时变和静态矩阵问题的自适应AZNN方法
Adapted AZNN Methods for Time-Varying and Static Matrix Problems
摘要：
我们提出了适应张神经网络（AZNN），其中指数衰减常数$η$和基本ZNN的启动阶段的长度的参数设置适合手头的问题。具体来说，我们研究实验与AZNN随时间变化的方阵分解为随时间变化的对称矩阵的产品和随时间变化的矩阵平方根的问题。不同于ZNN中通常使用的小$η$值和最小启动长度阶段，我们使用欧拉低精度有限差分公式调整基本ZNN方法以处理大的甚至巨大的$η$设置和任意长度的启动。这些调整提高了AZNN的收敛速度和降低其解决方案的误差界为我们所选择的问题显着接近机器常数，甚至更低的水平。变参数AZNN还允许我们可靠地找到静态矩阵的满秩对称化子，例如对于Kahan和Frank矩阵以及具有高度病态特征值和复杂Jordan结构的矩阵，从$n = 2$开始。这在满秩静态矩阵对称化子之前从未成功计算的情况下有所帮助。
英文摘要：
We present adapted Zhang Neural Networks (AZNN) in which the parameter settings for the exponential decay constant $η$ and the length of the start-up phase of basic ZNN are adapted to the problem at hand. Specifically we study experiments with AZNN for time-varying square matrix factorizations as a product of time-varying symmetric matrices and for the time-varying matrix square roots problem. Differing from generally used small $η$ values and minimal start-up length phases in ZNN, we adapt the basic ZNN method to work with large or even gigantic $η$ settings and arbitrary length start-ups using Euler's low accuracy finite difference formula. These adaptations improve the speed of AZNN's convergence and lower its solution error bounds for our chosen problems significantly to near machine constant or even lower levels. Parameter-varying AZNN also allows us to find full rank symmetrizers of static matrices reliably, for example for the Kahan and Frank matrices and for matrices with highly ill-conditioned eigenvalues and complicated Jordan structures of dimensions from $n = 2$ on up. This helps in cases where full rank static matrix symmetrizers have never been successfully computed before.
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