By defining two important terms called basic perturbation vectors and obtaining their linear bounds, we obtain the linear componentwise perturbation bounds for unitary factors and upper triangular factors of the generalized schur decomposition. The perturbation bounds for the diagonal elements of the upper triangular factors and the generalized invariant subspace are also derived. From the former, we present an upper bound and a condition number of the generalized eigenvalue. Furthermore, with numerical iterative method, the nonlinear componentwise perturbation bounds of the generalized Schur decomposition are also provided. Numerical examples are given to test the obtained bounds. Among them, we compare our upper bound and condition number of the generalized eigenvalue with their counterparts given in the literature. Numerical results show that they are very close to each other but our results don't need to calculate the left and right eigenvectors.
翻译:通过定义两个重要术语,即基本的扰动矢量和获得其线性边框,我们获得了一个单一因素的线性部分扰动边框,以及一般沙丘分解的上三角因素的上三角因素的上边边边边框。也从上三角因素的对角元素和普遍变异子空间的扰动边框中得出。从前者中,我们呈现了一个上边框和通用电子元值的一个条件号。此外,通过数字迭接法,还提供了普遍舒尔分解法的非线性部分的扰动边框。提供了用于测试所获取界限的数值示例。其中,我们比较了我们通用电子元值的上边框和条件号与文献中的对应方。数字结果显示,它们彼此非常接近,但我们的结果并不需要计算左侧和右脑源。