A stable vector algorithm for the solution of the block bidiagonal linear systems is obtained combining a wrap-around partitioning (reordering of the unknowns in a system of linear equations) and the QR factorization. A step of the wrap-around partitioning transforms a linear system into a system of the \(q\) (super)blocks of \(p\) (submatrix) elements each. Then the orthogonal transformations is considered for elimination of the \(q-1\) subdiagonal \(p\times p\) block matrices. Each subdiagonal block row in the current pivotal block column can be eliminated independently of the other block rows. This observation provides the basis for vectorizing the computation. The final block is again block bidiagonal in the structure and so it is possible to use the recursive implementation of the wrap-around partitioning. Organization of the wrap-around partitioning and elimination of the subdiagonal blocks are discussed. A determination of the optimum transformation parameters \(p\) and \(q\) is considered. Certain implementation details are discussed and some conclusions are drawn. Using a simple model for the vectorization overhead it is shown that small block sizes give the best performance.