Hi! I'm trying to understand how the root finding of quartic polynomials works but I came to a dead end.
The method discovered by Euler is described here: http://mathforum.org/dr.math/faq/faq.cubic.equations.html To quote the relevant parts: z3 + (e/2) z2 + ((e2-4 g)/16) z - f2/64 = 0 (*) r = -f/(8 p q) x = p + q + r - a/4 x = p - q - r - a/4 x = -p + q - r - a/4 x = -p - q + r - a/4 But "_roots_quartic_euler" function from SymPy which claims to use the Descartes-Euler solution contains (very) different procedure: 64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0 (this is the same as above) (*) but: p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R x1 = sqrt(R) - sqrt(A + B*sqrt(R)) x2 = -sqrt(R) - sqrt(A - B*sqrt(R)) x3 = -sqrt(R) + sqrt(A - B*sqrt(R)) x4 = sqrt(R) + sqrt(A + B*sqrt(R)) and later: c1 = sqrt(R) c2 = sqrt(A + B) c3 = sqrt(A - B) Please, could someone explain a few things to me? Where do all the definitions of x1, x2, x3, x4 come from? Is there somewhere a paper which derives them? And how was "p" derived? In particular, what I'm trying to understand is how to go from using two roots of (*) to using just one (c1)? I.e., the description of Euler's method says, that one has to pick two roots of (*) - how is it possible to pick just one and still be able to calculate x1...x4? Thanks in advance! Tuom Larsen -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/9546da96-f0a7-4a1f-abf2-f41069e17d25%40googlegroups.com. For more options, visit https://groups.google.com/groups/opt_out.
