CALCULATE WITH CONFIDENCE

Averages & Means

Arithmetic–Geometric Mean Calculator

Iterate the arithmetic and geometric means of two non-negative values to their common limit.

Averages & Means

Iterate arithmetic and geometric means

Convergent two-value process

Arithmetic–geometric mean

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Enter valid values to see the result.

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Quick guide

How to use this calculator

  1. Enter two finite non-negative values.
  2. Inspect the starting arithmetic and geometric bounds.
  3. Read their common converged limit and iteration count.

Calculation method

Replace both values repeatedly

At each step, the larger arithmetic mean decreases while the smaller geometric mean increases. They converge rapidly to the arithmetic–geometric mean.

Worked example

AGM of 1 and 2

Starting at 1.5 and √2, repeated averaging converges near 1.45679.

aₙ₊₁=(aₙ+gₙ)/2; gₙ₊₁=√(aₙgₙ)

Supported inputs

Precision and limits

Domain

Both values must be finite and non-negative.

Convergence

Iteration stops when the two floating-point bounds agree within a scale-relative tolerance, with a 100-step safety cap.