 2 a n g n g n 1 a n g n 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 it is a multivalued function 1 example edit to find the arithmetic geometric mean of a 0 24 and g 0 6 iterate as follows a 1 1 2 24 6 15 g 1 24 6 12 a 2 1 2 15 12 13 5 g 2 15 12 13 416 407 8649 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 a n g n 0 24 6 1 1 5 1 2 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 7 45 176 983 217 305 13 458 171 481 7 06 053 858 316 334 5 13 458 171 481 725 615 420 766 8 20 13 458 171 481 725 615 420 766 8 06 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 13 458 171 481 725 615 420 766 813 156 974 399 243 053 838 8544 2 history edit the first algorithm based on this sequence pair appeared in the works of joseph louis lagrange its properties were further analyzed by carl friedrich gauss 1 properties edit 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 g 0 g 1 g 2 the arithmetic means are a decreasing sequence a 0 a 1 a 2 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 m x y π 2 0 π 2 d θ x 2 cos 2 θ y 2 sin 2 θ 1 π 0 d t t t x 2 t y 2 1 π 4 x y k x y x y 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 k k 0 π 2 d θ 1 k 2 sin 2 θ 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 θ 3 displaystyle theta _ 3 by 6 m 1 x θ 3 2 exp π m 1 x m 1 1 x 2 n z exp n 2 π m 1 x m 1 1 x 2 2 displaystyle begin aligned 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 end aligned which upon setting x 1 2 displaystyle x 1 sqrt 2 gives m 1 1 2 n z e n 2 π 2 displaystyle m 1 1 sqrt 2 left sum _ n in mathbb z e n 2 pi right 2 related concepts edit the reciprocal of the arithmetic geometric mean of 1 and the square root of 2 is gauss s constant 1 m 1 2 g 0 8346268 displaystyle frac 1 m 1 sqrt 2 g 0 8346268 dots in 1799 gauss proved note 1 that m 1 2 π ϖ displaystyle m 1 sqrt 2 frac pi varpi where ϖ displaystyle varpi is the lemniscate constant in 1941 m 1 2 displaystyle m 1 sqrt 2 and hence g displaystyle g was proved transcendental by theodor schneider note 2 7 8 the set π m 1 1 2 displaystyle pi m 1 1 sqrt 2 is algebraically independent over q displaystyle mathbb q 9 10 but the set π m 1 1 2 m 1 1 2 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 q displaystyle mathbb q in fact 11 π 2 2 m 3 1 1 2 m 1 1 2 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 edit the inequality of arithmetic and geometric means implies that g n a n displaystyle g_ n leq a_ n and thus g n 1 g n a n g n g n g n 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 lim n g n g displaystyle lim _ n to infty g_ n g however we can also see that a n g n 1 2 g n displaystyle a_ n frac g_ n 1 2 g_ n and so lim n a n lim n g n 1 2 g n g 2 g g 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 edit this proof is given by gauss 1 let i x y 0 π 2 d θ x 2 cos 2 θ y 2 sin 2 θ 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 θ displaystyle theta where sin θ 2 x sin θ x y x y sin 2 θ d sin θ d 2 x sin θ x y x y sin 2 θ cos θ d θ 2 x x y x y sin 2 θ x y x y sin 2 θ 2 cos θ d θ displaystyle begin aligned 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 end aligned cos θ x y 2 2 x 2 y 2 sin 2 θ x y 2 sin 4 θ x y x y sin 2 θ cos θ x y 2 cos 2 θ 4 x y x y x y sin 2 θ cos θ x y 2 cos 2 θ 4 x y sin 2 θ x y x y sin 2 θ displaystyle begin aligned 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 1ex frac cos theta sqrt x y 2 cos 2 theta 4xy x y x y sin 2 theta 1ex frac cos theta sqrt x y 2 cos 2 theta 4xy sin 2 theta x y x y sin 2 theta end aligned cos θ d θ cos θ x y 2 cos 2 θ 4 x y sin 2 θ x y x y sin 2 θ d θ 2 x x y x y sin 2 θ x y x y sin 2 θ 2 cos θ d θ displaystyle begin aligned 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 1ex 2x frac x y x y sin 2 theta x y x y sin 2 theta 2 cos theta d theta end aligned d θ x x y x y sin 2 θ x y x y sin 2 θ 2 d θ x y 2 cos 2 θ 4 x y sin 2 θ 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 x 2 cos 2 θ y 2 sin 2 θ x 2 x y 2 2 x 2 y 2 sin 2 θ x y 2 sin 4 θ 4 x 2 y 2 sin 2 θ x y x y sin 2 θ x x y x y sin 2 θ x y x y sin 2 θ displaystyle begin aligned sqrt x 2 cos 2 theta y 2 sin 2 theta frac sqrt x 2 left x y 2 2 x 2 y 2 sin 2 theta x y 2 sin 4 theta right 4x 2 y 2 sin 2 theta x y x y sin 2 theta frac x left x y x y sin 2 theta right left x y x y sin 2 theta right end aligned this yields d θ x 2 cos 2 θ y 2 sin 2 θ 2 d θ x y 2 cos 2 θ 4 x y sin 2 θ d θ x y 2 2 cos 2 θ x y 2 sin 2 θ displaystyle begin aligned 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 left frac x y 2 right 2 cos 2 theta left sqrt xy right 2 sin 2 theta end aligned gives i x y 0 π 2 d θ x y 2 2 cos 2 θ x y 2 sin 2 θ i x y 2 x y displaystyle begin aligned i x y int _ 0 pi 2 frac d theta sqrt left frac x y 2 right 2 cos 2 theta left sqrt xy right 2 sin 2 theta i left tfrac x y 2 sqrt xy right end aligned thus we have i x y i a 1 g 1 i a 2 g 2 i m x y m x y π 2 m x y 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 frac pi 2m x y end aligned the last equality comes from observing that i z z π 2 z displaystyle i z z pi 2z finally we obtain the desired result m x y π 2 i x y displaystyle m x y frac pi 2i x y applications edit the number π edit according to the gauss legendre algorithm 15 π 4 m 1 1 2 2 1 j 1 2 j 1 c j 2 displaystyle pi frac 4 m 1 1 sqrt 2 2 1 displaystyle sum _ j 1 infty 2 j 1 c_ j 2 where c j 1 2 a j 1 g j 1 displaystyle c_ j frac 1 2 left a_ j 1 g_ j 1 right with a 0 1 displaystyle a_ 0 1 and g 0 1 2 displaystyle g_ 0 1 sqrt 2 which can be computed without loss of precision using c j c j 1 2 4 a j displaystyle c_ j frac c_ j 1 2 4a_ j complete elliptic integral k sin α edit taking a 0 1 displaystyle a_ 0 1 and g 0 cos α displaystyle g_ 0 cos alpha yields the agm m 1 cos α π 2 k sin α displaystyle m 1 cos alpha frac pi 2k sin alpha where k k is a complete elliptic integral of the first kind k k 0 π 2 d θ 1 k 2 sin 2 θ displaystyle k k int _ 0 pi 2 frac d theta sqrt 1 k 2 sin 2 theta that is to say that this quarter period may be efficiently computed through the agm k k π 2 m 1 1 k 2 displaystyle k k frac pi 2m 1 sqrt 1 k 2 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 e x cos x sin x subsequently many authors went on to study the use of the agm algorithms 18 see also edit gauss legendre algorithm generalized mean landen s transformation 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 b a b displaystyle mathrm b a b is transcendental for all a b q z displaystyle a b in mathbb q setminus mathbb z such that a b z 0 displaystyle a b notin mathbb z _ 0 the fact that m 1 2 displaystyle m 1 sqrt 2 is transcendental follows from m 1 2 1 2 b 1 2 3 4 displaystyle m 1 sqrt 2 tfrac 1 2 mathrm b left tfrac 1 2 tfrac 3 4 right citations edit 1 2 3 4 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 cite journal cite uses deprecated parameter citeseerx help 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 hdl 2437 110605 arithmetic geometric mean process encyclopedia of mathematics ems press 2001 1994 weisstein eric w arithmetic geometric mean mathworld v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion aver...
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