Computing real square roots of a real matrix

Björck and Hammarling [1] describe a fast, stable Schur method for computing a square root X of a matrix A (X2 = A). We present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix. For a nonsingular real matrix A conditions are given for the existence of a real square root, and for the existence of a real square root which is a polynomial in A; the number of square roots of the latter type is determined. The conditioning of matrix square roots is investigated, and an algorithm is given for the computation of a well-conditioned square root. © 1987.