"Nearly" correctly rounnded solver for under-4th degree algebraic equations
Version 0.0: 2008-02-19 Tomonori Kouya
We have built the solver for under 4th degree algebraic equations, which can (probably :->) guarantee the appoximations of the roots which have user-required precision. This is based on IEEE754 double precision arithmetic and MPFR library. You can try to use it on the following address:
http://ex-cs.sist.ac.jp/~tkouya/try_cee_algebraic_eq.html
Our solver follows the procedures in order to obtain the user-required "u" decimal digits approximations of the roots.
1) It solves the equation with IEEE754 double precision arithmetic, and estimates the errors included. If they are more precise than the user-required precision "u", our solver returns the double precision values rounded to u decimal digits.
2) If not, then our solver set c = max(10, p + u / 10) as redundant digits (where p is the number of lost digits in previous IEEE754 double prec arithmetic), and calculate two kinds of appoximations, x_s in s (= u + c) digits precision arithmetic and x_l in l (= s + c) digits precision arithmetic. The differences between x_s and x_l are estimated as the precision of x_s. If they are more precise than the user-required precision "u", our solver returns the x_l values rounded to d decimal digits.
3) If not, then it sets c *= 2 and repeats 2) procedure. Of course, the whole repeated procedure is with MPFR.
We will appreciate your sending reports or comments on our solver. And we will stop the service immediately if any problems on security are found.
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Tomonori Kouya