I have a 16 by 16 matrix A with increasing powers in each row, and a vector b. When I try to use Mathematica to solve Ax=b (using LinearSolve) I get an error warning me there may be significant numerical errors because of badly conditioned matrix.
I don't quite understand this notion, but I am under the impression that it just means it is a very "sensitive" equation, so an error in A or B may cause significant errors in x. Given that A and b have integer components that I have not rounded, is this something to be concerned? Does the algorithm or rounding Mathematica does with LinearSolve affect this?
EDIT:
I am unable to post the code right now, will do so later. What confuses me is this (in wikipedia: Condition Number): "For example, the condition number associated with the linear equation Ax = b gives a bound on how inaccurate the solution x will be after approximation. Note that this is before the effects of round-off error are taken into account; conditioning is a property of the matrix, not the algorithm or floating-point accuracy of the computer used to solve the corresponding system. In particular, one should think of the condition number as being (very roughly) the rate at which the solution x will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x. On the other hand, if the condition number is small, then the error in x will not be much bigger than the error in b.". If A and b are exact, why does condition number matter?
For some reason, doing Inverse[A].b instead of LinearSolve[A,b] didn't yield this error, so I believe it solves the problem in my case.