A linear least squares problem based on the QR factorization is referred to as the recursive least squares problem if it is desired to solve the same problem with one or more rows added or deleted from the data. The authors introduce the concept of rank-\(k\) modifications for recursive least squares problems, with the emphasis on developing numerically accurate rank-\(k\) downdating algorithms because downdating is numerically harder than updating. They extend known rank-1 downdating algorithms, such as corrected seminormal equations (CSNE), the LINPACK method, and downdating via classical Gram-Schmidt (CGS) with re-orthogonalization and develop some new rank-\(k\) downdating algorithms, e.g. based on modified-Gram-Schmidt (MGS) and block-Gram-Schmidt (BGS). They provide experimental results comparing the numerical accuracy of the various algorithms and other computational aspects of the algorithms. On the basis of the computer experiments, they are able to state, that a single rank-\(k\) downdate for the various methods is faster than \(k\) consecutive rank-1 downdates and the Gram-Schmidt methods are faster methods but not numerically accurate for ill-conditioned problems.