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.