Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function $f$. Independently and under the milder hypothesis that $f$ is even and monotonic over $[0,\pi]$, matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is extremely better for the computation of the extreme eigenvalues.
翻译:在适当的技术假设下,简单略略理论可以推断出托普利茨矩阵由函数美元产生的电子价值的各种无症状扩展。 独立地和在更温和的假设下,美元是均的,单调的,美元是$[0,\pi]美元,在适当的技术假设下,为大型托普利茨矩阵的快速电子价值计算开发了无矩阵算法,在矩阵顺序的线性复杂度范围内:在这种算法效率高的背后,有简单略位理论预测的扩展,加上外推理论。在这里,我们集中关注变量的变化,随后是新变量的无症状扩展,我们根据考虑的新设置调整了无矩阵的算法。 数值实验显示了更高的精度(机器精度)和相同的线性计算成本,而与相关文献中已经提供的无矩阵程序相比。 我们简洁地提到以下优点:当简单略函数的系数在分析上为人所知时,我们集中关注变量的变化,然后是新的变量的无症状扩展,然后是新的变量的无症状扩展式算法,我们将没有矩阵的算法的算法根据考虑考虑新的新环境。 与最精确的计算法进行最差的计算。