Need resources for solving nonlinear matrix equations

[Kosh Naranek]

In my work I've encountered equations of the type:

(Ax).*(Bx) + Cx = d

Where A,B and C are non-unitary square matrices, x and d column vectors and .* denote component-wise multiplication.

I have a few books which discuss nonlinear matrix equations, but not of this kind. Any suggestions?

 

[Stephen Tashi]

(Ax).*(Bx) + Cx = d

Where A,B and C are non-unitary square matrices, x and d column vectors and .* denote component-wise multiplication.

If you want to solve for the vector x, I wouldn't call the equation a "matrix" equation since there is no unknown matrix. It would be equivalent to a system of multivariate quadratic equations in the components x1,x2,.. of the vector x.

Systematic ways to find symbolic solutions to multivariate polynomial equations are Buchberger's algorithm and Wu's method (called "Wu's method of the characteristic set" in the Wikipedia and "Wu's method of elimination" in other sources).

Of course it's an interesting mathematical question whether the fact that your equation is compactly stated as an equation with matrix coefficients implies that there is something special about the simultaneous quadratic equations, something that would make their solutions have a special form, perhaps a form that could be expressed concisely using matrices. I don't know whether that's the case.

 

[Kosh Naranek]

Thank you. Perhaps I should mention that I can find a solution using linearization and iteration. The physical origins is in a branch of fluid dynamics. Typically one assume only one solution. However I think that there should be several solutions, at least if |C| is small compared to |A| and |B|.

 

[DuncanM]

Hi, Kosh.

That equation looks very similar to a Quadratic Programming problem:
minimize F(x) = (1/2) x^T[H]x + c^T x + α .
Except in your case you are trying to zero the function, not minimize it.
I wonder if it could be turned into such a problem with a little manipulation.

Alternately, it might be a problem appropriate for multi-dimensional root-finding.
Given N equations in N unknowns, in which the equations are non-linear, find a zero.
Do a search for "multi-dimensional root-finding". You will find a lot of information, and several algorithms.

 

[blue_raver22]

There are a few resources that may be helpful for solving nonlinear matrix equations of this type. One option is to consult with colleagues or experts in the field who have experience with similar equations. They may be able to provide insights or suggest specific methods for solving this type of equation.

Another option is to search for articles or research papers that have addressed similar equations. This can be done through academic databases or by attending conferences in your field. These sources may provide specific techniques or algorithms for solving nonlinear matrix equations.

Additionally, there are software packages and programming languages that have built-in functions for solving nonlinear matrix equations. Some examples include MATLAB, Mathematica, and Python. These tools may have resources and forums where you can seek guidance or ask questions about your specific equation.

It may also be helpful to consult textbooks or online resources on nonlinear equations and linear algebra. These sources may provide a general understanding of the concepts and techniques needed to solve your equation.

Overall, the best approach may be to combine various resources and techniques to find the most efficient and accurate solution for your specific equation. It may also be beneficial to consult with a mentor or supervisor for guidance and support in solving this type of equation.