Arithmetic-Geometric Mean (AGM) Calculator
Compute the arithmetic-geometric mean (AGM) of two positive numbers a and b by iterating the arithmetic and geometric means until they converge, with the full iteration table.
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Input
Enter two positive numbers to iterate the arithmetic and geometric means until they converge and compute the arithmetic-geometric mean (AGM).
Both a and b must be positive numbers.
Result
AGM of 1 and 2
1.45679103
Arithmetic mean
1.5
Geometric mean
1.41421356
Iterations
4
Iteration steps
Each row shows the arithmetic mean, the geometric mean, and their difference per iteration. The closer the difference is to zero, the more it has converged.
| Step | Arithmetic mean | Geometric mean | Difference |
|---|---|---|---|
| 1 | 1.5 | 1.41421356 | 0.08578644 |
| 2 | 1.45710678 | 1.45647532 | 0.00063147 |
| 3 | 1.45679105 | 1.45679101 | 0.00000003 |
| 4 | 1.45679103 | 1.45679103 | 0 |
Each step computes a'' = (a + b) / 2 and b'' = the square root of ab. Since the arithmetic mean is always at least the geometric mean, the two values converge quadratically to a common limit.
How it works
- The arithmetic-geometric mean of two numbers a and b is the common limit reached by repeatedly replacing them with their arithmetic mean (a+b)/2 and geometric mean the square root of ab until the two values agree.
- Because the arithmetic mean is always at least the geometric mean, the gap between the two values shrinks quadratically each step, so the iteration converges to high precision in only a few steps.
- The arithmetic-geometric mean was studied by Gauss and is used to compute the complete elliptic integral of the first kind very efficiently.
- Both inputs a and b must be positive. When a and b are equal, the arithmetic-geometric mean is simply that value.
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