In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the identity matrix, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced.

翻译：本文聚焦于使用Gauss-Legendre求积规则计算矩阵对数时产生的误差。这些公式可解释为合适的高斯超几何函数的Padé逼近。经验观察表明，当矩阵不接近单位矩阵时，这些求积的收敛速度变慢，因此建议使用逆缩放和平方方法以获得具有该性质的矩阵。本文的创新之处在于引入误差估计，可用于先验地选择达到给定精度所需的Legendre点数以及需要执行的逆缩放和平方次数。我们通过数值实验展示了所引入估计的可靠性。