研讀isospectral flow線性代數演算法及其穩定性分析

Abstract

在這篇文章裡，主要是探討QR algorithm以及isospectral flow，在文章的一開始我們會先簡介傳統的QR method以及從離散型中建構單變數的QR algorithm；接著會介紹一些QR algorithm的性質以及其對應的一些微分方程。 其實在一般常見的分解都有這種isospectral的性質，因此我們會介紹isospectral flow需要滿足哪些微分方程，以及isospectral flow的 Similarity Prop.、Decomposition Prop.以及Reversal Prop. 最後我們會討論 algorithm的穩定性以及收斂速度；我們在這篇文章中最主要的研究是用矩陣的內積定義出isospectral flow的矩陣運算，這個isospectral flow收斂的條件和 algorithm相似，然而收斂速度比 algorithm還要更快。

In this thesis, we study the isospectral flows of matrix valued differential equations. First, we introduce some properties of the QR algorithm and its corresponding differential equations, known as the Toda flow, which is generated to complex-valued, full and nonsymmetric matrices. The next, we consider the genetated abstract g1g2 decomposition which is corresponding to the isospectral flow. Finally, we expand the differentiable function f(X) by using Taylor series and discuss the stability analysis of QR algorithm by solving eigenvalue problems. We define matrix inner product to find the isospectral flow k(X). We also study the stability for this isospecial flow. In this thesis, our main contribution is to find the operation of the isospectral flow which is better than QR flow in convergence.