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> You should be suspicious of using floats and expecting the solver to detect if a solution is real. Consider the polynomial $x^2 + C$ for $C \approx 0$. Are al

> You should be suspicious of using floats and expecting the solver to detect if a solution is real. Consider the polynomial $x^2 + C$ for $C \approx 0$. Are all of its roots real or complex? The problem is that even a very small change in C, maybe caused by rounding errors, can easily change a root from real to not real. C \approx 0? The imaginary parts in the first bug are nowhere close to zero.

ogogmad 6y ago
I was actually responding to the following claim: > I would then recommend they not have an option for "real" if they can't get it right. Users will expect it to work (unless the docs point out it is unreliable). It can't work reliably if the coefficients of the polynomial consist of floats. At least not in general. This point is a more fundamental one than your example. Also, regarding your particular example: When solving cubics using the cubic formula, it is often the case that you will end up computing with complex numbers even if the result ends up real. If you introduce floats into this, it's impossible to guarantee that the imaginary parts will cancel out completely. Instead, they may cancel up to 10^-21 or something like that, which is precisely what happens in your example.
- mehrdadn 6y ago
  You're missing my point entirely. They don't need to solve this by keeping the variable real at every step or anything like that. What they need to do is to ensure the final solutions are real. Which means they need to solve it in the complex domain entirely and then filter out non-real solutions at the end, where they have the final magnitudes of the real and imaginary components. The only place that would run into trouble is where the imaginary components are small but nonzero, where you'd just expect it to use some tolerance threshold as the cutoff, but that obviously isn't the case in the first example (hence my comment). There's no reason that the algorithm must fail on univariate real polynomials if it can't work equally well on arbitrary functions with arbitrary domains. It's trivial for a symbolic engine to recognize a univariate polynomial. And practically speaking it's the job of a computer algebra system to recognize different well-known classes of functions and treat them as well as it can. What I suspect likely happened here is they've been just so busy dealing with the more general problems (which are far more difficult) that they haven't gotten around to implementing many better methods for the special cases (like polynomials). Which I sympathize with, as I'd have virtually no clue how to solve some of the more general problems to begin with.

ogogmad 6y ago

I was actually responding to the following claim: > I would then recommend they not have an option for "real" if they can't get it right. Users will expect it to work (unless the docs point out it is unreliable). It can't work reliably if the coefficients of the polynomial consist of floats. At least not in general. This point is a more fundamental one than your example. Also, regarding your particular example: When solving cubics using the cubic formula, it is often the case that you will end up computing with complex numbers even if the result ends up real. If you introduce floats into this, it's impossible to guarantee that the imaginary parts will cancel out completely. Instead, they may cancel up to 10^-21 or something like that, which is precisely what happens in your example.

mehrdadn 6y ago
You're missing my point entirely. They don't need to solve this by keeping the variable real at every step or anything like that. What they need to do is to ensure the final solutions are real. Which means they need to solve it in the complex domain entirely and then filter out non-real solutions at the end, where they have the final magnitudes of the real and imaginary components. The only place that would run into trouble is where the imaginary components are small but nonzero, where you'd just expect it to use some tolerance threshold as the cutoff, but that obviously isn't the case in the first example (hence my comment). There's no reason that the algorithm must fail on univariate real polynomials if it can't work equally well on arbitrary functions with arbitrary domains. It's trivial for a symbolic engine to recognize a univariate polynomial. And practically speaking it's the job of a computer algebra system to recognize different well-known classes of functions and treat them as well as it can. What I suspect likely happened here is they've been just so busy dealing with the more general problems (which are far more difficult) that they haven't gotten around to implementing many better methods for the special cases (like polynomials). Which I sympathize with, as I'd have virtually no clue how to solve some of the more general problems to begin with.