Troubleshooting Mathematica Eigen Problem: Analytical vs. Numerical Results"

[guidob]

Hi! I'm having some trouble with this. In the code I define two matrices, then calculate its eigenvalues/eigenvectors, and finally I plot the absolute values of the components of the eigenvectors. The results are consistent with (I'm quite sure) the analytical ones but only below "t approx 30". Above this value, (only) the first plot corresponding to one of the vectors shows some strange behaviour.

Any idea? I've tried to solve it changing the precision options, but I couldn't.

Thanks!

 

[djelovin]

You should avoid to letting Mathematica works with lots of symbols. Just absorb exponential factor in constant b and redefine it later.

Clear[a,b,w,g,t]
ρt={{a,b},{b*,1-a}};
{e1,e2}=Eigenvalues[ρt]; 
{v1,v2}=Eigenvectors[ρt];

Here you return exponential factor since it is conugated just like constant b.

a=0.6`1000; 
w=1`1000; 
g=1`1000;
b=(1`1000+2`1000*I) Exp[-I*w*t-g*t];
v1n=Normalize[N[v1,1000]]; 
v2n=Normalize[N[v2,1000]];
Abs1={Abs[v1n[[1]]],Abs[v1n[[2]]]};
Abs2={Abs[v2n[[1]]],Abs[v2n[[2]]]};

 

[guidob]

Thanks djelovin! It seems that I could fix it doing what you told me and also defining the values of the parameters after the calculations. I'm new to this. Thanks again.

 

[Bill Simpson]

I believe the underlying problem can be seen if you get rid of all your decimal points and N and attempts to fix this with thousand digit precision and just look at the exact values that you are attempting to calculate with approximate math.

For example, you can see your Abs1 for t==50 has -(E^50)/5+Sqrt[20+(E^100)/25] in both your numerator and denominator.

In[1]:= Table[Abs1,{t,50,50}]

Out[1]= {{(-E^50/5 + Sqrt[20 + E^100/25])/ (2*Sqrt[5*(1 + (-E^50/5 + Sqrt[20 + E^100/25])^2/20)]), 1/Sqrt[1 + (-E^50/5 + Sqrt[20 + E^100/25])^2/20]}}

FabulouslyCloseToZeroButApproximated/ FabulouslyCloseToZeroButApproximated and then throwing this at Plot doesn't seem so surprising when I make tables of those values and see that when t gets large enough that your error bits fall on the wrong side of the scale sometimes.

In other words your problem is what I believe I remember they call "ill posed."

If I'm remembering the right title, there is a nice little book from about 20 years ago titled "Real Computing Made Real: Preventing Errors In Scientific And Engineering Calculations" that tries to teach how to reformulate ill posed problems so that you get much greater accuracy in the results. Ah! Dover has republished that. I highly recommend that one to anyone pushing bits around.

 

[alphy]

Hello! Thank you for sharing your issue with Mathematica's eigenvalues and eigenvectors. It's great to see that you have already compared your results with the analytical ones and have found consistency up to a certain point. This indicates that your code is functioning correctly up to that point.

Based on your description, it seems that the issue may lie in the numerical calculation of the eigenvalues and eigenvectors. When dealing with numerical calculations, it is important to consider the precision and accuracy of the results. You mentioned that you have tried changing the precision options, but were unable to solve the issue. In that case, I would recommend exploring other options such as increasing the working precision or using alternative methods for calculating the eigenvalues and eigenvectors.

Another factor to consider is the stability of the algorithm used for the numerical calculations. Sometimes, certain algorithms may not perform well for certain types of matrices, leading to unexpected results. It may be helpful to try different algorithms and see if that improves the accuracy of your results.

Additionally, it would be beneficial to check for any errors or warnings in your code or results. This can give you a better understanding of where the issue may be arising from.

Overall, troubleshooting numerical problems in Mathematica can be challenging, but with careful examination of your code and results, as well as exploring different options, you should be able to identify and resolve the issue. I hope this helps and good luck with your research!