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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク condition number ： ウィキペディア英語版 condition number In the field of numerical analysis, the condition number of a function with respect to an argument 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 $f(x)\; =\; y,$ one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. In linear regression the condition number 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. 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 the backward stable algorithms. As a rule of thumb, if the condition number $\backslash kappa(A)\; =\; 10^k$, then you may lose up to $k$ 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). ==Matrices== For example, the condition number associated with the linear equation ''Ax'' = ''b'' gives a bound on how inaccurate the solution ''x'' will be after approximation. Note that this is before the effects of round-off error are taken into account; conditioning is a property of the matrix, not the algorithm or floating point accuracy of the computer used to solve the corresponding system. In particular, one should think of the condition number as being (very roughly) the rate at which the solution, ''x'', will change with respect to a change in ''b''. Thus, if the condition number is large, even a small error in ''b'' may cause a large error in ''x''. On the other hand, if the condition number is small then the error in ''x'' will not be much bigger than the error in ''b''. The condition number is defined more precisely to be the maximum ratio of the relative error in ''x'' divided by the relative error in ''b''. Let ''e'' be the error in ''b''. Assuming that ''A'' is a nonsingular matrix, the error in the solution ''A''−1''b'' is ''A''−1''e''. The ratio of the relative error in the solution to the relative error in ''b'' is : $\backslash frac\; b\; \backslash right\backslash Vert\; \}\; .$ This is easily transformed to : $\backslash left(\; \backslash left\backslash Vert\; A^\; e\; \backslash right\backslash Vert\; /\; \backslash left\backslash Vert\; e\; \backslash right\backslash Vert\; \backslash right)\; \backslash cdot\; \backslash left(\; \backslash left\backslash Vert\; b\; \backslash right\backslash Vert\; /\; \backslash left\backslash Vert\; A^\; b\; \backslash right\backslash Vert\; \backslash right)\; .$ The maximum value (for nonzero ''b'' and ''e'') is easily seen to be the product of the two operator norms: : $\backslash kappa(A)\; =\; \backslash left\backslash Vert\; A^\; \backslash right\backslash Vert\; \backslash cdot\; \backslash left\backslash Vert\; A\; \backslash right\backslash Vert\; .$ The same definition is used for any consistent norm, i.e. one that satisfies : $\backslash kappa(A)\; \backslash ge\; 1\; .\backslash ,$ When the condition number is exactly one (which can only happen if ''A'' is a linear isometry), then a solution algorithm can find (in principle, meaning if the algorithm introduces no errors of its own) an approximation of the solution whose precision is no worse than that of the data. However, it does not mean that the algorithm will converge rapidly to this solution, just that it won't diverge arbitrarily because of inaccuracy on the source data (backward error), provided that the forward error introduced by the algorithm does not diverge as well because of accumulating intermediate rounding errors. The condition number may also be infinite, but this implies that the problem is ill-posed (does not possess a unique, well-defined solution for each choice of data -- that is, the matrix is not invertible), and no algorithm can be expected to reliably find a solution. Of course, the definition of the condition number depends on the choice of norm, as can be illustrated by two examples. If $\backslash left\backslash |\; \backslash cdot\; \backslash right\backslash |$ is the norm (usually noted as $\backslash left\backslash |\; \backslash cdot\; \backslash right\backslash |\_2$) defined in the square-summable sequence space ℓ2 (which matches the usual distance in a standard Euclidean space), then : $\backslash kappa(A)\; =\; \backslash frac\; ,$ where $\backslash sigma\_(A)$ and $\backslash sigma\_(A)$ are maximal and minimal singular values of $A$ respectively. Hence * If $A$ is normal then *: $\backslash kappa(A)\; =\; \backslash left|\backslash frac\backslash right|\; ,$ where $\backslash lambda\_(A)$ and $\backslash lambda\_(A)$ are maximal and minimal (by moduli) eigenvalues of $A$ respectively. * If $A$ is unitary then *: $\backslash kappa(A)\; =\; 1\; .\backslash ,$ The condition number with respect to ''L2'' arises so often in numerical linear algebra that it is given a name, the condition number of a matrix. If $\backslash left\backslash |\; \backslash cdot\; \backslash right\backslash |$ is the norm (usually denoted by $\backslash left\backslash |\; \backslash cdot\; \backslash right\backslash |\_\backslash infty$) defined in the sequence space ℓ∞ of all bounded sequences (which matches the maximum of distances measured on projections into the base subspaces), and $A$ is lower triangular non-singular (i.e., $\backslash forall\; i,\; a\_\; \backslash ne\; 0\; \backslash ,$) then : $\backslash kappa(A)\; \backslash geq\; \backslash frac\; .$ The condition number computed with this norm is generally larger than the condition number computed with square-summable sequences, but it can be evaluated more easily (and this is often the only practicably computable condition number, when the problem to solve involves a ''non-linear algebra'', for example when approximating irrational and transcendental functions or numbers with numerical methods.) If the condition number is not too much larger than one (but it can still be a multiple of one), the matrix is well conditioned which means its inverse can be computed with good accuracy. If the condition number is very large, then the matrix is said to be ill-conditioned. Practically, such a matrix is almost singular, and the computation of its inverse, or solution of a linear system of equations is prone to large numerical errors. A matrix that is not invertible has condition number equal to infinity. == Non-linear == Condition numbers can also be defined for nonlinear functions, and can be computed using calculus. The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「condition number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.