Sorry, but it is often the case that people are sloppy in their work. That is my expectation here.
You gave us A and x_sol, but not b. But since A is non-singular, and it is not too poorly conditioned, I can get b as:
b = A*x_sol
11.4615398028092 - 49.2105988921614i
-0.105039111494943 + 0.00717362381530284i
-0.132209750840726 - 0.0184213587768616i
-6.39310652188872 + 28.1486882505365i
-0.177682675601164 - 0.0177507775692129i
So I'll assume that is what you stared with.
Now , compute the LU, as
Finally, while I have no idea why you wanted to do a comparison this way (it seems to be a bit silly to me.) norm(A*x_sol - b) is the common way to do a comaprison of the accuracy. Why you are using an LU anyway is beyond me. Just use backslash.
inv(L)*[A*x_sol - b]
U*x_sol - inv(L)*b
0 + 0i
0 + 0i
0 - 8.88178419700125e-16i
-8.88178419700125e-16 + 0i
0 + 8.88178419700125e-16i
I fail to see the problem. There is a difference in the least significant bits. You should never trust the least significant bits of a vector when it is computed in different ways, even if the mathematics tell you the result should be the same.
Remember that floating point arithmetic is only an APPROXIMATION of real mathematics. It is not exact.
But if you got something different, then you made a mistake in what you are telling us that you did. For example, I see this in your question:
[U*x_sol - invL*b]
0.000000000000000 - 0.000000000000007i
-0.000000000000000 + 0.000000000000001i
-0.000037006789095 - 0.000015944272153i
-0.000970975504200 + 0.004054164127652i
0.000106135310812 + 0.000332352007079i
I'd bet a decent sum of money that the invL you used there is NOT the same matrix as inv(L), perhaps computed from a different problem. It happens. It happens a LOT, especailly with novice users, who tend to be sloppy in their code. Are you one of them? Maybe. After all, you used invL there instead of inv(L).
Again, that is a mistake that people make all the time, then they get all upset, and think there is a problem in MATLAB, when it was in fact the user who screwed things up.