We consider the numerical evaluation of the quantity $Af(A^{-1}B)$, where $A$ is Hermitian positive definite, $B$ is Hermitian, and $f$ is a function defined on the spectrum of $A^{-1}B$. We study the conditioning of the problem, and we introduce several algorithms that combine the Schur decomposition with either the matrix square root or the Cholesky factorization. We study the numerical behavior of these algorithms in floating-point arithmetic, assess their computational costs, and compare their numerical performance. Our analysis suggests that the algorithms based on the Cholesky factorization will be more accurate and efficient than those based on the matrix square root. This is confirmed by our numerical experiments.

翻译：我们研究了数值计算量$Af(A^{-1}B)$的问题，其中$A$为埃尔米特正定矩阵，$B$为埃尔米特矩阵，$f$为定义在$A^{-1}B$谱上的函数。我们分析了该问题的条件数，并提出了若干结合舒尔分解与矩阵平方根或Cholesky分解的算法。我们研究了这些算法在浮点运算中的数值行为，评估了其计算成本，并比较了它们的数值性能。我们的分析表明，基于Cholesky分解的算法比基于矩阵平方根的算法具有更高的精度和效率。数值实验证实了这一结论。