[eigen] Accepting different matrix type in the decompositions

["ESCANDE Adrien 222264" <adrien.escande@xxxxxx>]

Hello again,

the different decompositions (QR, LU...) I looked at are all templated by a type MatrixType and have, among others, a method compute(const MatrixType& matrix) with a line m_matrix = matrix, and a constructor
Decomposition(const MatrixType& matrix) calling compute.
I think it wouldn't hurt to template these two methods by a type InputMatrixName:
template<typename MatrixType>
template<typename InputMatrixType>
Decomposition<MatrixType>::Decomposition(const InputMatrixType& matrix)

and

template<typename MatrixType>
template<typename InputMatrixType>
Decomposition<MatrixType>::compute(const InputMatrixType& matrix)

this way, when InputMatrixType is different from MatrixType, there is no need to create a temporary object for the conversion, and this conversion is only made at the line m_matrix  = matrix.

In the usual case where the user use the .decomposition() method of MatrixBase<Derived> (for example matrix.colPivHouseholderQR()), we have MatrixType = Derived::PlainObject and InputMatrixBase = Derived, and there's no need to call eval() within decomposition.

And this simplify the writing in other cases. My use case (a complete orthogonal decomposition):
ColPivHouseholderQR<MatrixXd> LQ(M.transpose());
HouseholderQR<MatrixXd> qr(LQ.matrixQR().topLeftCorner(LQ.rank(), LQ.cols()).triangularView<Upper>().transpose()); 
I don't even want to imagine the MatrixType I should have wrote for the HouseholderQR...
By adding the InputMatrixType template, I avoid there two temporaries.

I wanted to add a static assert to check for the compatibility of MatrixType and InputMatrixType, but ei_is_same_type<MatrixType, InputMatrixType::PlainObject>::ret seems to be a too strict condition. I just test the same scalar type for now.

Adrien