Issue finding numerical error (problem with MATLAB's precision?)

I'm trying to recreate some code that a TA showed me the other day. Suppose we have the system Ax=b. Let's say A = [2,1,1,0;4,3,3,1;8,7,9,5;6,7,9,8]. And let's say b = [1;1;-1;-3]. I saw a trick in MATLAB that can solve this equation using the "\" symbol. So A \ b should yield the following:

ans =

    1.0000
    0.0000
   -1.0000
   -0.0000

If you subtract this from the true solution:

(A \ b) - [1;0;-1;0]

ans =

   1.0e-15 *

   -0.2220
    0.2538
         0
   -0.0912

Then you can actually see the error in MATLAB's solution versus the actual solution. My issue is that I cannot recreate this. When I do A \ b I get this:

ans =

     1
     0
    -1
     0

>> (A \ b) - [1;0;-1;0]

ans =

     0
     0
     0
     0

Therefore I cannot see the error, or this answer is just very imprecise-I'm not sure. How can I recreate the correct output? Is there another way I can see numerical error from an approximated solution versus the actual solution in MATLAB?

Answer 1

[Andrew Newell]

That pesky MATLAB - getting more accurate all the time! Here is an example that I got from the blog article Floating Point Comparisons in Matlab:

x = 0.8-0.7;
x-0.1

On my computer, that returns 8.3267e-17 (which is still pretty good!).

[John Alperto]

I didnt really see anything in the article that could help me here. I'm still unable to detect the numerical error in the approximated solution versus the true solution.

[Andrew Newell]

Did you try this example from the same article?

K = 7;
s = zeros(K,1);
for k = 1:K
  s(k) = sum((10^-k)*ones(10^k,1));
end
s
e = abs(s - 1)

Answer 2

[Jan]

Which format is active in the command window? What do you get with this:

format long g
(A \ b) - [1;0;-1;0]

Are A and B arrays of type double?

[John Alperto]

When I use your code, I get:

>> format long g
>> (A \ b) - [1;0;-1;0]

ans =

     0
     0
     0
     0

I can see that A and b are of type double.

[John Alperto]

Sorry I forgot to add a new line there before pasting the output.

[Andrew Newell]

I fixed that for you. You should be able to click on "Edit" and fix it yourself, though.