[/
  Copyright Nick Thompson, John Maddock, 2020
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]

[section:agm Arithmetic-Geometric Mean]

[h4 Synopsis]

    #include <boost/math/tools/agm.hpp>

    namespace boost::math::tools {

    template<typename Real>
    Real agm(Real a0, Real g0);
 
    } // namespaces


The function `agm` produces the limiting value of the sequence

[$../equations/agm_sequence.svg]


A basic usage is

    double G = boost::math::tools::agm(sqrt(2.0), 1.0);

The AGM inequality

[$../equations/agm_sequence.svg]

shows that

[$../equations/agm_bound.svg]

We use this condition internally to measure convergence; however, there is no need to worry about putting arguments in the correct order since we extend `agm` to a symmetric function by definition.
Both arguments must be non-negative, as the sequence becomes complex for negative arguments.
(We have not implemented the complex-valued AGM sequence.)
The function `agm` is "essentially" one-dimensional, as the scale invariance `agm(k*x, k*y) == k*agm(x,y)` always allows us to take one argument to be unity.
The following ULP plot has been generated with the function `agm(x, Real(1))`:

[$../graphs/agm_ulps_plot.svg]

The graph above shows an ulps plot of the Boost implementation of  `agm(x, Real(1))`.
An ~2 ULP bound is to be expected.

A google benchmark for various types is available in `boost/libs/math/reporting/performance/test_agm.cpp`; some results on a consumer laptop are provided for convenience:

```
Run on (16 X 2300 MHz CPU s)
CPU Caches:
  L1 Data 32K (x8)
  L1 Instruction 32K (x8)
  L2 Unified 262K (x8)
  L3 Unified 16777K (x1)
Load Average: 2.02, 2.14, 2.00
-------------------------------------------------------------------------------
Benchmark                                     Time             CPU   Iterations
-------------------------------------------------------------------------------
AGM<float>                                 8.52 ns         8.51 ns     59654685
AGM<double>                                13.5 ns         13.5 ns     51709746
AGM<long double>                           30.6 ns         30.6 ns     18745247
AGM<boost::multiprecision::float128>       2332 ns         2332 ns       299303
```

If any inputs are NaNs, the result is a NaN.
If any inputs are +∞, the result is +∞, unless the other argument fails NaN or negative validation.

[heading References]

* Steven R. Finch. ['Mathematical Constants] Cambridge, 2003.


[endsect]
