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Lets say you have a linear system $Ax = b$ you want to solve and you use LU factorization for it. Can this system ever become numerically unstable?

We know that if the condition number of $A$ is $1$, the system is numerically stable but if the condition number is high, does this mean that the system is unstable, how do you even define numerical unstability? And at which point does it become unstable?

I guess the higher the condition number is, the more unstable the system is, and there is no clear way of defining that it is numerically stable only up to a certain value.

osk
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    I believe numerical instability is a subjective measurement - of course, bigger condition numbers do give worse degrees of instability. – Sean Roberson Aug 09 '17 at 16:57
  • I recommend looking at a Numerical Analysis book, where such things are a topic of considerable interest. I like Trefethen and Bao, but others may have other suggestions. – John Hughes Aug 09 '17 at 17:40

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The conclusion is that numerical stability is subjective, so instability increases as the condition number increases however determining at what point it is considered unstable is subjective.

osk
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