Numerical experimentation · from two means to a new mean
Let two means meet
Choose two powers, then drag the dot. Updating both coordinates simultaneously squeezes a bracket around their common limit.
One power mean
P₁(x, y)At the dot
The Gauss composition
M₁,₋₁(x, y)Common-limit estimate
A closing bracket
—lower upperiteration n →
Drag either heatmap, edit x and y, or focus a heatmap and use the arrow keys (Shift: larger steps; Home: reset). Both heatmaps share equal axes and the same color scale.
- Floating-point bracket
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- Bracket half-width
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- What this pair produces
- Geometric mean
Computing the bracket…
Read the numerical trace
| n | xₙ | yₙ | half-width |
|---|
Pₚ(x, y) = ((xᵖ + yᵖ)/2)¹/ᵖ, with P₀ = √(xy). Each update uses the same old pair: (xₙ₊₁, yₙ₊₁) = (Pₚ(xₙ, yₙ), Pq(xₙ, yₙ)). In exact arithmetic, the limit lies between min(xₙ, yₙ) and max(xₙ, yₙ), so the midpoint error is at most the bracket half-width. The displayed bracket uses ordinary floating-point arithmetic, not certified interval arithmetic; its half-width excludes rounding error. A collapsed or stalled floating-point bracket is reported separately. Selected points stop at relative half-width 10⁻¹² or 64 updates. The 111 × 111 heatmap uses 10⁻⁹; colors and contours are raw mean values on [0.08, 2]². No animation runs in the background.