Constructing a 2D Matrix for Solving Equations with Multiple Unknowns

[Hypatio]

I am trying to figure out how you would write out a matrix for the solution to an equation such as the following:

$$\alpha x^{rr}-(2\alpha+2\gamma) x^r+(\alpha+\gamma) x^i+(\beta+\gamma) x^{br}-\beta z^r -(\beta+2\eta) z^b+(\beta+\eta) z^i +\eta z^{bb}=0$$

where alpha, beta, gamma, and eta are coefficients, and x and z are unknowns, and superscripts simply indicate the relative locations of the nodes.

I do not understand how such an equation can possibly have an invertible matrix because for each equation I need to know the x and z values. If I only needed solutions to x or z values, the matrix might look something like this, depending on the values of alpha, beta, etc.:

http://www.eecs.berkeley.edu/~demmel/cs267/lecture17/DiscretePoisson.gif

but I have no idea how to construct a matrix when you need to know two values for each node location.

 

[AlephZero]

I don't understand your notation. Why do x and z sometimes have two superscripts and sometimes only one?

 

[Hypatio]

I don't understand your notation. Why do x and z sometimes have two superscripts and sometimes only one?

Sorry, this is a notation which is easier for me to understand for index references. b=i,j-1, bb= i,j-2,, r=i+1,j, br=i+1,j-1, i=i,j, rr=i+2,j. Think of it as bottom, bottom right,etc.

 

[AlephZero]

I would write it using 2 matrices A and B, in the form
A x + B z = 0
where x is the vector (x11 ... x1n, x21 ... x2n, ... xn1 ... xnn)
and similarly for z.

Clearly x = z = 0 is always a solution. Whether there are non-trivial solutions will depend on the structure of A and B and mayve on the values of the constants alpha, beta, etc.

 

[Hypatio]

I would write it using 2 matrices A and B, in the form
A x + B z = 0
where x is the vector (x11 ... x1n, x21 ... x2n, ... xn1 ... xnn)
and similarly for z.

Clearly x = z = 0 is always a solution. Whether there are non-trivial solutions will depend on the structure of A and B and mayve on the values of the constants alpha, beta, etc.

If I could solve an equation of the form Ax+Bz=0 that would be fantastic, but I don't see how to approach such a solution. Is this a special type of linear system that has been studied or is it not much different from an equation of the form Ax=B? In particular, I do not see how it is consistent to use two separate matrices. How are the matrices associated such that I can, ideally, perform gaussian elimination over it to arrive at the solution? I apologize if this is actually trivial.

Thank you very much for the help. I will be ecstatic if I can figure out how to solve such equations. I can then move on to solving stokes flow and problems of linear elasticity :D

 

[AlephZero]

If z and x are both unknowns, so far you have n^2 equations in 2n^2 variables. You need another n^2 equations from somewhere.