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

From: Steven G. Johnson
Subject: [Help-gsl] Re: (in)accuracy of gsl_poly_complex_solve for repeated roots?
Date: Fri, 10 Jun 2005 15:50:14 -0400
User-agent: 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 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.

Steven G. Johnson

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