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Text of the page (random words):
ithmetic 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 average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model baye...