how to find the roots of polynomial equations with coefficients of variable is given in matrix form?

Hi,

I am a beginner in using MATLAB. I would like to know how do you write in MATLAB the polynomial equation below? I am solving for the values (i.e. roots) of z:

Az^3 + Bz^2 +Cz = 0

Note: A, B, C & D are in matrix form, as shown below:

A = [2,1;1,2] B = [-3,0;-2,-5] C = [-2,-4;0,2] D = [0,0;0,0]

Thanks in advance for your help.

[John D'Errico]

A,B,C are all 2x2 matrices. However, I would note that D, although you have posed the matrix, is never used. Perhaps you intended to use D as the constant term in that equation. We cannot know your intention if you don't tell us though.

Is z a scalar variable? If so, then you effectively have 4 simultaneous quadratic equations in one unknown. There will in general never be any solution, EXCEPT zero in this case. Of course, z==0 is obviously always a solution.

So I think you need to be clear about your question. Is z a scalar variable? Is it a 2x2 matrix?

[Lawrence Galvez]

Thanks for the reply John.

D is constant which is a 'zero matrix'. It was just written in 2x2 to emphasize the same mxn size.

Yes, z is a scalar variable. and yes, zero is one of the obvious solution.

Please see attached image of the solution if done in hand calculation.

I am thinking if I can use MATLAB to get the same answer which are 0 and 2.

[John D'Errico]

So as I suggested, the result is a set of 4 cubic equalities in one unknown. You can freely divide by z, since you KNOW that z==0 is one root in common. So in effect, you have 4 quadratics. But cubics work too.

You COULD just compute the roots of each quadratic polynomial. Then look to see if there is any root in common. This would not be terribly difficult to do. But you would need to use a loop. Not hard. But...

Even easier is to try using the symbolic toolbox. Again, WHY NOT TRY IT! I'll start you out:

 syms z

Now, can you use solve? If you don't try it, you will never learn to use the tools. Come on. This takes only ONE line of code to try. So try using solve here. You might be surprised.

[Lawrence Galvez]

Thanks much for your valuable reply, I will try it. This is just my second day of 'seriously' exploring MATLAB, so still searching the help topics and tutorials. Cheers!

[John D'Errico]

Try it. See what happens if you just give that problem to solve. Solve might be smart enough to get the answer. But rather than me showing you how to use solve, you will learn way more by trying to use it yourself. I'll check back in case you had a problem.

[Lawrence Galvez]

I agree, that is better way of learning things, thanks.

[Lawrence Galvez]

my first attempt to try 'syms' & 'solve' functions as attached.

still learning the 'parameter' & 'Returnconditions', maybe will yield to find the complete solutions.

Also reading these subtopics: - "Solve Multivariate Equations and Assign Outputs to Variables" - "Solve Polynomial Equations of High Degree"