A Padé Approximation Method for Square Roots of Symmetric Positive Definite Matrices

A numerical method for computing the square root of a symmetric positive definite matrix is developed in this paper. It is based on the Padé approximation of $\sqrt{1+x}$ in the prime fraction form. A precise analysis allows us to determine the minimum number of terms required in the Padé approximation for a given error tolerance. Theoretical studies and numerical experiments indicate that the method is more efficient than the standard method based on the spectral decomposition, unless the condition number is very large. A numerical method for computing the square root of a symmetric positive definite matrix is developed in this paper. It is based on the Padé approximation of $\sqrt{1+x}$ in the prime fraction form. A precise analysis allows us to determine the minimum number of terms required in the Padé approximation for a given error tolerance. Theoretical studies and numerical experiments indicate that the method is more efficient than the standard method based on the spectral decomposition, unless the condition number is very large.