Re: [eigen] Machine precision<> too coarse

[Benoit Jacob <jacob.benoit.1@xxxxxxxxx>]

2009/2/3 Keir Mierle <mierle@xxxxxxxxx>:
> On Tue, Feb 3, 2009 at 8:59 AM, Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
> wrote:
>>
>> Also, look at things this way:
>>
>> Suppose that you have a NxN matrix of doubles, with entries between -1
>> and 1, except in one row where entries are smaller than 1e-12.
>>
>> For example (N=2):
>> 1 0
>> 0 1e-12
>>
>> Do you want to consider this matrix invertible or not?
>
> The condition number is bad, but the matrix is certainly invertible to
> machine precision.

Indeed. Yet the condition number is so bad that in practice, in most
cases where this happens, the matrix is meant to be non-invertible.

What this shows, is that "invertible to machine precision" is not a
particularly useful notion.

To take a more extreme example:
1 0
0 1e-15
is also "invertible to machine precision" and there it's 99.99%
guaranteed that the 1e-15 is noise (and hence it should be considered
non invertible), as this imprecision can be realized by just a few
floating point operations.

Cheers,
Benoit