
From:  Steven G. Johnson 
Subject:  [Helpgsl] Re: (in)accuracy of gsl_poly_complex_solve for repeated roots? 
Date:  Fri, 10 Jun 2005 15:50:14 0400 
Useragent:  Mozilla Thunderbird 1.0 (Macintosh/20041206) 
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 2e16 ~ 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
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