We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. For a k-parameter eigenvalue problem with matrices of sizes \(n_1,\ldots ,n_k = O(n)\), fiber product homotopy method requires deformation of O(1) linear equations, while existing homotopy methods for MEPs require O(n) nonlinear equations. We show that the fiber product homotopy method theoretically finds all eigenpairs of an MEP with probability one. It is especially well-suited for a class of problems we call dimension-deficient singular problems that are generic with respect to intrinsic dimension, as the fiber product homotopy method is provably convergent with probability one for such problems as well, a fact borne out by numerical experiments. More generally, our numerical experiments indicate that the fiber product homotopy method significantly outperforms the standard Delta method in terms of accuracy, with consistent backward errors on the order of \(10^{-16}\) without any use of extended precision. In terms of speed, it significantly outperforms previous homotopy-based methods on all problems and outperforms the Delta method on larger problems, and is also highly parallelizable. We show that the fiber product MEP that we solve in the fiber product homotopy method, although mathematically equivalent to a standard MEP, is typically a much better conditioned problem.

我们开发了一种新的同伦方法来解决多参数特征值问题 (MEP)，称为纤维积同伦方法。对于矩阵大小为\(n_1,\ldots ,n_k = O(n)\)的k参数特征值问题，纤维积同伦方法需要O (1) 线性方程的变形，而现有的 MEP 同伦方法需要O ( n) 非线性方程。我们表明，纤维积同伦方法在理论上以概率 1 找到 MEP 的所有特征对。它特别适用于一类问题，我们称之为维数缺陷奇异问题，这些问题对于内在维数是通用的，因为纤维积同伦方法对于此类问题也可证明收敛于概率为 1，这一事实由数值实验。更一般地说，我们的数值实验表明，纤维积同伦方法在精度方面明显优于标准 Delta 方法，具有\(10^{-16}\)数量级的一致后向误差不使用任何扩展精度。在速度方面，它在所有问题上都明显优于以前的基于同伦的方法，在更大的问题上优于 Delta 方法，并且具有高度的并行性。我们表明，我们在纤维积同伦方法中求解的纤维积 MEP，虽然在数学上与标准 MEP 等效，但通常是一个条件更好的问题。