Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms
Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms
复制标题
DOI：
10.1109/tac.1987.1104549
复制
发表时间：
1987-02
影响因子：
6.8
通讯作者：
A. Laub;M. Heath;C. Paige;R. Ward
中科院分区：
文献类型：
--
作者：
A. Laub;M. Heath;C. Paige;R. Ward
本文提出了一种直接从状态空间实现计算状态空间平衡变换的算法。该算法不需要“平方”或不必要的矩阵乘积。详细讨论了各种算法方面。该算法的一个关键特征是通过计算某个矩阵乘积的奇异值分解来确定反对变换，而无需显式形成乘积。还描述了其他相反变换应用。进一步表明，可以采用涉及广义奇异值分解的类似方法来解决经典的同时对角化问题。这些基于 SVD 的联立对角化算法为解决某些类别的对称正定广义特征值问题的现有方法提供了一种计算替代方案。
An algorithm is presented in this paper for computing state-space balancing transformations directly from a state-space realization. The algorithm requires no "squaring up" or unnecessary matrix products. Various algorithmic aspects are discussed in detail. A key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product. Other contragredient transformation applications are also described. It is further shown that a similar approach may be taken, involving the generalized singular value decomposition, to the classical simultaneous diagonalization problem. These SVD-based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems.