矩陣 Riccati 方程式之疊代解的收歛速率

Abstract

為了要解決一系列的矩陣 Riccati 方程式,我們提供了四種疊代的技巧, 分別是 Gauss-Jacobi, Gauss-Seidel, JOR, SOR 等方法; 其中方程式內 包含兩個重要變數 C 和 α。C 和 α 均為介於 0,1 間之實數。C代表粒 子每次碰撞後發生散射的比例,而 α 代表角度之改變量。此一系列方程 式是從所謂 "簡單轉移模式" 而來,並滿足線性粒子轉移之反射矩陣。本 論文的目的是在於分析 C 和 α 對於疊代解之收歛速率的影響,此外我們 並要比較 Gauss-Jacobi 和 Gauss-Seidel 兩種方法在解決此一問題時的 收歛速率。 We provide four iterative techniques the Gauss-Jacobi,eidel, JOR, and SOR methods for solving a certain class of algebraic matrix Riccati equations with two parameters, C ( 0≦C≦1 ) and α ( 0≦α≦1 ). Here C denotes then of scattering per collision, and α is an angular shift. Equations of this class are induced, via invariant imbedding and the shifted Gauss- Legendre quadrature formula, from a "simple-transport model" and are satisfied by the reflection matrix for linear particle transport in a half- space. The purpose of this paper is to describe the effects of the parameters C and α on the convergence rates of the iterative solutions. We also compare the convergence rates of the Gauss-Jacobi and Gauss-Seidel methods.