gr_mat_qr, gr_mat_lq, gr_mat_lq_generic, gr_mat_lq_gso, gr_mat_lq_recursive

The entries should be real (a subfield of $\mathbb{R}$ or real floating-point numbers); for other rings the functions may succeed but the normalisations need not be meaningful. A full QR decomposition with a square $Q$ is obtained by extending $Q$ with an orthonormal basis of the complement and padding $R$ (or $L$) with zeros. The gso algorithm is Gram-Schmidt orthogonalisation of the rows and the recursive algorithm a block recursion that benefits from fast matrix multiplication. gr_mat_qr is a wrapper of the LQ decomposition, which works on rows because the matrices are stored by rows. gr_mat_lq calls the ring method GR_METHOD_MAT_LQ (overridden by nfloat). Aliasing of $A$ and $Q$ is handled in place; aliasing of $A$ with $R$ or $L$ uses a temporary copy. The functions return GR_DOMAIN for incompatible shapes, rank-deficient input, or when the divisions or square roots of the normalisation are not possible in the ring.