Status: New Owner: asmeurer Labels: Type-Enhancement Priority-Medium Polynomial Solvers
New issue 1546 by asmeurer: Implement quasi-symmetric quartic equations http://code.google.com/p/sympy/issues/detail?id=1546 If we implement the method described here: http://en.wikipedia.org/wiki/Quartic_polynomial#Quasi- symmetric_equations , then the solutions of equations that fit that form using that methods would be much simpler than the ones given now. For example, the solution given now for x**4 + x**3 + x**2 + x + 1 is quite large, but if you use the method described there and apply sqrtdenest() to the answer, you get the very small solution set [(1/4)*sqrt(5)-1/4+(1/4*I)*sqrt(2)*sqrt(5+sqrt(5)), - (1/4)*sqrt(5)-1/4+(1/4*I)*sqrt(2)*sqrt(5-sqrt(5)), -(1/4)*sqrt(5)-1/4-(1/4*I)*sqrt(2)*sqrt(5-sqrt(5)), (1/4)*sqrt(5)-1/4-(1/4*I)*sqrt(2)*sqrt(5+sqrt(5))] (the same as Maple returns). The basic method is, if a quartic is of the type a0*x**4 + a1*x**3 + a2*x**4 + a1*m*x + a0*m**2 for some m, then you can divide by x**2 and make the variable substitution z == x + m/x, which will give you a quadratic in z, and z == x + m/x is a quadratic in x, so you end up solving the first quadratic in z, and the second with the roots of the first. Apparently, it is then a good idea to apply sqrtdenest to the final roots to simplify them. I have no idea it this is applicable to polynomials of other degrees. I just learned it by seeing it on that wikipedia page. Also note that x**4 + x**3 + x**2 + x + 1 is a cyclotomic polynomial (the roots are all primitive 5th roots of unity), so it could potentially be solved more efficiently that way too. However, not all polynomials of the above form are cyclotomic. -- You received this message because you are listed in the owner or CC fields of this issue, or because you starred this issue. You may adjust your issue notification preferences at: http://code.google.com/hosting/settings --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "sympy-issues" group. To post to this group, send email to sympy-issues@googlegroups.com To unsubscribe from this group, send email to sympy-issues+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sympy-issues?hl=en -~----------~----~----~----~------~----~------~--~---
