id: 05969224
dt: j
an: 05969224
au: Li, Tiexiang; Chu, Eric King-Wah; Juang, Jong; Lin, Wen-Wei
ti: Solution of a nonsymmetric algebraic Riccati equation from a
    one-dimensional multistate transport model.
so: IMA J. Numer. Anal. 31, No. 4, 1453-1467 (2011).
py: 2011
pu: Oxford University Press, Oxford
la: EN
cc: 
ut: algebraic Riccati equation; doubling algorithm; fixed-point iteration;
    Newton’s method; reflection; transport theory
ci: 
li: doi:10.1093/imanum/drq034
ab: Summary: For the steady-state solution of a differential equation from a
    one-dimensional multistate model in transport theory, we shall derive
    and study a nonsymmetric algebraic Riccati equation $B^{ - } - XF^{ - }
    - F^{+}X + XB^{+}X = 0$, where $F ^{\pm } \equiv (I - F)D ^{\pm }$ and
    $B ^{\pm } \equiv BD ^{\pm }$ with positive diagonal matrices $D ^{\pm
    }$ and possibly low-ranked matrices $F$ and $B$. We prove the existence
    of the minimal positive solution $X^{*}$ under a set of physically
    reasonable assumptions and study its numerical computation by
    fixed-point iteration, Newton’s method and the doubling algorithm. We
    shall also study several special cases. For example when $B$ and $F$
    are low ranked then $X^*=\varGamma \circ \left
    (\sum^4_{i=1}U_iV_i^{\text T}\right )$ with low-ranked $U_{i}$ and
    $V_{i}$ that can be computed using more efficient iterative processes.
    Numerical examples will be given to illustrate our theoretical results.
rv: