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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]));
}
----- Original Message -----
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.
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