Existence of algebraic matrix Riccati equations arising in transport theory
type: Item
instance of: scholarly article
title: Existence of algebraic matrix Riccati equations arising in transport theory
zbMATH Open document ID: 0839.15006
DOI: 10.1016/0024-3795(93)00366-8
author: Jonq Juang
published in: Linear Algebra and its Applications
publication date: 1996-06-19T00:00:00Z
review text: The iterative procedures, one proposed by \textit{A. Shimizu} and \textit{K. Aoki} [Applications of invariant embedding to reactor physics (1972)] and another by \textit{J. Juang} and \textit{Z. T. Lin} [Transp. Theory Stat. Phys. 21, No. 1/2, 87-100 (1992; Zbl 0759.65022)] for finding the minimal positive solution of the matrix Riccati equation are investigated. The form of the equation is \(B - AS - SD + SDS = 0\), in which \[ \begin{aligned} A_{N^-\times N^-} & = \text{diag} \left[ {1 \over c (w^-_1 + \alpha)}, {1 \over c (w_2^-+ \alpha)}, \dots, {1 \over c (w^-_{N^-}+ \alpha)} \right] \\ & - \left[ \begin{matrix} 1 \\ \vdots \\ 1 \end{matrix} \right] \left[ {c^-_1 \over 2 (w^-_1+ \alpha)}, {c^-_2 \over 2 (w^-_2 + \alpha)}, \dots, {c^-_{N^-} \over 2 (w^-_{N^-} + \alpha)} \right] \\ & :=D_A-ia^T, \qquad\text{where } i=[1,\dots,1]^T;\end{aligned} \] \[ \begin{aligned} D_{N^+\times N^+} & =\text{diag}\left[{1\over c(w_1^+-\alpha) }, {1 \over c (w^+_2 - \alpha)}, \dots, {1 \over c (w^+_{N^+} - \alpha)} \right] \\ & -i \left[ {c^+_1 \over 2 (w^+_1 - \alpha)}, {c^+_2 \over 2 (w^+_2 - \alpha)}, \dots, {c^+_{N^+} \over 2 (w^+_{N^+} - \alpha)} \right] \\ & : = D_D - id^T; \end{aligned} \] \(B = ii^T\); \(C = da^T\). Though the procedures have proved to be quite effective in practice, the convergence of iteration has not been fully investigated yet. The authors of this article completely solve the convergence problem for \(0 \leq c \leq 1\) and \(0 \leq \alpha \leq 1\) by first finding a direct method for establishing the existence of positive solutions of the above equation and then prove the convergence of the corresponding iterations to the minimal positive solutions \(S_{\max}\), and its uniqueness is proved subsequently.
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 826710
zbMATH Keywords: matrix Riccati equation
zbMATH Keywords: convergence
zbMATH Keywords: iteration
zbMATH Keywords: positive solutions
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)00366-8
OpenAlex ID: W2052323297
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description: scientific article; zbMATH DE number 826710