Double precision limit with norm

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Zoltán Csáti
Zoltán Csáti 2013-10-29
编辑： Matt J 2013-10-29
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.)

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Matt J
Matt J 2013-10-29
编辑：Matt J 2013-10-29
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)).
  2 个评论
Zoltán Csáti
Zoltán Csáti 2013-10-29
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
Matt J 2013-10-29
编辑：Matt J 2013-10-29
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!

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