konditionstal

konditionstal (condition number) is a measure of the sensitivity of a function's output to small changes in its input. In numerical analysis, it is used to gauge how errors from input, rounding, or data propagate through computations. The concept applies to functions, matrices, and linear systems.

For a nonsingular matrix A, the condition number with respect to a chosen norm is cond(A) = ||A|| ||A^{-1}||. In the common 2-norm, cond_2(A) equals the ratio of the largest to smallest singular value, σ_max/σ_min. A high condition number indicates an ill-conditioned problem.

In solving Ax = b, if the input b is perturbed to b + Δb, the relative change in the solution x changes by at most about cond(A) times the relative change in b (for small perturbations). More generally, perturbations in A and b satisfy bounds that involve cond(A).

Condition number depends on the norm; different norms give different cond values. It can be computed via singular value decomposition for the 2-norm, or via norms of A and A^{-1} for the 1- and ∞-norms. Practically, a condition number close to 1 suggests stable numerics; values much larger indicate sensitivity to errors.

Applications include linear systems, eigenvalue problems, and optimization; it is a key indicator of numerical stability and error amplification. The concept is closely related to backward error and forward error analyses.