  Let the scalars $A^{(j)}_n$ be defined via the linear equations
$$A_l=A^{(j)}_n+\sum^n_{k=1}\bar{\alpha}_ku_{k+l-1},\ \ l=j,j+1,\ldots,j+n\ .$$
Here the $A_i$ and $u_i$ are known and the $\bar{\alpha}_k$ are additional
unknowns, and the quantities of interest are the $A^{(j)}_n$. This problem
arises, for example, when one computes infinite-range integrals by the
higher-order $G$-transformation of Gray, Atchison, and McWilliams. One
efficient procedure for computing the $A^{(j)}_n$ is the rs-algorithm of Pye
and Atchison. In the present work, we develop yet another procedure that
combines the FS-algorithm of Ford and Sidi and the qd-algorithm of Rutishauser,
and we denote it the FS/qd-algorithm. We show that the FS/qd-algorithm has a
smaller operation count than the rs-algorithm. We also show that the FS/qd
algorithm can also be used to implement the transformation of Shanks, and
compares very favorably with the $\varepsilon$-algorithm of Wynn that is
normally used for this purpose.
