The reverse bordering method

The bordering method allows recursive computation of the solution of a system of linear equations by adding one new row and one new column at each step of the procedure. When some of the intermediate systems are nearly singular, it is possible, by the block bordering method, to add several new rows and columns simultaneously. However, in that case, the solutions of some of the intermediate systems are not computed. The reverse bordering method allows computation of the solutions of these systems afterwards. Such a procedure has many applications in numerical analysis, that include orthogonal polynomials, Padé approximation, and the progressive forms of extrapolation processes.