Concept
How to compute reliable answers on finite machines — the algorithms behind every simulation.
Understand it in one breath
Find an approximation together with a defensible account of its error when a closed form is unavailable or impractical. Differential-equation solvers, fluid calculations, optimization, and numerical weather prediction use such methods. Stability, convergence assumptions, and roundoff differ by algorithm; Newton's method itself can diverge from a poor starting point or near a vanishing derivative.
At a glance
Iteration n | xₙ | |xₙ − √2| | Matching decimal digits |
|---|---|---|---|
0 | 1.000000 | 0.4142 | 0 |
1 | 1.500000 | 0.0858 | 0 |
2 | 1.416667 | 0.0025 | 2 |
3 | 1.414216 | 0.0000022 | 5 |
4 | 1.414214 | < 10⁻¹² | 12 |
5 | 1.414214 | Machine precision | ~16 |
Computing √2 with Newton’s method: x ← (x + 2/x) / 2. Once sufficiently close to the root and under the right conditions, the error is roughly squared at each step—quadratic convergence—so this example reaches double-precision limits in only a few iterations.
Key formula
Key moments
Babylonia — an approximation to √2
A Babylonian clay tablet records √2 to remarkable accuracy in sexagesimal notation, evidence of a systematic numerical procedure thousands of years ago.
Euler — Euler’s method
Euler introduced a simple step-by-step approximation for differential equations, a rough method that became a starting point for numerical ODE solvers.
von Neumann and the Monte Carlo method
Random sampling was put to work on calculations arising from the Manhattan Project, helping establish modern computational simulation.
Cooley and Tukey — the FFT
The fast Fourier transform reduced the work of spectral computation to O(n log n), making large-scale signal processing practical.
Modern applications
Numerical weather prediction, aircraft and vehicle simulation, financial models, molecular drug simulation, film physics, and matrix operations in machine learning.
Beyond MathVoyage
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