kondíciószám

A kondíciószám, often translated as "condition number," is a measure of how sensitive the solution of a linear system of equations is to small changes in the input data. It is particularly relevant in numerical linear algebra and scientific computing. A small condition number indicates that the system is well-conditioned, meaning that small perturbations in the matrix or the right-hand side vector will result in small changes in the solution. Conversely, a large condition number signifies an ill-conditioned system, where even minor errors can lead to significantly inaccurate solutions.

The condition number is typically defined using a matrix norm. For a non-singular square matrix A, its condition number with respect to a norm ||.|| is given by cond(A) = ||A|| ||A⁻¹||. Different norms (e.g., the 1-norm, 2-norm, or infinity-norm) can be used, leading to different condition numbers. The 2-norm condition number, for instance, is the ratio of the largest singular value of A to the smallest singular value of A.

In practice, a high condition number suggests that numerical algorithms attempting to solve the linear system may encounter difficulties, potentially yielding unreliable results. This can arise from various sources, such as nearly linearly dependent rows or columns in the matrix, or issues inherent to the problem being modeled. Therefore, understanding and, if possible, improving the condition number of a system is crucial for obtaining accurate and stable numerical solutions. Techniques like preconditioning are often employed to reduce the condition number and enhance the performance of iterative solvers.