After #167 we need to see how we would apply boundary conditions.
My guess is that for the boundary conditions we have in mind, that we basically just need to stack up the operators with the applied boundary conditions together. The reason is that the gaussian elimination steps do not involve 2 discrete states simulataneously, so we would
I would double-check this but then,
[x] Add in a new subsection to the larger section on applying boundary conditions (Resolved by e461ea5)
[x] Show with our 2 state discrete problem the gaussian elimination steps to get operator applied on the boundary. (Resolved by e461ea5)
Remember that there are 4 boundary conditions there, which means you eliminate 4 columns to become zeros
You will notice that 2 of the columns are interior to the matrix. They should be at the 1, M+1, M+2, and M+4 positions I think.
But note that since the boundary conditions shouldnt involve the q in any way, we can just apply the boundary conditions first to the operators for the continuous space and then do the expansion across the multiple states
If the q enter the gaussian elimination steps, then either I have made a mistake, or you have lined them up incorrectly in the stacking of the extension operator
If we demonstrate this, I don't think we need to do a formal proof that it extends to arbitrary sizes N.
[x] Show the full, composed setup with boundary conditions applied for the general n = 1, ... N discrete states.
Given the operators with boundary conditions already applied as L^B_1, ... L^B_N I think that a typical row of this looks either exactly or something like
[q_1 * I ... q_(n-1) *I L^B_n - q_n * I q_(n+1) * I .... q_N * I]
where I is the identity matrix of size MxM. (Resolved by f335fbc)
After #167 we need to see how we would apply boundary conditions. My guess is that for the boundary conditions we have in mind, that we basically just need to stack up the operators with the applied boundary conditions together. The reason is that the gaussian elimination steps do not involve 2 discrete states simulataneously, so we would
I would double-check this but then,
qin any way, we can just apply the boundary conditions first to the operators for the continuous space and then do the expansion across the multiple statesqenter the gaussian elimination steps, then either I have made a mistake, or you have lined them up incorrectly in the stacking of the extension operatorL^B_1, ... L^B_NI think that a typical row of this looks either exactly or something likewhere
Iis the identity matrix of size MxM. (Resolved by f335fbc)