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What is condition number?
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.
What is Cond A?
C = cond( A ) returns the 2-norm condition number for inversion, equal to the ratio of the largest singular value of A to the smallest. example. C = cond( A , p ) returns the p -norm condition number, where p can be 1 , 2 , Inf , or ‘fro’ .
What is Frobenius condition number?
The Frobenius condition number is defined by . In the Frobenius inner product is defined by. for which we have the associated norm that satisfies . The Frobenius inner product allows us to define the cosine of the angle between two given real matrices as.
How is the condition number related to the solution?
Condition number. 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.
How is the condition number of a function defined?
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.
What is the condition number of a matrix?
A condition number for a matrix measures how sensitive the answer is to perturbations in the input data and to roundoff errors made during the solution process. I should point out that there are many different condition numbers. In general, a condition number applies not only to a particular matrix, but also to the problem being solved.
How is the condition number used in linear algebra?
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.