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which happens as a result of tiny errors in coefficients if the roots are very close together, even with exact methods of solution, and in this case the imaginary parts are smaller than the errors in the real parts, so I'm doubtful that the presence of nonzero imaginary parts is evidence of instability.
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There are two kinds of unstabilities that must not be confused here: Unstabilities that result from approximations made during the computation and unstabilities that result from approximations of input coefficients. The former unstabilities result generally from subtraction of two close numbers, and
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that the estimates are not that bad, really, as far as absolute error is concerned. However, whether that means that the numerical stability is not too bad, or just that the example given isn't a very good one to illustrate the point, I don't know. As for producing complex roots, that is something
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The problem is that your coefficients are wrong. The correct coefficients of your polynomial are : 1 x^4 -8.006 x^3 + 24.036011 x^2 -32.072044006 x + 16.048044012. Because your roots are so close to each other, rounding the coefficients produce the wrong results, which are technically not that far
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the
Ferrari's method and the one based on the depressed quartic are both known to be numerically unstable (e.g. see Refs. and ). For example, consider the polynomial: 16.048 - 32.072 x + 24.036 x^2 - 8.006 x^3 + x^4 whose roots are 2.003, 2.002, 2.001 and 2. By using the present method you will
943:) have a better stability and are more efficient. Indeed, for applying algebraic methods, one needs an application of Newton's method for each square or cube root appearing in the formula. This is one reason for not mentioning this work in Knowledge. A stronger reason is that it is
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The whole point of the concept of numerical instability is that the results are highly sensitive to small numerical inaccuracies such as rounding of coefficients, so that is not a valid criticism of the original post by
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S. L. Shmakov, A universal method of solving quartic equations. Int. J. Pure Appl. Math. 71, 2 251–259 (2011) A.Orellana and C. De
Michele ACM Transactions on Mathematical Software, Vol. 46, No. 2, 20 (2020),
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I you plot your equation (with the rounded coefficients), you will find that 2.006000, 2.000071 and 1.999857 are precisely where the curve meets with the Y axis. So the estimates are not that bad after all.
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I have tested the formula on several thousand cases, and it worked correctly for all of them. I think, therefore, you must have made a mistake of some kind. I agree, therefore, with
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Apologies. Please delete my comment. It seems to have been caused by rounding errors. And yes, I did mean “discriminant,” not “determinant.” Thank you for catching that!
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I tried the same equation at wolframalpha.com, and without rounding, it finds the right roots, but the slightest rounding, even by 1 digit, will produce complex roots.
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is the determinant,” but I found (at least a few) examples where this equality doesn’t hold. Is there something wrong with the formulas given to compute
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584:'s suggestion that you provide your examples, so that we can check them. (Incidentally, you mean "discriminant", not "determinant".)
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is an example of he latter unstabilities, where a very small change of one coefficient changes dramatically the nature of the roots.
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Nevertheless, the discussion on numerical stability of
Ferrari's method and its variants is totally useless since the general
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As
Ferrari's method contains additions and subtractions, it is normal that it is unstable when applied numerically.
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for a thorough discussion and other tests see Ref. . For a numerically stable and efficient quartic solver, see
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from the real ones... But try again with the full precision of coefficient values, and see what it does.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Please, provide your examples for allowing to search where is the error.
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can sometimes be avoided by changing the algorithm. For example, when a
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has a root that is close to zero, the roots are better approximated as
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An editor has asked for a discussion to address the redirect
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441:{\displaystyle \Delta _{1}^{2}-4\Delta _{0}^{3}=-27\Delta }
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obtain the following very bad estimates of quartic roots:
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