Double precision limit with norm

I wrote a Taylor series approach for calculating the natural logarithm of matrix A. When I compared my result with the built-in function logm ( norm(myResult-logm(A),2) ), I got the following: 1.3878e-016. However eps = 2^(-52) = 2.2204e-016 when we use double precision. How can MATLAB determine this value if its maximum precision is lower than the result? Is it a bug? (The estimated norm with normest is 1.9626e-016.)

Answer 1

[Matt J]

eps() gives a relative precision limit.

>> eps(1)

ans =

     2.2204e-16

>> eps(.001)

ans =

     2.1684e-19

Presumably your logm(A) have values much less than 1. You should really be comparing to eps(logm(A)) or maybe to eps(norm(logm(A)).

[Zoltán Csáti]

For my case:

logm(A) =  -0.1438    0.5493         0
            0.5493   -0.1438         0
                 0         0         0

These entries are not much less than 0 (in absolute value).

[Matt J]

They're clearly small enough:

 >> logmA=[ -0.1438    0.5493         0
            0.5493   -0.1438         0
                 0         0         0];

   >> eps(norm(logmA,2))

ans =

     1.1102e-16

Looks like your calculation is pretty accurate!