[Help-gsl] Re: (in)accuracy of gsl_poly_complex_solve for repeated roots?

[Steven G. Johnson]

Brian Gough wrote:
> Thanks, I will add a note about it in the manual.  Higher multiplicity
> roots are always more sensitive to numerical error as there is a
> factor of (macheps)^(1/n) in the error for a root of multiplicity n.

Note that the method of http://www.neiu.edu/~zzeng/multroot.htm achieves 
greater accuracy.  In my simple test case (1 + 4x + 6x^2 + 4x^3 + x^4), 
it correctly detects that the root is -1.0 with a multiplicity of 4.  I 
also gave it a case (x + sqrt(2))^4 where the polynomial and solution 
aren't exactly representable, and it correctly found that -sqrt(2) (with 
an error of 2e-16 ~ machine precision) was a root of multiplicity 4.

So, you should be aware that there are apparently more accurate 
algorithms out there, which someone might want to implement in GSL at a 
future date.  You also might want to reference Zeng et al. in the 
manual, for the same reason.

Cordially,
Steven G. Johnson