A numerical algorithm for block-diagonal decomposition of matrix $${*}$$-algebras with application to semidefinite programming
A numerical algorithm for block-diagonal decomposition of matrix $${*}$$-algebras with application to semidefinite programming
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DOI：
10.1007/s13160-010-0006-9
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发表时间：
2010-05
影响因子：
0.9
通讯作者：
K. Murota;Y. Kanno;M. Kojima;Sadayoshi Kojima
中科院分区：
文献类型：
--
作者：
K. Murota;Y. Kanno;M. Kojima;Sadayoshi Kojima
最近的兴趣在半定规划的群对称性的启发，我们提出了一个数值方法找到一个最好的同时块对角化的有限数量的矩阵，或等价的不可约分解的生成矩阵代数。该方法由数值线性代数计算组成，如特征值计算，并自动充分利用基本的代数结构，这通常是给定矩阵的物理或几何对称性，稀疏性和结构或数值退化的结果。本文提出的方法的主要问题的一些假设下，而配套文件给出了一个算法具有充分的通用性。桁架和框架设计的数值例子。
Motivated by recent interest in group-symmetry in semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of matrices, or equivalently the irreducible decomposition of the generated matrix-algebra. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. The main issues of the proposed approach are presented in this paper under some assumptions, while the companion paper gives an algorithm with full generality. Numerical examples of truss and frame designs are also presented.