In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. It describes how errors in the input data propagate through the algorithm. In a stable method, the errors due to the approximations get damped out (or remain the same) as the computation proceeds. In an unstable method, any errors in processing get magnified as the calculation proceeds. Unstable methods quickly generate garbage and are useless for numerical processing. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics) using basic arithmetical operations like addition. ... Flowcharts are often used to represent algorithms. ...

The numerical stability of a method together with the condition number (in matrix computations) contribute to the accuracy of the result one can get when using approximate methods to calculate a specific mathematical problem. In numerical analysis, the condition number associated with a numerical problem is a measure of that quantitys amenability to digital computation, that is, how well-posed the problem is. ...

Sometimes a single calculation can be achieved in several ways, all of which are algebraically equivalent in terms of ideal real or complex numbers, but in practice yield different results as they have different levels of numerical stability. One of the common tasks of numerical analysis is to try to select algorithms which are robust — that is to say, have good numerical stability. Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics) using basic arithmetical operations like addition. ...

Given an algorithm f(x), with x the input data and ε the error in the input data, one says that the algorithm is numerically stable for the absolute error if In the mathematical subfield of numerical analysis the approximation error in some data is the difference between the exact value and the value used. ...

and numerically stable for the relative error if In the mathematical subfield of numerical analysis the approximation error in some data is the difference between the exact value and the value used. ...

One says that an algorithm is numerically unstable for the absolute error if

and numerically unstable for the relative error if

Notes

In calculating numerical solutions to certain partial differential equations, stability is sometimes achieved by including numerical diffusion. Numerical diffusion is a mathematical term which ensures that roundoff and other errors in the calculation get spread out and do not add up to cause the calculation to "blow up". In mathematics, a partial differential equation (PDE) is an equation relating the partial derivatives of an unknown function of several variables. ... Numerical diffusion is a difficulty with computer simulations of continuous systems such as fluids or plasmas. ...

Numerical stability is the reason why one usually cannot test a numerical algorithm such as a climate simulation by running it backward. Running the algorithm forward includes numerical methods to ensure that the approximation errors become less and less important as the calculation proceed insuring numerical stability. Running the code backward causes those mechanisms to magnify those errors generating useless results. Climate models use quantitative methods to simulate the interactions of the atmosphere, oceans, land surface, and ice. ...

When solving a numerical problem with an approximated method, two types of errors can occur:

• Truncation errors: One can only make a finite number of calculations. Examples: calculating a transcendental function using its Taylor expansion, integrating using a sum of finite rectangles.
• Round-off errors: Certain numbers need an infinite number of digits to be represented (such as π), when rounding these numbers the round-off errors will propagate through the calculation.