What does the "condition number" indicate in numerical analysis?

The condition number in numerical analysis is a critical concept that quantifies the sensitivity of a function's output to changes in its input. It essentially measures how small perturbations in the input can affect the accuracy of the results produced by a mathematical function or algorithm. A high condition number indicates that even a tiny change in the input can lead to significant variations in the output, suggesting that the problem is ill-conditioned and might lead to inaccurate results when computed using finite precision arithmetic.

In contrast, a low condition number suggests that the function is well-conditioned, meaning that the outputs would change only slightly in response to small changes in the inputs. This property is especially important when dealing with numerical algorithms, as it provides insight into the reliability and stability of the results produced. Understanding the condition number helps engineers and scientists to evaluate and improve the robustness of computational methods when applying them to real-world problems.