Explanation on the particular result of MATLAB determinant function

Question

I observed a simple, yet particular behavior when using the determinant function in MATLAB and I would like to get some explanations as I didn't find anything about it in the function help documentation.

I'm generating a random unitary matrix Q with the following code:

[Q, R] = qr(randn(3));

After that, I evaluate the determinant of Q with the det function:

det(Q)

I would expect the result to be -1.000 or 1.000. However, the format doesn't seem to be constant. So when I do something like this:

detResults = zeros(100,1);
for ii = 1:100
    [Q, R] = qr(randn(3));
    detResults(ii,1) = det(Q);
end

the detResults vector contains 1.000 and sometime 1. Is it just a print format issue or is it caused by something else?

Answer 1

It's related to floating-point precision. Each time you iterate through that loop, even though you will theoretically get a determinant of 1 for the Q matrix, the numbers in the matrices themselves are irrational so theoretically the only way you will get the value of 1 is when your numbers are represented with infinite precision. Sometimes there are enough digits so that MATLAB can safely round to 1 with assurance. Also, you're not getting the full picture. The reason why you see 1.0000 and 1 is also related to the print format. The default print format only shows up to five decimal places but it may be prudent to show more decimal places to appreciate the bigger picture.

Here's a small example with using only 10 iterations instead of 100.

Using the default print format:

>> detResults

detResults =

    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000

Using a format of increased precision (just for display purposes) with format long g:

>> format long g;
>> detResults

detResults =

                     1
     0.999999999999999
                     1
                     1
     0.999999999999999
                     1
                     1
     0.999999999999999
                     1
     0.999999999999999

Internally, it really depends on what the Q matrix is and what you get out of the bag when you generate the random matrices. However, as far as precision goes for using these for further calculations, 0.999... is very close to 1 so for all intents and purposes you should consider it equal to 1.

Answer 2

I believe that you are observing the effect of the finite precision of the floating point number representation. By default, MATLAB uses 64-bit floating point numbers. So only a finite set of number, with at most 2^64 unique elements, can be represented by this system exactly. All other numbers, resulting during intermediate computations, are rounded to the nearest representable values. These rouding operations result in errors, which are negligible for most, but not all, applications.

You can estimate the errors in your results by appending this line to your code:

err = detResults - 1;

A simple example to observe the finite-precision artifact is:

2-(sqrt(2))^2

Obviously, this should be exactly 0. But MATLAB would return a non-zero small number because of rounding errors in the square-root and squared steps.