# What is the relation between the condition of a problem and the condition of the underlying mathematical model?

This is a more general question about the theory of condition in numerical analysis. Can one classify an abstract problem as good or bad conditioned?

Lets take the following abstract problem: Take two sufficiently parallel lines and find the intersection of the two lines.

Can you say that for sufficiently parallel lines this problem is bad conditioned?

If the answer is "yes", is it possible to find an underlying mathematical model which solves our problem but is good conditioned?

• Hi, welcome to the site. Have you ever hear of the term "condition number" ? Dec 1, 2021 at 15:15
• Yes, I have heard of them in terms of matrices. Dec 1, 2021 at 15:31
• OK good. For any linearizable system, the matrix definition extends to the non linear system (using the jacobian). For other systems, I'm not sure how to define conditionning Dec 1, 2021 at 15:56
• Your questions strongly suggest to me that you are confusing the conditioning of a problem with the stability of an algorithm. It is a common mistake, so this is easily forgiven. However, the conditioning of the underlying mathematical model is not an expression that makes sense without additional words. My best guess is that you are asking: "Given that the problem is ill-conditioned, can we find a stable algorithm to solve it?" Dec 2, 2021 at 13:59
• I am not sure if this is the question I actually had in mind here but I am interested in your answer to that question. Dec 2, 2021 at 16:03