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A defect correction formula for quadratic matrix equations of the kind$A_1X^2+A_0X+A_{-1}=0$ is presented. This formula, expressed by means of aninvariant subspace of a suitable pencil, allows us to introduce a modificationof the Structure-preserving Doubling Algorithm (SDA), that enables refining aninitial approximation to the sought solution. This modification providessubstantial advantages, in terms of convergence acceleration, in the solutionof equations coming from stochastic models, by choosing a stochastic matrix asthe initial approximation. An application to solving random walks in thequarter plane is shown, where the coefficients $A_{-1},A_0,A_1$ arequasi-Toeplitz matrices of infinite size.

A defect-correction algorithm for quadratic matrix equations, with applications to quasi-Toeplitz matrices