A Schur method for the square root of a matrix

AbstractA fast and stable method for computing the square root X of a given matrix A (X2 = A) is developed. The method is based on the Schur factorization A = QSQH and uses a fast recursion to compute the upper triangular square root of S. It is shown that if α = ∥X∥2/∥A∥ is not large, then the computed square root is the exact square root of a matrix close to A. The method is extended for computing the cube root of A. Matrices exist for which the square root computed by the Schur method is ill conditioned, but which nonetheless have well-conditioned square roots. An optimization approach is suggested for computing the well-conditioned square roots in these cases