---
tag: B6E54YTR64
kind: concept
name: Numerical analysis
msc: [65-01, 65G50]
sources:
  - url: https://en.wikipedia.org/wiki/Numerical_analysis
same_as:
  wikidata: Q11216
  wikipedia: Numerical analysis
---

## Summary

Numerical analysis is the study of algorithms that compute approximate solutions to problems of continuous mathematics.
It designs methods for tasks such as solving equations, computing integrals, finding eigenvalues and solving differential equations.
It also proves how large the error of such a method can be, and how fast the method converges.
Two typical sources of error are the [[truncation error|4AKCQERM89]] of a finite method and the [[round-off error|CPN6SD3BTV]] of [[floating-point arithmetic|RTKRSC3KPH]].

## Explanations

### L1 human

Many equations have no solution that one can write down with a formula, such as $\cos x = x$.
Numerical analysis gives step-by-step recipes that produce a number as close to the true answer as one wishes.
A computer works with a finite number of digits, so each step can introduce a tiny error.
Part of the subject is to check that these tiny errors stay small and do not grow out of control.
For example, the [[bisection method|7VMYC7MB3Q]] halves an interval around the solution again and again.
After $20$ steps the solution is known to within about one millionth of the starting length.