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- Subject: Re: [eigen] Problem inverting a Matrix4f with Eigen 2.0.0
- From: Benoit Jacob <jacob.benoit.1@xxxxxxxxx>
- Date: Sat, 17 Jul 2010 11:53:48 -0400
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2010/7/17 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:
> is actually doing "LU condition numbers". Namely, it is calling SGECON,
> 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")
> From the above, we already see that it's not "well accepted" by
> LAPACK! I'm not even sure if any LAPACK function is using this notion
> of condition number.
To be clear: I do agree that the notion of condition number that you
mention here is the only "geometrically" meaningful one in general,
and it is the same as I had mentioned in my previous email; however,
we see that LAPACK uses decomposition-specific notions of condition
numbers instead of this one. Which makes sense because 1) this notion
of condition number requires computing a SVD in general and 2) for
many decompositions, other more decomposition-specific notions are
> Here is yet another LAPACK function doing totally different condition
> numbers than what we saw above:
> * SENSE (input) CHARACTER*1
> * Determines which reciprocal condition numbers are computed.
> * = 'N': None are computed;
> * = 'E': Computed for average of selected eigenvalues only;
> * = 'V': Computed for selected right invariant subspace only;
> * = 'B': Computed for both.
> * If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
>> It is also offered in
> Hm, these aren't very big authorities in numerical linear algebra,
> compared to LAPACK...
>> I know that condition number refers to problems, not specific
>> operators, however the common parlance, at least in American textbooks
>> and numerical computing software, is to treat "condition number", as
>> an alias for "condition number of matrix inversion".
> The problem is that even "condition number of matrix inversion" is
> very vague, because there are different ways to invert a matrix, and
> depending one the one that you use, different notions of condition
> number are relevant. For example, if you invert a matrix using LU,
> what matters is to avoid having near-zero coefficients on the diagonal
> of U. If you invert a matrix using SVD, what matters is the ratio of
> max/min singular values. If you invert using cofactors, what matters
> is the ratio between the determinnat and the cofactors.
>> Hope this helps,
>> On Sat, Jul 17, 2010 at 6:01 PM, Benoit Jacob <jacob.benoit.1@xxxxxxxxx> wrote:
>>> 2010/7/17 Robert Lupton the Good <rhl@xxxxxxxxxxxxxxxxxxx>:
>>>> Thanks for the quick response and analysis.
>>>>> Your matrix is indeed not very well conditioned for floats. Are you
>>>>> using fixed size matrices (Matrix4f) or dynamically sized ones
>>>>> (MatrixXf) ? I'm asking because there is a special path to invert 4x4
>>>>> fixed size matrices, and as far as I remember there was bug for some
>>>>> special configurations in the early versions of Eigen 2.0, so I would
>>>>> recommend you to upgrade it to the latest.
>>>> Fixed size, Matrix4f (I thought I included that in the post; sorry). Yes, I should upgrade but that's a little inconvenient in quite a large software system. Doable, and will be done!
>>>>> determinant != condition:
>>>>> Matlab says the condition of your matrix is 3.8e8, which is pretty bad. (btw: is there a way to calculate condition with Eigen?)
>>>> I know that I wanted a condition number, and quoted the determinant as I couldn't find any eigen way of finding either the condition number, or whether it thinks that the inversion succeeded.
>>> Eigen doesn't offer any "conditionNumber()" function because there is
>>> no such thing as the "condition number" of a matrix in general. There
>>> are many mutually inequivalent things that are called "the condition
>>> number of a matrix" in different contexts.
>>> The most general notion of condition number, the only one to make
>>> sense for all matrices, is as the quotient of the biggest singular
>>> value over the smallest singular value.
>>> However, in the setting of LU/LLT decomposition and inverting
>>> matrices, this is NOT what is usually called the "condition number".
>>> Instead, in the context of LU, one usually calls "condition number"
>>> the quotient of the biggest absolute value of a diagonal coeff of U,
>>> over the smallest. Thus this "condition number" doesn't have a simple
>>> meaning in terms of the original matrix, and is only interesting to
>>> know in the context of LU (it indicates how instable LU decomposition
>>> is, without full pivoting).
>>> By the way, I don't think that LAPACK offers any "condition number"
>>> computation either.
>>> In short, it seems to be a case where it's best to let the user
>>> compute his own "condition number" for himself, which is easy from the
>>> decompositions, as that ensures that he understands what he's
>>> All that being said, a condition number of 3e+8 for a float matrix
>>> (so, bigger than 1/epsilon) really means that you shouldn't try
>>> inverting it; it's true for both notions of condition number discussed