---
title: "Conditioning, Stability and Floating-Point Error"
description: "Conditioning says how much the answer moves when the data moves, and stability says how much an algorithm adds on top of that. Rounding error is tiny, and those two amplifiers are what make it visible"
canonical: https://lightmysky.com/learn/mathematics/conditioning-stability-and-floating-point-error-mt_q8IoCElXIV
source: https://lightmysky.com/learn/mathematics/conditioning-stability-and-floating-point-error-mt_q8IoCElXIV.md
retrieved: 2026-09-12
---

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# Conditioning, Stability and Floating-Point Error

Conditioning says how much the answer moves when the data moves, and stability says how much an algorithm adds on top of that. Rounding error is tiny, and those two amplifiers are what make it visible.

Subject: Mathematics · Area: Calculus & Analysis · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/conditioning-stability-and-floating-point-error-mt_q8IoCElXIV

## Ready when they can

- Compute a condition number and read it as an error amplifier
- Tell an ill-conditioned problem apart from an unstable algorithm
- Show catastrophic cancellation on a concrete subtraction

## Lesson: Sensitive problems and honest algorithms

Every stored number is rounded, and floating point rations about fifteen decimal digits. Most fractions refuse exact binary form, so 0.1 arrives already approximated. Adding a tiny number to a huge one may change nothing at all.

**Example.** Conditioning says how the answer moves when the data moves. The condition number multiplies the norm of the matrix by the norm of its inverse, and it reads as an error amplifier. With norms 4 and 0.5 the number is 2, so input error at most doubles; a nearly singular system loses digits even under a stable algorithm.

Stability says how much the algorithm adds on top of that problem sensitivity. An ill-conditioned problem swings wildly by itself, while an unstable algorithm ruins even calm problems. A stable method on an ill-conditioned problem can still return a poor answer, and that is nobody's coding bug.

**Tip.** Subtracting nearly equal quantities exposes the rounding dust, which is called catastrophic cancellation. With 4-digit arithmetic, 1.0001 minus 1.0000 is the dangerous pair: the leading digits cancel and the result lives on leftovers. Pairs far apart are safe, and extra digits help only after the algebra is reworked.

**Recap.** Rounding is tiny, the condition number amplifies it, stability adds its own share, and cancellation of near twins is the classic trap.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Gaussian Elimination and Row Echelon Form](https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm)

## Opens up

- [Iterative Methods for Large Linear Systems](https://lightmysky.com/learn/mathematics/iterative-methods-for-large-linear-systems-mt_FKkxKYVGYJ)
- [Fitting a Model to Data: Least Squares, Chi-Square and Goodness of Fit](https://lightmysky.com/learn/science/fitting-a-model-to-data-least-squares-chi-square-and-goodness-of-fit-mt_NN0WYlp0md)