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 -