Existence of algebraic matrix Riccati equations arising in transport theory
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.
scientific article; zbMATH DE number 826710