|Re: [eigen] Machine precision<> too coarse|
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- To: eigen@xxxxxxxxxxxxxxxxxxx
- Subject: Re: [eigen] Machine precision<> too coarse
- From: Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
- Date: Tue, 3 Feb 2009 19:32:07 +0100
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2009/2/3 Keir Mierle <mierle@xxxxxxxxx>:
> On Tue, Feb 3, 2009 at 8:59 AM, Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
>> 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:
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.