Dear All,
Derive seems to have this problem. May we know, from the inventors of Derive, the algorithm used for the factoring programme? It seems following Ignacio Larrosa Cañestro suggestion we should get the following 8th degree polynomial whose roots are all eight possible sums of + -sqrt(a), +- sqrt(b), +-sqrt(c), with a, b, c non negative real. But if you input a =2 and any b and c of the form b =p and c = p +2 which are consecutive prime, then we shall consistently getting errors, first only 4 real roots, then 8 incorrect real roots. The algorithm might have become unstable.
x^8 - 4·x^6·(a + b + c) + 2·x^4·(3·a^2 + 2·a·(b + c) + 3·b^2 + 2·b·c + 3·c^2) - 4·x^2·(a^3 - a^2·(b + c) - a·(b^2 - 10·b·c + c^2) + (b + c)·(b^2 - 2·b·c + c^2)) + a^4 - 4·a^3·(b + c) + 2·a^2·(3·b^2 + 2·b·c + 3·c^2) - 4·a·(b + c)·(b^2 - 2·b·c + c^2) + b^4 - 4·b^3·c + 6·b^2·c^2 - 4·b·c^3 + c^4
I have tried some values with a = prime > 2 and b and c any odd integers > 2, Derive seems to have no problem. But with a = even square free integer, b and c odd natural numbers, the same problem occurs. However, with any two of a, b and c even, the third number may be even and odd, factorization proceeds correctly.
Can Albert Rich give us some insight into this problem? Thank you.
Ng Tze Beng
-----Original Message-----
From: DERIVE computer algebra system on behalf of Ignacio Larrosa Cañestro
Sent: Thu 11/13/2003 8:28 AM
To: [log in to unmask]
Cc:
Subject: Error when factoring
I get the polinomial correctly
x^8 - 40x^6 + 352x^4 - 960x^2 + 576 (#1)
by squaring from three times, isolating th roots.
x = sqrt(2) + sqrt(3) + sqrt(5)
But when I factorize #1 in radicals with Derive, it put:
(x + <(20·<6 + 4·<114))·(x - <(20·<6 + 4·<114))·(x + <(20·<6 - 4·<114))·(x -
<(20·<6 - 4·<114))·(x^2 + x·<(40·<6 - 48) + 24)·(x^2 - x·<(40·<6 - 48) + 24)
= 0
('<' is 'sqrt')
And in complex,
(x + <(20·<6 + 4·<114))·(x - <(20·<6 + 4·<114))·(x + <(20·<6 - 4·<114))·(x -
<(20·<6 - 4·<114))·(x + <(10·<6 - 12) + î·<(36 - 10·<6))·(x + <(10·<6 -
12) - î·<(36 - 10·<6))·(x - <(10·<6 - 12) + î·<(36 - 10·<6))·(x - <(10·<6 -
12) - î·<(36 - 10·<6)) = 0
getting four real roots, any of them sqrt(2) + sqrt(3) + sqrt(5), and four
complex conjugates ones.
But substituing x ---> sqrt(t), we get
t^4 - 40·t^3 + 352·t^2 - 960·t + 576
that derive factorice correctly to:
(t + 2·<15 + 2·<10 - 2·<6 - 10)·(t + 2·<15 - 2·<10 + 2·<6 - 10)·(t - 2·<15 +
2·<10 + 2·<6 - 10)·(t - 2·<15 - 2·<10 - 2·<6 - 10)
or, aproximating
(t - 0.8284574727)·(t - 3.679609142)·(t - 6.522431886)·(t - 28.96950149)
getting four linear real factors, as It expects.
My version of Derive is 5.06 in Spanish.
Saludos,
Ignacio Larrosa Cañestro
A Coruña (España)
[log in to unmask]
