Re: [eigen] Implementation of M.transpose()*M & M*M.transpose()

On Sat, Feb 20, 2010 at 5:38 PM, Jinglei Hu <jingleihu@xxxxxxxxx> wrote:

Well, by trivial I mean one can write something as follows(not using Eigen):

Matrix M_Mult_MT(const Matrix& Rhs) {
// [m][n] * [n][m] --> [m][m]
register int N(Rhs.num_rows());
double v[N][N];
double** v2d(Rhs);
for (int i = 0; i < N; ++i) {
    double* vi = &(v[i][0]);
    double* pi = v2d[i];
    for (int j = i; j < N; ++j) {
      double sum = 0.0;
      double* pj = v2d[j];
      for (int k = 0; k < Rhs.num_cols(); ++k)
        sum += pi[k] * pj[k];
      vi[j] = v[j][i] = sum;
    }
}
return Matrix(N, N, &(v[0][0]));
}

yes but that would be terribly slow because here you do not take care of the caches, vectorization, etc.

gael

----- Original Message -----

From: Gael Guennebaud

To: eigen@xxxxxxxxxxxxxxxxxxx

Sent: 2010年2月20日 16:27

Subject: Re: [eigen] Implementation of M.transpose()*M & M*M.transpose()

2010/2/20 Jinglei Hu <jingleihu@xxxxxxxxx>

Hi,

I'm curious how M.transpose()*M and M*M.transpose() is implemented in Eigen by using _expression_ Template. Or it's just the same as the usual multiplication of two matrices?

Currently A = M.transpose()*M involves a full matrix multiplication. If you want to compute only an half, with the devel branch you can do:

A.setZero()
A.selfadjointView<Upper>().rankUpdate(M); // A += M.transpose()*M

and you can compute A * B as follow:

A.selfadjointView<Upper>() * B

This is the preferred way if you only need to have one half (in most of the case it is enough). If you really need to have the full result then check you really really need it ;) Ok so if you really really need it then do:
A.triangularView<Lower>() = A.transpose();

Actually, it is quite easy to detect that case in the matrix product, so in the future you can expect that A = M.transpose()*M will be automatically optimized as in Atlas for instance.

gael

The fact that the resultant matrix is symmetric can be used to save computation cost especially for large matrix. For example, the Hessian matrix, often used in nonlinear optimization/minimization, is actually obtained as Jacobian.transpose() * Jacobian. It is trivial to write a efficient function to do this by considering the symmetry.

not that trivial when you consider cache performance as we do... have a look at the file in Eigen/src/Core/products/SelfadjointProduct.h.

But anyway it is less elegant in the sense that the code does not look nice, e.g. Matrix hessian = MT_Mult_M(Jacobian).

Any idea to do this efficiently by sticking to _expression_ Template? A different but somewhat similar case is the multiplication of normal matrix by diagonal matrix.

this one is already optimized.

Cheers,

Jinglei Hu