Solving an inequality in Matlab

Question

Cédric Cavents 2016-3-17

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https://ww2.mathworks.cn/matlabcentral/answers/274031-solving-an-inequality-in-matlab

编辑： Cédric Cavents 2016-3-17

Hello!

I need to solve a quartic inequality. However, it is impossible to do it by hand and therefore I would like to use Matlab. The inequality is: q(a,b) < s sqrt(v)

where s and v are known, b = 1.08148*a^2-eps and eps = 10^(-6). Now, q(a,b) is the greatest real root (which should exist) of the quartic polynomial: (48*a^2+16*b)x^4 - (40*a^3+168*a*b)x^3 + (-45*a^4+225*a^2*b+72*b^2)*x^2+(27*a^3*b-162*a*b)*x+27*b^3

The function a = (x-3)*sqrt(v)/s. I need to solve the inequality for x. To that end, I did the following:

syms x z
av = ((x-3)*sqrt(v))/s;
Q = max(roots([48*z^2+16*(1.08148*z-eps), -40*z^3-168*z*(1.08148*z-eps), -45*z^4+225*z^2*(1.08148*z-eps)+72*(1.08148*z-eps)^2, 27*z^3*(1.08148*z-eps)-162*z*(1.08148*z-eps)^2, 27*(1.08148*z-eps)^3]));
F = compose(Q,av);
solve(F-skew*sqrt(var)<0, x)

However, that does not really work. Matlab gives the following error

Error using sym/max (line 97)
Input arguments must be convertible
to floating-point numbers.
Error in Testt (line 13)
Q =
max(roots([48*z^2+16*(1.08148*z-eps),
-40*z^3-168*z*(1.08148*z-eps),
-45*z^4+225*z^2*(1.08148*z-eps)+72*(1.08148*z-eps)^2,
27*z^3*(1.08148*z-eps)-162*z*(1.08148*z-eps)^2,
27*(1.08148*z-eps)^3]));

Is there perhaps an easier way? The best way would be to not work with the roots function and to use an explicit expression for the root. However, an explicit expression for the root of the quartic is horrendous.

Thanks in advance! Cheers