Large-Scale Stein and Lyapunov Equations 3 2 Large-Scale Stein Equations

We consider the solution of large-scale Lyapunov and Stein equations with numerically low-ranked solutions. For Stein equations, the structure-preserving doubling algorithm will be adapted, with the iterates for A not explicitly computed but in the recursive form Ak = A 2 k−1 = A 2 . Lyapunov equations will be first treated with the Cayley transform before doubling is applied to the resulting Stein equations. With n being the dimension of the algebraic equations, the resulting algorithms are of an efficient O(n) complexity per iteration and converge essentially quadratically. Some numerical results will be presented. For instance, the Stein equation in Example 4 in Section 6.2, of dimension n = 79841 with 3.19 billion variables in the solution X, was solved using MATLAB on a MacBook Pro within 48 seconds (10 iterations) to machine accuracy.

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