[Mathematica] Subnormal numbers handling

[guerom00]

Hello all :)

I have a matrix which I diagonalize giving me a list of eigenvalues lambda. Those are complex numbers.
After that, I calculate Exp[-lambda] and my code goes on.

For some parameters in my code, some lambdas have very large magnitude so that the exponentiation leads to quantities involving subnormal numbers (for Mathematica, those are numbers which are smaller than $MinMachineNumber). At this point, the code becomes _very_ slow (in addition to inherent hardware difficulties performing floating point operations with subnormal numbers, I understand Mathematica treat them in arbitrary precision !). After a _very_ long time though, Mathematica gives me the right answer (that can be said at least

To circumvent this situation, I apply the function If[Abs[#]<1.*^-50,0,#]& just after the exponentiation. So far, this seems to give me the right answer with no slowing down of the code.

One question though : I've chosen the value 1.*^-50 quite arbitrarily and I'm afraid that what I do might be a little bit “rough”.
Is there a rigorous and/or elegant way to avoid generation of subnormal numbers by the exponentiation?

Thanks in advance :)

 

[mmwave]

Hello there,

Thank you for sharing your experience with us. Dealing with subnormal numbers can be a challenge in scientific computing, but there are some ways to avoid them and improve the efficiency of your code.

One approach is to use the logarithm of the matrix instead of directly exponentiating the eigenvalues. This can help avoid the issue of subnormal numbers and also improve the numerical stability of your calculations. You can then take the exponential of the resulting matrix to get the final answer.

Another option is to use a higher precision data type, such as arbitrary precision arithmetic, to perform your calculations. This can help avoid the issue of subnormal numbers and provide a more accurate result. However, it may also slow down your code significantly, so it's important to weigh the trade-offs.

Finally, you can also consider using specialized libraries or packages for matrix diagonalization and exponentiation. These may have optimized algorithms for handling subnormal numbers and can improve the efficiency of your code.

I hope these suggestions are helpful. Good luck with your research!