Calculator
Example Data
These examples illustrate typical AGM magnitudes and convergence.
| a₀ | b₀ | Approx. AGM | Typical iterations (tol 1e‑10) |
|---|---|---|---|
| 1 | 2 | 1.4567910310 | ~5–7 |
| 24.5 | 7.2 | 14.4741 (rounded) | ~6–9 |
| 100 | 0.5 | 14.1724 (rounded) | ~8–12 |
Formula Used
The arithmetic‑geometric mean starts from two non‑negative values a₀ and b₀.
It forms two sequences that converge to the same limit:
- Arithmetic update:
aₙ₊₁ = (aₙ + bₙ) / 2 - Geometric update:
bₙ₊₁ = √(aₙ · bₙ) - Stopping rule: stop when
|aₙ − bₙ| ≤ tolor when max iterations is reached.
For positive inputs, the AGM converges rapidly; each iteration typically doubles correct digits.
How to Use This Calculator
- Enter your starting values a₀ and b₀ (non‑negative).
- Choose a tolerance to control the stopping accuracy.
- Set max iterations as a safety cap for difficult cases.
- Select decimal places for display and exported files.
- Enable Show iteration table if you need full steps.
- Press Compute AGM; results appear above the form.
- Use Download CSV or Download PDF for reporting.
Professional Article
1) What the AGM Represents
The arithmetic‑geometric mean (AGM) is the common limit obtained by repeatedly combining two non‑negative numbers with two complementary averages. Starting from a₀ and b₀, the process quickly drives both values toward a single stable mean used in precision mathematics. It blends additive smoothing and multiplicative structure into one convergent procedure.
2) Iteration Rules Implemented
The method updates aₙ₊₁ = (aₙ + bₙ)/2 and bₙ₊₁ = √(aₙ·bₙ). The difference |aₙ − bₙ| is tracked each step and compared with a user‑chosen tolerance. This gives a clear stopping point and a reliable convergence audit trail. You can also choose rounding to match reporting and downstream numeric expectations.
3) Convergence Speed in Practice
For positive inputs, convergence is typically quadratic: once aₙ and bₙ are close, correct digits grow rapidly per iteration. With tolerance 1e‑10, many practical pairs converge in under ten steps, even when a₀ and b₀ start far apart. When inputs are already close, two to four iterations can be enough.
4) Stability and Input Ranges
The arithmetic update is robust under rounding, while the geometric update requires non‑negative inputs to keep the square root real. Extremely large or tiny numbers can stress floating‑point limits; if needed, rescale inputs and use a tolerance consistent with your precision. Set max iterations as a safeguard when experimenting with extreme scales.
5) Why Step Tables Improve Trust
A single AGM value is useful, but a trace is often essential for reporting. The iteration table records aᵢ, bᵢ, and |aᵢ − bᵢ|, helping you confirm monotonic tightening, compare tolerances, and document exactly how the result was produced. This is valuable for teaching, QA checks, and reproducible computational notes.
6) Connection to Elliptic Integrals
AGM has a celebrated link to the complete elliptic integral of the first kind. This relationship enables fast evaluation of elliptic integrals and appears in high‑precision algorithms for mathematical constants. It explains why AGM is embedded in many numerical libraries. In practice, it supports efficient computation of complete elliptic integrals K(k) for many k.
7) Applied Use Cases and Data
Beyond theory, AGM supports efficient computation in models involving elliptic integrals, calibration tasks, and numeric benchmarking. Inputs may range from fractions (0.5) to large values (100) and still converge quickly. Logging iterations provides a measurable performance metric. Iteration count can also act as a quick indicator of numerical difficulty for a dataset.
8) Reading Results and Exports
The reported AGM is based on the final aₙ and bₙ once the tolerance is met. CSV exports are ideal for analysis and plotting, while the PDF provides a compact report with inputs, results, and a short iteration snapshot for sharing and documentation. Keep settings consistent when comparing runs across studies, dashboards, or publications.
FAQs
1) Can I use negative inputs?
No. This calculator expects non‑negative values so the geometric mean stays real. If you need signed variants, transform the problem or use a complex‑valued method.
2) What does tolerance control?
Tolerance sets the stopping threshold for |aₙ − bₙ|. Smaller tolerances usually increase iterations but improve agreement between arithmetic and geometric sequences.
3) Why do aₙ and bₙ become equal?
Each step pulls the larger value down via averaging and pushes the smaller up via geometric growth. For positive inputs, both sequences converge to the same limit, the AGM.
4) How many iterations are typical?
Often 5–12 iterations are enough for 1e‑10 tolerance, depending on how different a₀ and b₀ are. Very uneven pairs may take a few more steps.
5) What if one input is zero?
If either a₀ or b₀ is zero, the geometric mean becomes zero and the AGM converges to zero. The calculator flags this case with a note.
6) Why export CSV and PDF?
CSV is ideal for spreadsheets, plots, and further analysis. PDF is convenient for sharing a compact report containing inputs, results, and a short iteration snapshot.
7) Is the AGM always between a₀ and b₀?
Yes for non‑negative inputs. Each arithmetic mean lies between the previous pair, and the geometric mean never exceeds the arithmetic mean. The common limit therefore stays within the initial bounds.