[Eigen] Issues when using Eigen

1. Efficient Expression

Refer to this post http://eigen.tuxfamily.org/dox/TopicWritingEfficientProductExpression.html, for dense matrix calculation.

But what about sparse matrix product??

What does Eigen do when:

- [Enhancement, unsolved] Sparse matrix product –> Sparse m? dense matrix and vector? Does it make itself dense first?

Is this expression fast when using SpMat? When a_, y_, lambda_*_,  is a (dense) vector, penalty_ a scalar?

a_ = - penalty_ * (A_.transpose() * y_)
    + A_.transpose() * lambda_y_
    + Q_.transpose() * lambda_stf_;

or this one better following the post above?

a_ = - penalty_ * (A_.transpose() * y_);
a_.noalias() += A_.transpose() * lambda_y_;
a_.noalias() += Q_.transpose() * lambda_stf_;

- [Enhancement, unsolved] Will it be faster to use sparse vector when do multiplication with sparse matrix, instead of dense one?

- [Bugs, unsolved] When I ran the following I got bugs:

VX dual_res_v         = -lambda_stf_.transpose() * (Q_ - Q_new);
dual_res_v.noalias() += - penalty_ * (y_ - y_prev).transpose() * A_;
dual_res_v.noalias() += - (lambda_y_ - lambda_y_prev).transpose() * A_;
dual_res_v.noalias() += penalty_ * x_.transpose() * (Q_ - Q_new).transpose() * Q_;

Where VX is dense vector in Eigen, and lambda_*_, y_* and x_* are dense vector, Q_ and A_ are sparse matrix.

The index is compatible. The following lines work fine in my code:

VX dual_res_v = -lambda_stf_.transpose() * (Q_ - Q_new)
    - penalty_ * (y_ - y_prev).transpose() * A_
    - (lambda_y_ - lambda_y_prev).transpose() * A_
    + penalty_ * x_.transpose() * (Q_ - Q_new).transpose() * Q_;

But I don’t know what’s wrong with the new one.