All solutions to Quadratic matrix polynomials

Question

I am after two things:

1- algorithms for finding all solutions of possibly large quadratic matrix equations of the form $AX^2+BX+C=0$

2- (if possible) software implementing the algorithms

Everything except Grobner basis based methods.

Thanks,

Pat

Answer 1

Normally, $X$ is a vector. So, $X^2$ does not make sense much.

I hack your question a bit.

$$A^T X + X A - X B R^{-1} B^T X + Q = 0$$

It is called Algebraic Riccati Equation.

What software for solving it?

C++ or MATLAB