The k th Fréchet derivative of a matrix function f is a multilinear operator from a cartesian product of k subsets of the space \(\mathbb {C}^{n\times n}\) into itself. We show that the k th Fréchet derivative of a real-valued matrix function f at a real matrix A in real direction matrices E₁, E₂, \(\dots \), E_k can be computed using the complex step approximation. We exploit the algorithm of Higham and Relton (SIAM J. Matrix Anal. Appl. 35(3):1019–1037, 2014) with the complex step approximation and mixed derivative of complex step and central finite difference scheme. Comparing with their approach, our cost analysis and numerical experiment reveal that half and seven-eighths of the computational cost can be saved for the complex step and mixed derivative, respectively. When f has an algorithm that computes its action on a vector, the computational cost drops down significantly as the dimension of the problem and k increase.

矩阵函数f的第k个Fréchet导数是空间\（\ mathbb {C} ^ {n \ times n} \）的k个子集的笛卡尔积到自己的多线性算子。我们表明，在实方向矩阵E ₁，E ₂，\（\ dots \），E _k上的实值矩阵A处的实值矩阵函数f的第k个Fréchet导数可以使用复数阶跃近似来计算。我们利用海厄姆和Relton（的算法SIAM J.矩阵分析。申请35 （3）：1019–1037，2014），其中复步近似和复步与中心有限差分方案的混合导数。与他们的方法相比，我们的成本分析和数值实验表明，对于复杂步骤和混合导数，分别可以节省一半和八分之二的计算成本。当f具有计算其对向量的作用的算法时，随着问题的规模和k的增加，计算成本将大大下降。