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.

如果代数Riccati方程（are）如果其矩阵变量被对角矩阵部分平移后可以转化为母马，则该方程称为移位M矩阵代数Riccati方程（mare）。这样一个被可以从计算马尔可夫的不变密度出现调制布朗运动。为充分必要条件是将被移位的母马获得。根据条件，建立了交替方向倍增算法（adda）的高精度实现，以计算移动母马的极值解。，以及计算应用中不变密度所需的数量，且具有较高的入门相对精度。数值例子说明了该理论和算法。