Editing Numerical analysis (section)

==Generation and propagation of errors==
{{further|Error propagation}}
The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.

===Round-off===
[[Round-off error]]s arise because it is impossible to represent all [[real number]]s exactly on a machine with finite memory (which is what all practical [[digital computer]]s are).

===Truncation and discretization error===
[[Truncation error]]s are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a [[discretization error]] because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of <math>3x^3+4=28</math>, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01.

Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type {{tmath|a+b+c+d+e}} is even more inexact.

A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.

===Numerical stability and well-posed problems===
An algorithm is called ''[[numerically stable]]'' if an error, whatever its cause, does not grow to be much larger during the calculation.<ref name="stab">{{harvnb|Higham|2002}}</ref> This happens if the problem is ''[[well-conditioned]]'', meaning that the solution changes by only a small amount if the problem data are changed by a small amount.<ref name="stab"/> To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error.<ref name="stab"/>
Both the original problem and the algorithm used to solve that problem can be well-conditioned or ill-conditioned, and any combination is possible.
So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem.