How to enforce element-wise condition in matrix operations?

[poster]

I am trying to use \ for Euler's backward method:

where dt and lambda are constant numbers like:

dt = 0.01

lambda = 0.5

and L is a sparse square matrix with the same size as x_old and x_new

Here is what I have tried:

x_new = (speye(size(x_old,2))- dt * lambda * L) \ x_old;

I had three questions:

1. How to avoid the division in the elements of L that are 0?


2. Is it OK to move (speye(size(x_old,2))- dt * lambda * L) to the
   other side, or should I maintain the formula's structure?.


3. Am I supposed to use .* or .\ anywhere?

For the first question, I tried this but didn't work, because of the the matrix dimensions:

x_new = (speye(size(x_old,2))- dt * lambda * L(L~=0)) \ x_old;

For example, for these matrices:

full(L) = [4.6188    1.1547    1.1547    1.1547    1.1547;

    1.1547    2.3094    0.5774         0    0.5774;

    1.1547    0.5774    2.3094    0.5774         0;

    1.1547         0    0.5774    2.3094    0.5774;

    1.1547    0.5774         0    0.5774    2.3094]



x_old = [-0.4000         0    0.5000;

    0.1000   -0.5000    0.5000;

    0.1000         0    1.0000;

    0.1000    0.5000    0.5000;

    0.1000         0         0]

the result is like this:

x_new = [-0.407106857205753 8.52491160610484e-19    0.523921051079662;

0.0993707696612247  -0.505840945396493  0.511891327878594;

0.0993707696612247  3.20963195216710e-19    1.01773227327509;

0.0993707696612247  0.505840945396493   0.511891327878594;

0.0993707696612247  5.22659462097013e-19    0.00605038248210024]

which in command window looks like this:

x_new =



   -0.4071    0.0000    0.5239

    0.0994   -0.5058    0.5119

    0.0994    0.0000    1.0177

    0.0994    0.5058    0.5119

    0.0994    0.0000    0.0061

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