We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model and evaluating quantum transition amplitudes for large many-body Hamiltonians of sizes up to $2^{64} \times 2^{64}$ on a single workstation. We also discuss the application of the method to matrix inverses. We relate and compare our method to the state-of-the-art and demonstrate its advantages. We also discuss practical applications of our method.

中文翻译：

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使用除法计算矩阵函数的元素

我们介绍了一种计算矩阵函数各个元素的方法。我们的技术利用了一种新颖的级数扩展，用于矩阵函数对基向量的作用，即使对于非常大的矩阵也是内存有效的。我们通过计算横向场 Ising 模型的指数矩阵元素并评估大小高达 $2^{64} \times 2^{64}$ 的大型多体哈密顿量的量子跃迁幅度来展示我们的方法工作站。我们还讨论了该方法在矩阵求逆中的应用。我们将我们的方法与最先进的方法相关联并进行比较，并展示其优势。我们还讨论了我们方法的实际应用。