Accurate solutions of M-matrix algebraic Riccati equations

作者：Xue Jungong; Xu Shufang; Li Ren Cang^*

来源：Numerische Mathematik, 2012, 120(4): 671-700.

DOI：10.1007/s00211-011-0421-0

This paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an M-matrix Algebraic Riccati Equations (MARE) %26lt;br%26gt;XDX - AX - XB + C = 0 %26lt;br%26gt;by which we mean the following conformally partitioned matrix %26lt;br%26gt;(B -C -D A) %26lt;br%26gt;is a nonsingular or an irreducible singular M-matrix. It is known that such an MARE has a unique minimal nonnegative solution Phi. It is proved that small relative perturbations to the entries of A, B, C, and D introduce small relative changes to the entries of the nonnegative solution Phi. Thus the smaller entries Phi do not suffer bigger relative errors than its larger entries, unlike the existing perturbation theory for (general) Algebraic Riccati Equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute F as accurately as the input data deserve. Current study is based on a previous paper of the authors%26apos; on M-matrix Sylvester equation for which D = 0.

• 出版日期2012-4
• 单位北京大学; 复旦大学