The author presents a new variant of the Cabay-Meleshko algorithm for numerically computing pairs of basis polynomials, where the numerical orthogonality is explicitly monitored with the help of stability parameters. A corresponding error analysis is given. It is shown in one theorem, that by controlling the stability parameters, the author is able to monitor the size of the residual. The given new explicit bounds for the residuals enable finally to propose a reliable dynamical strategy for choosing the step size in the look-ahead process. The stability parameter is shown to reflect the condition number of the underlying Hankel matrix of moments. This enables to prove the weak and strong stability of the method, provided that the corresponding Hankel matrix is well conditioned.