TY - JOUR
T1 - Accurate Numerical Solution for Shifted M-Matrix Algebraic Riccati Equations
AU - Liu, Changli
AU - Xue, Jungong
AU - Li, Ren-Cang
PY - 2020/7
Y1 - 2020/7
N2 - An algebraic Riccati equation (ARE) is called a shifted M-matrix algebraic Riccati equation (MARE) if it can be turned into an MARE after its matrix variable is partially shifted by a diagonal matrix. Such an ARE can arise from computing the invariant density of a Markov modulated Brownian motion. Sufficient and necessary conditions for an ARE to be a shifted MARE are obtained. Based on the conditions, a highly accurate implementation of the alternating directional doubling algorithm (ADDA) is established to compute the extremal solution of a shifted MARE, as well as a quantity needed for computing the invariant density in the application, with high entrywise relative accuracy. Numerical examples are presented to demonstrate the theory and algorithms.
AB - An algebraic Riccati equation (ARE) is called a shifted M-matrix algebraic Riccati equation (MARE) if it can be turned into an MARE after its matrix variable is partially shifted by a diagonal matrix. Such an ARE can arise from computing the invariant density of a Markov modulated Brownian motion. Sufficient and necessary conditions for an ARE to be a shifted MARE are obtained. Based on the conditions, a highly accurate implementation of the alternating directional doubling algorithm (ADDA) is established to compute the extremal solution of a shifted MARE, as well as a quantity needed for computing the invariant density in the application, with high entrywise relative accuracy. Numerical examples are presented to demonstrate the theory and algorithms.
UR - https://doi.org/10.1007/s10915-020-01263-4
U2 - 10.1007/s10915-020-01263-4
DO - 10.1007/s10915-020-01263-4
M3 - Journal article
SN - 0885-7474
VL - 84
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 15
ER -