Graeffe's root squaring method
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11-23-2022, 07:10 PM
Post: #2
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RE: Graeffe's root squaring method
11-18-2022, 12:42 PM
Werner Wrote:- the Graeffe squaring process quickly overflows; in this case, you obtained 12+ digits of the correct answer - but you cannot know that unless you performed one or more additional steps, which would overflow. I tried it out with smaller degrees (in excel..) and I could only confirm about 6 digits before overflow. And then remains the Newton-Rhapson step to refine the value, but all I have is the abs value, not the estimate of the root, so I have n possible complex roots to verfiy. The link to Cleve Moler's article says just that. (thanks for that link BTW!) P(x) = 2 + 3x + 5x^2 + 7x^3 + 11x^4 + ... All P roots inside complex unit circle. We can assume min root real part closer to -1 than +1, to keep imag part small. Root squaring process gives min abs root ≈ 0.8, so I would first try x = 0.8*cis(3/4*pi ≈ 2.36) CAS> p := makelist(ithprime, 148+1,1,-1) :; CAS> f := horner(p, x) :; CAS> newton(f, x = 0.8*exp (3/4*pi*i)) −0.645758096347+0.483177676218*i CAS> polar(Ans) [0.806513599261, 2.49922322693] Magnitude matched root squaring produced min --> this (and its conjugate) is the root to seek. |
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