# agmean: Arithmetic-geometric Mean

### Description

The arithmetic-geometric mean of real or complex numbers.

### Usage

 1 agmean(a, b) 

### Arguments

 a, b vectors of real or complex numbers of the same length (or scalars).

### Details

The arithmetic-geometric mean is defined as the common limit of the two sequences a_{n+1} = (a_n + b_n)/2 and b_{n+1} = √(a_n b_n).

When used for negative or complex numbers, the complex square root function is applied.

### Value

Returns a list with compoinents: agm a vector of arithmetic-geometric means, component-wise, niter the number of iterations, and prec the overall estimated precision.

### Note

Gauss discovered that elliptic integrals can be effectively computed via the arithmetic-geometric mean (see example below), for example:

\int_0^{π/2} \frac{dt}{√{1 - m^2 sin^2(t)}} = \frac{(a+b) π}{4 \cdot agm(a,b)}

where m = (a-b)/(a+b)

### References

Arithmetic, geometric, and harmonic mean.

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ## Accuracy test: Gauss constant 1/agmean(1, sqrt(2))$agm - 0.834626841674073186 # 1.11e-16 < eps = 2.22e-16 ## Gauss' AGM-based computation of \pi a <- 1.0 b <- 1.0/sqrt(2) s <- 0.5 d <- 1L while (abs(a-b) > eps()) { t <- a a <- (a + b)*0.5 b <- sqrt(t*b) c <- (a-t)*(a-t) d <- 2L * d s <- s - d*c } approx_pi <- (a+b)^2 / s / 2.0 abs(approx_pi - pi) # 8.881784e-16 in 4 iterations ## Example: Approximate elliptic integral N <- 20 m <- seq(0, 1, len = N+1)[1:N] E <- numeric(N) for (i in 1:N) { f <- function(t) 1/sqrt(1 - m[i]^2 * sin(t)^2) E[i] <- quad(f, 0, pi/2) } A <- numeric(2*N-1) a <- 1 b <- a * (1-m) / (m+1) ## Not run: plot(m, E, main = "Elliptic Integrals vs. arith.-geom. Mean") lines(m, (a+b)*pi / 4 / agmean(a, b)$agm, col="blue") grid() ## End(Not run) 

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