Perturbation analysis and condition numbers of symmetric algebraic Riccati equations

Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations

“…In this section, we adopt the examples in [3,4,50] to illustrate the effectiveness of our methods. All the experiments were performed using Matlab 8.1, with the machine epsilon µ ≈ 2.2 × 10 −16 .…”

“…For the bound κ U (ϕ Re ), we set δ 1 = A F , δ 2 = sym(Q) 2 , δ 3 = sym(G) 2 . For κ U 1 (ϕ Re ) in [50] we choose δ 1 = A F , δ 2 = Q F , δ 3 = G F . Table 1 compares the accurate relative changes ∆X F / X F , ∆X max / X max and ∆X ⊘ X max obtained by MATLAB with the estimates obtained by our condition numbers.…”

“…Sun [47] defined the structured normwise condition numbers for CARE and DARE and showed that the expressions of structured normwise condition numbers are the same as their unstructured counterparts for both real and complex cases. Later, Zhou et al [50] performed componentwise perturbation analyses of CARE and DARE and obtained the exact expressions for mixed and componentwise condition numbers defined in [15] for the real case. However, in their paper, the perturbations on Q and G are general (unstructured).…”

“…Together with the knowledge of backward error, a good condition estimate can provide an estimate for the accuracy of the computed solution. Although the expressions of the condition numbers derived in [47], [50] and this paper are explicit, they involve the solution matrix and require extensive computation, especially for large size problems. As pointed out in [47,Page 260], practical algorithms for estimating the condition numbers of the algebraic Riccati equations are worth studying.…”

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

“…In this case, componentwise analysis can be one alternative approach by which much tighter and revealing bounds can be obtained. There are two kinds of alternative condition numbers called mixed and componentwise condition numbers, respectively, which are developed by Gohberg and Koltracht [17], and we refer to [16,22,34,35,[39][40][41][42][43] for more details of these two kinds of condition numbers.…”