论文标题
通过padé-type近似值计算矩阵对数的理论错误估计值
Theoretical error estimates for computing the matrix logarithm by Padé-type approximants
论文作者
论文摘要
在本文中，我们专注于使用高斯 - legendre正交规则计算矩阵对数时犯下的错误。这些公式可以解释为合适的高斯超几何函数的PADé近似值。经验观察告诉我们，当矩阵不接近身份矩阵时，这些四倍体的收敛性变得慢，从而表明使用逆缩放和平方方法用于获得此属性的矩阵。这项工作的新颖性是误差估计值的引入，可用于选择获得给定精度所需的legendre点的数量以及要执行的反向缩放和平方的数量。我们包括一些数值实验，以显示引入的估计值的可靠性。
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 Padé 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.
