Solution of Quadratic Matrix Equation

Hello!

I want to solve a Quadratic Matrix Equation of the form -

Solve for z: z' A z + B' z + D = 0

I did see a similar question here and David Goodmanson has an excellent answer for the z' C z + z + D = 0. However, I don't understand how he came up with the closed form solution or how to generalize it for my case. Any help would be greatly appreciated.

[Torsten]

What are the dimensions of the scalars/vectors/matrices involved ?

[Neilabh Banzal]

@Torsten, A is 6x6x6, B is 6x6, D is 6x1, z is 6x1.

[Torsten]

You will have to tell us how to multiply A with z.

[Neilabh Banzal]

@Torsten, I can compute Az as a function of z which is 6x6.

[Torsten]

Then you have 6 equations for 6 unknowns.

Use fsolve to solve for z1,...,z6.

[Neilabh Banzal]

I was trying to avoid using fsolve as I have to call it repeatedly. Is there no other way?

[Torsten]

Maybe - if you tell us how you plan to evaluate A*z to get a 6x6-matrix.

Answer 1

[Benjamin Thompson]

icare and idare are used to numerically solve the continuous or discrete forms of the Riccati equation.

[Neilabh Banzal]

@Benjamin Thompson, this is not a Riccati Equation. Note that there is a B' z term, but no z' B term.

Answer 2

[Benjamin Thompson]

You could look at this work on File Exchange:

https://www.mathworks.com/matlabcentral/fileexchange/26956-solve-bilateral-matrix-quadratic-equation?s_tid=srchtitle

Then you run it 6 times for each of the 6 6x6 matrices in the 6x6x6 A matrix. If you have specific values to post for your problem in a MAT file, please do so.

[Torsten]

But according to the OP, z is 6x1, not 6x6.

[Benjamin Thompson]

Ok, I don't understand his problem requirements either but hopefully this helps him out.