wrong roots

[cridemichel]

Hi,
this implementation is based on Shmakov method [1] which is known to be numerically unstable
(see [2] and [3]).
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, from your code you obtain the following
bad estimates of quartic roots:
``
❯ ./quartic 1 -8.006 24.036 -32.072 16.048

2.006000 + 0.000000i
2.000071 + 0.000126i
2.000071 + -0.000126i
1.999857 + -0.000000i
``
for a thorough discussion and other tests see Ref. [2],

best C.

[1] S. L. Shmakov, A universal method of solving quartic equations. Int. J. Pure Appl. Math. 71, 2 251–259 (2011)
[2] A.Orellana and C. De Michele ACM Transactions on Mathematical Software, Vol. 46, No. 2, 20 (2020), https://doi.org/10.1145/3386241
[3] C. De Michele, ACM Transactions on Mathematical Software, Vol. 48, No. 46, pp 1-3 (2022), https://doi.org/10.1145/3564270
@cridemichel