[R] Efficient testing for +ve definiteness

[Spencer Graves]

	  To reinforce Prof. Ripley's comment that, "Knowing the determinant 
does not tell you if the matrix is close to non-positive definite", note 
that the determinant of the negative of the identity matrix, (-diag(k)), 
is (-1)^k;  if k is even, the determinant is positive.  This silly 
example connects to real cases, as for example a 3x3 matrix of rank 1 
with eigenvalues (1, -1e-20, -1e-21).  The last two eigenvalues are 
buried in the noise relative to the largest (under standard double 
precision arithmatec).

	  spencer graves

Prof Brian Ripley wrote:

> On Wed, 13 Jul 2005, Makram Talih wrote:
> 
> 
>>Dear R-users,
>>
>>Is there a preferred method for testing whether a real symmetric matrix is
>>positive definite? [modulo machine rounding errors.]
>>
>>The obvious way of computing eigenvalues via "E <- eigen(A, symmetric=T,
>>only.values=T)$values" and returning the result of "!any(E <= 0)" seems
>>less efficient than going through the LU decomposition invoked in
>>"determinant.matrix(A)" and checking the sign and (log) modulus of the
>>determinant.
>>
>>I suppose this has to do with the underlying C routines. Any thoughts or
>>anecdotes?
> 
> 
> 
> It has to do with what exactly you want to test.  Knowing the determinant 
> does not tell you if the matrix is close to non-positive definite or not.
> For numerical work, a comparison of the smallest eigenvalue to the largest 
> is usually the most useful indication of possible problems in 
> computations.  An alternative is to try a Choleski decomposition, which 
> may be faster but is less informative.
> 
> Given how fast eigenvalues can be computed by current algorithms (and note 
> the comments in ?eigen) I would suggest not worrying about speed until you 
> need to (probably never).
> 

-- 
Spencer Graves, PhD
Senior Development Engineer
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