Help with Syntax to Make Iterative Arithmetic-Geometric Mean Like Functions

Question

In the arithmetic-geometric mean, we combine the arithmetic mean and the geometric mean like so, $$a_0=a,~b_0=b$$ $$a_1=\frac{a_0+b_0}{2},~b_1=\sqrt{a_0 b_0}$$ $$\vdots$$ $$a_{n+1}=\frac{a_n+b_n}{2},~b_{n+1}=\sqrt{a_n b_n}$$ $$AGM(a, b)=\lim_{n\to\infty} a_n=\lim_{n\to\infty} b_n$$ I would like to have a more general purpose function, something like

ComboMean[f,g,x]

gives the a similar kind of mean, but instead of combining the arithmetic mean and the geometric mean, we combine any two functions f and g using a similar iteration.

But I'm stumped on the syntax on how to accomplish this. Right now, I'm writing a specific function for each mean I'm trying to find.

For example:

f2[x_] := {Mean[x], Total[x^2]/Total[x]}
M2[x_] := Nest[f2, x, 10][[1]] // N

Answer 1

This is a simple implementation:

ComboMean[f_, g_, x_] := FixedPoint[{f@#,g@#}&, x][[1]];
(* define our own AGM function *)
agm[a_, b_] := ComboMean[ Mean, GeometricMean, {a, b}];
(* example of use *)
agm[1, 2.]
(* 1.45679103104691 *)

Notice that to avoid infinite loops you could use the third argument to FixedPoint to provide a limit for the number of iterations.

Answer 2

Since it's a recurrence, you can use

RecurrenceTable[
{
a[n] == (a[n-1] + b[n-1]) /2,
b[n] == Sqrt@(a[n-1] b[n-1]),
a[0] == a0,
b[0] == b0
},{a,b},{n,1,3}
]

for example

RecurrenceTable[{a[n] == (a[n-1]+ b[n-1]) /2, b[n] == Sqrt@(a[n-1] b[n-1]), a[0] ==1.,b[0] == 2.
},{a,b},{n,1,10}]

gives

{{1.5, 1.4142135623730951}, {1.4571067811865475, 1.4564753151219703}, {1.4567910481542587, 1.456791013939555}, {1.4567910310469068, 1.4567910310469068}, {1.4567910310469068, 1.4567910310469068}, {1.4567910310469068, 1.4567910310469068}, {1.4567910310469068, 1.4567910310469068}, {1.4567910310469068, 1.4567910310469068}, {1.4567910310469068, 1.4567910310469068}, {1.4567910310469068, 1.4567910310469068}}