condition number

function K of the input x of a well-posed problem which describes how much its variation influences the variation of the output g(x)

DBpedia resource is: http://dbpedia.org/resource/Condition_number

Abstract is: In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given one is solving for x, and thus the condition number of the (local) inverse must be used. In linear regression the condition number of the moment matrix can be used as a diagnostic for multicollinearity. The condition number is an application of the derivative, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several variables. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. In non-mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the independent variables) there is a large change in the answer or dependent variable. This means that the correct solution/answer to the equation becomes hard to find. The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability; in general, a backward stable algorithm can be expected to accurately solve well-conditioned problems. Numerical analysis textbooks give formulas for the condition numbers of problems and identify known backward stable algorithms. As a rule of thumb, if the condition number , then you may lose up to digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods. However, the condition number does not give the exact value of the maximum inaccuracy that may occur in the algorithm. It generally just bounds it with an estimate (whose computed value depends on the choice of the norm to measure the inaccuracy).

condition number is …
instance of (P31):
functionQ11348

External links are
P646Freebase ID/m/01_rm
P8313Lex IDkonditionstal
P2812MathWorld IDConditionNumber
P6366Microsoft Academic ID84545080
P10283OpenAlex IDC84545080
P1482Stack Exchange taghttps://math.stackexchange.com/tags/condition-number

P1571codomainset of complex numbersQ26851286
set of real numbersQ1174982
P2534defining formula K ( x ) g ( x ) x g ( x )  if  g  is differentiable. {\displaystyle K(x)\approx \lVert g'(x)\rVert {\frac {\lVert x\rVert }{\lVert g(x)\rVert }}{\text{ if }}g{\text{ is differentiable.}}}
K ( x ) = sup x D ( g ) { g ( x ¯ ) g ( x ) x ¯ x x g ( x ) } {\displaystyle K(x)=\sup _{x\in D(g)}\left\{{\frac {\lVert g({\overline {x}})-g(x)\rVert }{\lVert {\overline {x}}-x\rVert }}{\frac {\lVert x\rVert }{\lVert g(x)\rVert }}\right\}}
P1269facet ofmathematicsQ395
well-posed problemQ1455898
P7235in defining formula K ( x ) {\displaystyle K(x)}
D ( g ) {\displaystyle D(g)}
P6104maintained by WikiProjectWikiProject MathematicsQ8487137
P2579studied innumerical analysisQ11216

Reverse relations

uri / http://www.wikidata.org/entity/L27184-S1L27184-S1item for this senseP5137
Q56999069A Survey of Condition Number Estimation for Triangular Matricesmain subjectP921

The articles in Wikimedia projects and languages

Catalan (ca / Q7026)Nombre de condiciówikipedia
      Podmíněnost maticewikipedia
      Kondition (Mathematik)wikipedia
      Condition numberwikipedia
      Número de condiciónwikipedia
      Häiriöalttius (matematiikka)wikipedia
      Conditionnement (analyse numérique)wikipedia
      Condizionamento (matematica)wikipedia
      条件数wikipedia
      조건수wikipedia
      Conditie (wiskunde)wikipedia
      Wskaźnik uwarunkowaniawikipedia
      Número de condicionamentowikipedia
      Число обусловленностиwikipedia
      Konditionstalwikipedia
      Число обумовленостіwikipedia
      条件数wikipedia

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