Wow now I think 85% fully understand, thanks so much. I love learning....

My two last question is:

1) which part of this code calculate the reliable digits? I want to try it on cloud.

2) There are no reliable digits? am I right? which part of your code show this?

Thanks again from your kind and advanced help.

Thanks so much, very nice.

We have one approximation solution:

Out[53]= {1.01850881228, 0.783151176337, 1.53552721482, 0.645697515412}

with this code:

In[54]:= diff = approxsol - sol

we get a vector with difference between the solution with approximation solution.

but I couldn't get this code:

In[55]:= Norm[diff]

Well, you could just print the intermediate results. That might make clear what are the vectors in use and what that norm is from.

In[53]:= approxsol = LinearSolve[mat, bv]

Out[53]= {1.01850881228, 0.783151176337, 1.53552721482, 0.645697515412}

In[54]:= diff = approxsol - sol

Out[54]= {0.0185088122791, -0.216848823663, 0.53552721482, \
-0.354302484588}

In[55]:= Norm[diff]

Out[55]= 0.678001207119

Numerically this is garbage anyway because the use of Round has made alterations a few digits in that, based on the conditioning, really should throw the result away from sol. To avoid this one might instead do as below.

mat = N[hm, 5]
bv = N[hm.sol, 5]

(* Out[56]= {{1.0000, 0.50000, 0.33333, 0.25000}, {0.50000, 0.33333, 
  0.25000, 0.20000}, {0.33333, 0.25000, 0.20000, 0.16667}, {0.25000, 
  0.20000, 0.16667, 0.14286}}

Out[57]= {2.0833, 1.2833, 0.95000, 0.75952} *)

approxsol = LinearSolve[mat, bv]

(* Out[47]= {1.0000, 1.0000, 1.000, 1.0000} *)

I should mention that, as on Mathematica.stackexchange.com, I was unable to make any sense of the initial paragraph in this query. Also it would be helpful to provide a link when cross-posting essentially the same query from a different forum.

Probably you are right about the purpose of that forced rounding.