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

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

2010/7/17 Aron Ahmadia <aja2111@xxxxxxxxxxxx>:
> Hi List,
>
> Just responding to this discussion on condition number functions and
> their availability in similar packages to Eigen..
>
> LAPACK's expert driver routines do calculate the condition number of
> the matrix (http://www.netlib.org/lapack/single/).

Sorry I shouldn't have written that "LAPACK offers any condition
number", instead I should have written that LAPACK doesn't standardize
on a single notion of condition number.

LAPACK does exposes the condition number of certain decompositions in
certain expert solvers. However, these condition numbers are mutually
inequivalent in different expert solvers, and are not the notion of
condition number that you describe below.

For example, the first link on that page mentioning condition numbers:

http://www.netlib.org/lapack/single/sgesvx.f

is actually doing "LU condition numbers". Namely, it is calling SGECON,

http://www.netlib.org/lapack/single/sgecon.f

which is doing this:

*  An estimate is obtained for norm(inv(A)), and the reciprocal of the
*  condition number is computed as
*     RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*
*  Arguments
*  =========
*
*  NORM    (input) CHARACTER*1
*          Specifies whether the 1-norm condition number or the
*          infinity-norm condition number is required:
*          = '1' or 'O':  1-norm;
*          = 'I':         Infinity-norm.

This is not the same as the condition number computed from the
quotient of singular values (that would be the above formula with the
spectral norm instead of L1 or Linf norm, which can be very different)

>  In Numerical
> Linear Algebra, I would consider the condition number's definition
> under the 2-norm as the largest singular value divided by the smallest
> to be well-accepted (see Trefethen and Bau, "Numerical Linear
> Algebra", Strang, "Introduction to Linear Algebra")