Eigen: inverting rectangular matrix (least norm solution)

Question

I have very simple question but unfortunately i cannot find answer in Eigen documentation

I have a "fat" matrix A (number of rows is less than number of cols) and i want to find least norm pseudoinverse of this matrix.

Ideally i want to find it via least norm QR decomposition as specified in this slides.

According slides i can use straight-forward approach to do this by using this formula

A.transpose() * (A * A.transpose()).inverse()

But i hope that there is more elegant solution in Eigen

PS Sorry for my English

Answer 1

If A is full rank then your formula is correct and you can also obtain it from the HouseholderQR decomposition of A.transpose():

MatrixXd A(3,6);
A.setRandom();
HouseholderQR<MatrixXd> qr(A.transpose());
MatrixXd pinv;
pinv.setIdentity(A.cols(), A.rows());
pinv = qr.householderQ() * pinv;
pinv = qr.matrixQR().topLeftCorner(A.rows(),A.rows()).triangularView<Upper>().transpose().solve<OnTheRight>(pinv);

If not, then you'll have to use Eigen::CompleteOrthogonalDecomposition that is much simpler to use because it's main purpose is to solve minimal-norm problems:

CompleteOrthogonalDecomposition<MatrixXd> cqr(A);
pinv = cqr.pseudoInverse();