论文标题
部分可观测时空混沌系统的无模型预测
Computing the extremal nonnegative solutions of the M-tensor equation with a nonnegative right side vector
论文作者
论文摘要
我们考虑了张量方程，其系数张量是一种非发音的M量量，其右侧向量是无负的。这种张量方程可能具有大量的非负溶液。众所周知，张量方程具有最大的非负溶液和最小的非负溶液（共同称为极端溶液）。但是，现有的证明并未显示如何计算极端解决方案。现有的数值方法可以找到一种非负解决方案，而无需知道计算的解决方案是否是极端解决方案。在本文中，我们为存在极端解决方案提供了新的证据。我们的证明比现有的证明要短得多，更重要的是，它们提供了可以计算极端解决方案的数值方法。这些数值方法的线性收敛也被证明在温和的假设下。我们的某些讨论还允许系数张量为Z量，或者允许右侧向量具有一些负元素。
We consider the tensor equation whose coefficient tensor is a nonsingular M-tensor and whose right side vector is nonnegative. Such a tensor equation may have a large number of nonnegative solutions. It is already known that the tensor equation has a maximal nonnegative solution and a minimal nonnegative solution (called extremal solutions collectively). However, the existing proofs do not show how the extremal solutions can be computed. The existing numerical methods can find one of the nonnegative solutions, without knowing whether the computed solution is an extremal solution. In this paper, we present new proofs for the existence of extremal solutions. Our proofs are much shorter than existing ones and more importantly they give numerical methods that can compute the extremal solutions. Linear convergence of these numerical methods is also proved under mild assumptions. Some of our discussions also allow the coefficient tensor to be a Z-tensor or allow the right side vector to have some negative elements.
