Various numerical linear algebra problems can be formulated as evaluating bivariate function of matrices. The most notable examples are the Fr\'echet derivative along a direction, the evaluation of (univariate) functions of Kronecker-sum-structured matrices and the solution of Sylvester matrix equations. In this work, we propose a recursive block diagonalization algorithm for computing bivariate functions of matrices of small to medium size, for which dense liner algebra is appropriate. The algorithm combines a blocking strategy, as in the Schur-Parlett scheme, and an evaluation procedure for the diagonal blocks. We discuss two implementations of the latter. The first is a natural choice based on Taylor expansions, whereas the second is derivative-free and relies on a multiprecision perturb-and-diagonalize approach. In particular, the appropriate use of multiprecision guarantees backward stability without affecting the efficiency in the generic case. This makes the second approach more robust. The whole method has cubic complexity and it is closely related to the well-known Bartels-Stewart algorithm for Sylvester matrix equations when applied to $f(x,y)=\frac{1}{x+y}$. We validate the performances of the proposed numerical method on several problems with different conditioning properties.
翻译:各种数字线性代数问题可以用来评价矩阵的双变函数。 最显著的例子有:Fr\'echet衍生物沿一个方向, Kronecker- 和结构矩阵的( 单变) 函数的评价, Sylvester 矩阵方程式的解决方案。 在这项工作中, 我们提出一个循环的区块分解算法, 用于计算中小矩阵的双变函数, 适合使用密度线性代数。 算法将阻塞策略( 如Schur- Parlett 方案) 和对对正方块的评估程序结合起来。 我们讨论后方块的两个执行程序。 第一个是基于 Taylor 扩展的自然选择, 而第二个是无衍生物, 并依赖于多精度的 perturb 和 diagonaliz 方法。 特别是, 适当使用多谱性能性能保证后向稳定, 而不影响普通案件的效率。 这使第二种方法更加坚固。 整个方法与众所周知的 Bartel- Stewarxx 等 和不同的性变压方法密切相关, 。