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
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Arithmetic–geometric mean

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Enter valid values to see the result.

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Understand the subject

What is the arithmetic–geometric mean?

The arithmetic–geometric mean (AGM) starts with two non-negative values and repeatedly replaces them with their arithmetic and geometric means. The two results rapidly meet at one shared limit.

The relationship

The defining relationship

an is the arithmetic sequence and gn is the geometric sequence. With unequal positive inputs, one moves down and the other moves up.

See the idea

Two bounds close in on one value

Worked example

AGM of 1 and 2

Start with arithmetic mean 1.5 and geometric mean √2 ≈ 1.4142. Repeating the two updates gives the common value 1.45679….

Interpret with care

Choose the relationship that fits

AGM is a special numerical and mathematical construction, not a replacement for an ordinary average. Zero in either input gives a zero limit in this calculator’s supported model.

For a side-by-side view of the classical means, use the Mean Comparison Calculator.

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.