Matlab - How to implement backward Euler's method?

[poster]

I am trying to implement these formulas:

Forward Euler's method:

this is what I have tried:

x_new = (speye(nv)+ dt * lambda * L) * x_old;

Is there anything wrong with this? How can I calculate this using sparse operation?

Backward Euler's method:

I have tried this:

x_new = (speye(nv)- dt * lambda * L) \ x_old;

How to implement the backward part where the existing x is calculated based on the new x? Is it OK to use division?

---

L is a sparse matrix like this:

full(L) =



   -1.0000    0.2500    0.2500    0.2500    0.2500

    0.3333   -1.0000    0.3333         0    0.3333

    0.3333    0.3333   -1.0000    0.3333         0

    0.3333         0    0.3333   -1.0000    0.3333

    0.3333    0.3333         0    0.3333   -1.0000

also for other variable we have something like this:

nv = 5;

dt = 0.01;

lambda = 0.5;



x_old =   [-4 0 5;

            1 -5 5;

            1 0 1;

            1 5 5;

            1 0 0]

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