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luca
Posts:
11
Registered:
5/19/10


Re: Computation of the matrix exponential
Posted:
May 25, 2010 4:49 AM


On 25 Mag, 07:59, Chip Eastham <hardm...@gmail.com> wrote: > On May 24, 8:37 pm, luca <luca.frammol...@gmail.com> wrote: > > > > > > > Hi, > > > i have the following problem: given a 3x3 real matrix, compute exp(A). > > > I need a really fast way to do this. I have searched a bit with > > google, but it seems to me that > > computing the matrix exponential is not so simple, at least if your > > matrix does not have a special > > structure (for example A=diagonal matrix). > > > I have found a simple method that use the diagonalization of A. If A > > has 3 distinct eigenvalues, than compute > > A=PDP^1, where P is the matrix of the eigenvectors, D is a diagonal > > matrix (whose diagonal elements are > > the eigenvalues of A). Than, exp(A) = P exp(D) P^1. Since P^1 is > > fast enough and exp(D) is simple > > to compute, this should be a fast method. > > > But, the problem is: i am not sure that the matrix A will always have > > 3 distinct eigenvalues...what happens > > if this does not happen? Can i use that formula even if 2 (or all > > three) eigenvalues are equal? > > > Are there any other ways to compute exp(A) in a fast way? > > > Thank you, > > Luca > > You'll probably find the "updated" discussion of matrix > exponentiation by Cleve Moler and Charles Van Loan to be > helpful: > > [Nineteen Dubious Ways to Compute the Exponential of a > Matrix, TwentyFive Years Later]http://www.cs.cornell.edu/cv/ResearchPDF/19ways+.pdf > > Your statement that you need to compute the matrix > exponential "really fast" suggests that you intend > to do this repeatedly. A bit more information about > the reason for doing this might lead to a more focused > evaluation of algorithms. > > regards, chip Nascondi testo citato > >  Mostra testo citato 
Hi Chip,
thank you for your replay. Yes, i need to compute the matrix exponential during an iterative algorithm.
The algorithm is a secondorder minimization one.
Suppose that you have a vector x (of size 8 x 1), computed during the iterations of the algorithm. When  x  <= eps, with eps a small real number, there is convergence.
The method is used to find a homography (3x3 matrix). In practice, one starts with an estimate of the homography H', compute the vector x (i will not give other details on this since it would require too much space), update the homography using H' = H' * exp(A(x)).
What is A? A is a 3x3 matrix of the form:
A(x) = sum_i=1..8 (x_i*A_i)
where the A_i are 3x3 matrices. They are chosen to be a basis for the Lie algebra sl(3). sl(3) is the algebra associated to the group SL(3). The projective transformation H is in SL(3). The papers says that :"the exponential map is a homeomorphism between a neighborhood of I in SL(3) and a neighborhood of the null matrix 0 of sl(3)".
The A_i are constant matrices. They have trace null. The matrix A_i will have:
1] only one nondiagonal element non null (1) or 2] two diagonal elements nonnull (1 and 1)
Since the methods is such that the vector x will tends to have smaller and smaller components, if we start with a x sufficiently near to the null vector, we can use the above parameterization of H.
This is all. If you are interested i can give you the paper of reference. I am working in the field of Computer Vision.
Luca



