Re: [eigen] Problem inverting a Matrix4f with Eigen 2.0.0

[Aron Ahmadia <aja2111@xxxxxxxxxxxx>]

LAPACK is a set of algorithms for direct solution of matrices with
very straightforward structure (dense, banded, etc...).  You get a
matrix and you compute the solution in finite time.  I agree that you
can check the quality of your result simply by direct comparison.

When you get into iterative methods such as Conjugate Gradients,
GMRES, etc..., the condition number plays a huge role in the
effectiveness of the method, how quickly it converges, and frequently,
whether it will converge at all.  That said, it is usually impossible
or very expensive to calculate the condition number of these large
matrices, they are usually estimated.

I hope this is taken in the spirit of information, and not argument :)

A

On Sat, Jul 17, 2010 at 7:18 PM, Benoit Jacob <jacob.benoit.1@xxxxxxxxx> wrote:
> But here as usual, I think it's LAPACK who is right, not MATLAB and NumPy!
>
> Even if the mutually inequivalent notions of condition numbers used by
> LAPACK feel complicated and inelegant, they are actually more relevant
> to the problems at hand, than a single unified notion of condition
> number can be.
>
> Another reason why we decided not to expose condition numbers in Eigen
> is that for the main purpose they're used for, namely checking if a
> result is reliable, there is a better approach which is: check the
> result itself. For example, if you want to check how accurate your
> matrix inverse is, just compute matrix*inverse and see how close it is
> to the identity matrix. Nothing beats that! When it comes to more
> general solving with potentially non full rank matrices, this is even
> better, because the condition number of the lhs matrix alone doesn't
> tell all you need to know (it also depends on your particular rhs), so
> the approach we're recommending in Eigen, to compute lhs*solution and
> compare with rhs, is the only way to know for sure how good your
> solution is.
>
> Benoit
>
> 2010/7/17 Aron Ahmadia <aja2111@xxxxxxxxxxxx>:
>> I admit that pointing to LAPACK was a bad example (I am least familiar
>> with that package of those mentioned), however, MATLAB and NumPy are
>> far more common these days as computational interfaces than LAPACK
>> (even if they are built on top of its routines).
>>
>> A
>>
>> On Sat, Jul 17, 2010 at 6:55 PM, Aron Ahmadia <aja2111@xxxxxxxxxxxx> wrote:
>>> Hi Benoit,
>>>
>>> Sorry, I meant the inverse in this sense, this is something that
>>> arises when solving the two problems:
>>>
>>> Ab = x
>>> Ax = b
>>>
>>> Where I leave the unknown as x, and the fixed as b.  Both problems can
>>> be bound by a condition number that depends on the perturbations of x
>>>
>>> \kappa = ||A||*||b||/||x||     <= ||A||*||A^-1|| (forward)
>>> \kappa = ||A^-1||*||b||/||x|| <= ||A||*||A^-1|| (backward)
>>>
>>> The term ||A||*||A^-1||, since it arises in both forward and backward
>>> problems, is called the condition number of A.  This is pretty solidly
>>> in the literature, and you wouldn't confuse anybody if you had a
>>> general "calculate the condition number of a matrix" function and more
>>> specialized ones for calculating the condition numbers of other
>>> specific operations.
>>>
>>> A
>>>
>>
>>
>>
>
>
>