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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``computer solution''
by ayjara on 2007-04-25 10:04:37 |
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| Hi,
"The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer."
I think we could compute the roots almost "easily" which this procedure:
%%%% begin MATLAB code %%%%
a3 = a;
a2 = b;
a1 = c;
a0 = d;
T1 = -a3/4;
T2 = a2^2 - 3*a3*a1 + 12*a0;
T3 = (2*a2^3 - 9*a3*a2*a1 + 27*a1^2 + 27*a3^2*a0 - 72*a2*a0)/2;
T4 = (-a3^3 + 4*a3*a2 - 8*a1)/32;
T5 = (3*a3^2 - 8*a2)/48;
R1 = sqrt(T3^2 - T2^3);
R2 = cubic_root(T3 + R1);
R3 = (1/12)*(T2/R2 + R2);
R4 = sqrt(T5 + R3);
R5 = 2*T5 - R3;
R6 = T4/R4;
r1 = T1 - R4 - sqrt(R5 - R6);
r2 = T1 - R4 + sqrt(R5 - R6);
r3 = T1 + R4 - sqrt(R5 + R6);
r4 = T1 + R4 + sqrt(R5 + R6);
%%%% end MATLAB code %%%
I does some tests and I think this code is correct.
My (not ease) job was to extract the Ts and Rs from your set of equations...
The cubic_root function must be
%%%% MATLAB code %%%
function v = cubic_root(u)
if imag(u) == 0 & real(u) < 0
v = -(abs(u)^(1/3));
else
v = u^(1/3);
end
%%%% end MATLAB code %%%
I think this may be helpful.
Regards,
Adalberto Ayjara Dornelles Filho
Caxias do Sul, Brasil
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