Summary: A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverse \(A^\dagger\) of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverse \(A^\dagger\) . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well as \(A^\dagger\). It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge to \(A^\dagger\); and (iii) the trace of a sequence of approximations multiplied by \(A\) is a monotone increasing function converging to the rank of \(A\). A practical way of compensating the self-correcting feature in the computation of \(A^\dagger\) is devised by using the duality of the algorithms. Comparison with Ben-Israel’s method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel’s method.
{After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitled “Pseudo lnverses and Conjugate Gradients”. This paper treated the same subject and appeared in Commun. ACM 18, 40–43 (1975; Zbl 0304.65029).}