Condition numbers for the nonlinear matrix equation and their statistical estimation

In this paper, we consider four different types of condition numbers, i.e., two normwise ones, mixed and componentwise ones, for the nonlinear matrix equation X+A⋆F(X)A=QX+A⋆F(X)A=Q, and derive their explicit expressions. Using the probabilistic spectral norm estimator and the statistical condition estimation method, these condition numbers can be estimated with high reliability. Two numerical examples are provided to illustrate the difference between the normwise condition numbers and the mixed and componentwise condition numbers, and the efficiency of the estimators, respectively.