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.