Spectral decomposition of matrices is a recurring and important task in applied mathematics, physics and engineering. Many application problems require the consideration of matrices of size three with spectral decomposition over the real numbers. If the functional dependence of the spectral decomposition on the matrix elements has to be preserved, then closed-form solution approaches must be considered. Existing closed-form expressions are based on the use of principal matrix invariants which suffer from a number of deficiencies when evaluated in the framework of finite precision arithmetic. This paper introduces an alternative form for the computation of the involved matrix invariants (in particular the discriminant) in terms of sum-of-products expressions as function of the matrix elements. We prove and demonstrate by numerical examples that this alternative approach leads to increased floating point accuracy, especially in all important limit cases (e.g. eigenvalue multiplicity). It is believed that the combination of symbolic algorithms with the accuracy improvements presented in this paper can serve as a powerful building block for many engineering tasks.

翻译：矩阵的光谱分解是应用数学、物理学和工程学中反复出现的重要任务。许多应用问题要求考虑三号大小矩阵,其光谱分解大于实际数字。如果必须保留对矩阵元素的光谱分解功能依赖性,那么就必须考虑封闭式解决办法。现有的封闭式表达方式基于主要矩阵变异物的使用,这些变异物在有限精确算术框架内评估时存在若干缺陷。本文介绍了一种替代形式,用于计算作为矩阵元素功能的产品总和表达方式所涉变异物(特别是差异性)中的产品总和表达形式。我们用数字实例证明和证明,这种替代方法可提高浮点的准确性,特别是在所有重要的限值案例中(例如,eigenvalue 多重性)。相信,将象征性算法与本文件提出的精度改进相结合,可以作为许多工程任务的强大基石。