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An analytical model for solving generalized interval eigenvalue problem

机译：广义区间特征值问题的解析模型

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摘要

Previous algorithms on calculating the eigenvalue bounds of generalized interval eigenvalue problems usually require preconditions difficult to be met or their efficiency and accuracy are unguaranteed. To overcome these defects, an exact analytical model for obtaining the interval solution set of generalized eigenvalue problem is proposed by rigorous derivations. The eigenvalue set of generalized interval eigenvalue problem is firstly characterized according to the linear interval system and interval arithmetic. Then by introducing the Rayleigh quotient the formulation of mathematical programming on maximizing and minimizing the eigenvalue is derived for the generalized interval eigenvalue problem. The Kuhn-Tucker theorem is subsequently applied to deduce the theorem on computing the upper and lower bounds of interval solution set. By applying the proposed model in real engineering problems, it is demonstrated that the positive definiteness or non-negative decomposition of the matrix pair required by the existing analytical methods cannot be satisfied in some engineering situations where the proposed model is applicable.
机译：先前的用于计算广义区间特征值问题特征值边界的算法通常需要满足一些先决条件，否则便无法保证其效率和准确性。为了克服这些缺陷，通过严格的推导提出了一种精确的解析模型，用于获得广义特征值问题的区间解集。首先根据线性区间系统和区间算法对广义区间特征值问题的特征集进行了刻画。然后，通过引入瑞利商，针对广义区间特征值问题，推导了关于最大化和最小化特征值的数学规划公式。随后将Kuhn-Tucker定理应用于计算区间解集的上下界。通过将所提出的模型应用于实际工程问题，证明了在某些适用于所提出模型的工程情况下，不能满足现有分析方法所要求的矩阵对的正定性或非负分解。

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