arbitrary precision linear algebra

[Ben123]

On Mar 2, 10:21 am, Arthur Mc Coy <1984docmc... at gmail.com> wrote:
> On Mar 2, 5:26 pm, Ben123 <ben.is.loca... at gmail.com> wrote:
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> > On Mar 2, 9:04 am, Arthur Mc Coy <1984docmc... at gmail.com> wrote:
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> > > What do you mean by "arbitrary precision" ? Each method of calculating
> > > of something has its own precision...
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> > If you are unfamiliar with arbitrary precision, I'm referring tohttp://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic
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> > Suppose I find the eigenvalues of a matrix and the eigenvalues range
> > from 1 to 0.0001. This can be handled by numpy in Python because the
> > smallest eigenvalue is larger than then numerical precision of 1E-19.
> > However, if the range of eigenvalues is 1 to 1E-40, then I will need
> > to increase the precision of all calculations leading up to finding
> > the eigenvalues.
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> > I am working with complex valued matrices and I expect to get real
> > eigenvalues back (based on the physics of the system). The precision
> > of numpy is apparent from the imaginary component of the eigenvalues I
> > find, currently 1E-19 or 1E-20. I need better precision for small
> > eigenvalues.
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> > In case you are curious, the complex-valued matrices are 20x20.
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> > Thanks
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> You probably have to change the method of finding eigenvalues.
> Which one do you use? Power or algebraic ?

I'm not sure what you mean by this. As I mentioned, in Python I am
using linalg.eig() from numpy on complex matrices. I have not
investigated how this is implemented.

> Do you use Gaussian method to simplify matrices ?

No

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> Languages can't support infinitely large or small numbers, so try to
> multiply the inner variables by 10^n to increase their values if this
> will not involve on the method. For example, I did this when was
> calculating geometric means of computer benchmarks.

Currently I have values between 1 and 1E-300 (not infinitely small). I
don't see how scaling by powers of 10 will increase precision.

> In such way you will be storing the number of zeros as n.

Are you saying python cares whether I express a number as 0.001 or
scaled by 10^5 to read 100? If this is the case, I'm still stuck. I
need the full range of eigenvalues from 1 to 1E-300, so the entire
range could be scaled by 1E300 but I would still need better precision
than 1E19

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> Yes, interesting what are you calculating.