Singular value decomposition for the Takagi factorization of symmetric matrices
Singular value decomposition for the Takagi factorization of symmetric matrices
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DOI：
10.1016/j.amc.2014.01.170
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发表时间：
2014-05-15
影响因子：
4
通讯作者：
Teretenkov, Alexander E.
中科院分区：
文献类型：
--
作者：
Chebotarev, Alexander M.;Teretenkov, Alexander E.
我们根据辅助酉矩阵的平方根和 Lambda 的奇异值分解，描述了对称矩阵 A = U Lambda U-T 的 Takagi 分解的简单实现，其中 U 为酉且对角线 Lambda >= 0。该方法基于代数精确表达式。对于参数化族 A(epsilon) = A + epsilon R = U-epsilon Lambda U-epsilon(epsilon)T，具有不同奇异值的 epsilon >= 0，如果 A 的奇异值是多个，但复合，则酉矩阵 U-epsilon 在 epsilon = 0 点处不连续 U-epsilon Lambda U-epsilon(epsilon)T 保持数值稳定并收敛于 A。分解表示为快速且紧凑的算法。其 Wolfram Mathematica 演示版本和交互式数值测试可在互联网上获得。 (c) 2014 Elsevier Inc. 保留所有权利。
We describe a simple implementation of the Takagi factorization of symmetric matrices A = U Lambda U-T with unitary U and diagonal Lambda >= 0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of Lambda. The method is based on an algebraically exact expression.For parameterized family A(epsilon) = A + epsilon R = U-epsilon Lambda U-epsilon(epsilon)T, epsilon >= 0 with distinct singular values, the unitary matrices U-epsilon are discontinuous at the point epsilon = 0, if the singular values of A are multiple, but the composition U-epsilon Lambda U-epsilon(epsilon)T remains numerically stable and converges to A.The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available on Internet. (c) 2014 Elsevier Inc. All rights reserved.