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You seem to be confusing the mathematical definition of matrix multiplication with an explicit algorithm to compute it. The wiki page you linked is almost entir

You seem to be confusing the mathematical definition of matrix multiplication with an explicit algorithm to compute it. The wiki page you linked is almost entirely about the abstract mathematical definition and talks about algorithms only for a single paragraph, which is about computational complexity. If you look at the wiki page for a particular algorithm (e.g. Strassen: https://en.wikipedia.org/wiki/Strassen_algorithm https://en.wikipedia.org/wiki/Strassen_algorithm) then of course they talk about stability in comparison to the naive algorithm (although the page could be improved quite a bit). If your college textbooks do not mention error analysis, conditioning and stability then they are not numerical linear algebra books worthy of the name. Check out a reference like Trefethen and Bau's Numerical Linear Algebra for example. This book has a whole part (out of the 7 parts in the book) talking about conditioning and stability, and these ideas are present throughout other parts as well. Once more the type of analysis I'm talking about is emphatically not implementation dependent. It is a property of the algorithm itself. For an example of the sort of statement I mean check out theorem 3.1 of this paper: https://arxiv.org/abs/math/0603207 https://arxiv.org/abs/math/0603207. If you disagree with me then please indicate what sort of "implementation factors" appear in the statement of the theorem.

DannyBee 4y ago
Correctness of a mathematical algorithm is of course defined by whether it meets a mathematical definition. In this case, correctness of matrix multiplication is only defined by whether it meets the mathematical definition of matrix multiplication. That's it. That's the whole thing. Correctness of all computer science algorithms is defined by whether they meet a particular algorithmic specification. That's literally the definition of correctness: https://en.wikipedia.org/wiki/Correctness_(computer_science) https://en.wikipedia.org/wiki/Correctness_(computer_science) "In theoretical computer science, an algorithm is correct with respect to a specification if it behaves as specified" The specification is the matrix multiplication definition given right on that Wikipedia page. this algorithm meets it. It is therefore correct. (again, "error" does not appear on the page here either). There is no separate, special definition for "correctness (mathematical)" or "correctness (eigenket)". You really seem to want to there to be one, but it ain't there. You really really really don't want to let this go, but the problem is - nothing, anywhere, agrees with you that correctness and usefulness are the same thing. Nor can you cite any reference, like i just did, to correctness that requires it do anything other than meet the specified mathematical definition. Your papers don't do it, my books don't do it. Nothing does it. Because it's not a thing. My college textbooks talk about error analysis. I did not claim otherwise. They talk about it in the context of how to make an algorithm useful for a particular purpose, not about correctness. They are not making a ridiculous claim like you are. I'm not going down this path anymore with you. Believe what you want, the rest of us will continue to not confuse it, and sources that people look at (textbooks, wikipedia, etc) will continue to not lead them astray. I can only hope that at some point, you too stop trying to do so.

DannyBee 4y ago

Correctness of a mathematical algorithm is of course defined by whether it meets a mathematical definition. In this case, correctness of matrix multiplication is only defined by whether it meets the mathematical definition of matrix multiplication. That's it. That's the whole thing. Correctness of all computer science algorithms is defined by whether they meet a particular algorithmic specification. That's literally the definition of correctness: https://en.wikipedia.org/wiki/Correctness_(computer_science) https://en.wikipedia.org/wiki/Correctness_(computer_science) "In theoretical computer science, an algorithm is correct with respect to a specification if it behaves as specified" The specification is the matrix multiplication definition given right on that Wikipedia page. this algorithm meets it. It is therefore correct. (again, "error" does not appear on the page here either). There is no separate, special definition for "correctness (mathematical)" or "correctness (eigenket)". You really seem to want to there to be one, but it ain't there. You really really really don't want to let this go, but the problem is - nothing, anywhere, agrees with you that correctness and usefulness are the same thing. Nor can you cite any reference, like i just did, to correctness that requires it do anything other than meet the specified mathematical definition. Your papers don't do it, my books don't do it. Nothing does it. Because it's not a thing. My college textbooks talk about error analysis. I did not claim otherwise. They talk about it in the context of how to make an algorithm useful for a particular purpose, not about correctness. They are not making a ridiculous claim like you are. I'm not going down this path anymore with you. Believe what you want, the rest of us will continue to not confuse it, and sources that people look at (textbooks, wikipedia, etc) will continue to not lead them astray. I can only hope that at some point, you too stop trying to do so.