A stability-enhancing scaling procedure for Schur—Riccati solvers

An exact line search method for solving generalized continuous-time algebraic Riccati equations

“…Example 1: This example is contrived to demonstrate a disastrous first step. (A similar example appears in [16] and [17].) For with 0 < < 1; let A = S = 0; E = C = B = R = I2; and In this paper we show how to minimize kR(X)kF along the search direction at little additional cost.…”

“…Although Algorithm 1 ultimately converges quadratically, rapid convergence occurs only in a neighborhood of X 3 : Automatic stabilizing procedures like those proposed in [1], [27], and [28] may give choices of X 0 that lie far from the solution X 3 : Sometimes the first Newton step N0 is disastrously large and many iterations are needed to find the region of rapid convergence [16], [17]. If the Lyapunov equation is ill-conditioned it may be difficult to compute an accurate Newton step, and the exact-arithmetic convergence theory breaks down.…”

“…Example 4: One of the situations in which defect correction or iterative refinement [16], [17] has the most to offer is when the Riccati equation is ill-conditioned. Rounding errors make it unlikely that any Riccati solver will produce much accuracy, but with its excellent structure-preserving rounding error properties, Newton's method is likely to squeeze out as much accuracy as possible [2], [16], [17].…”

“…The accuracy of Algorithm 1 is ultimately limited only by the accuracy to which R(X j ) and the sum X j +N j are calculated. Of the many methods for solving Riccati equations, Algorithm 1 usually squeezes out the maximum possible accuracy [2], [16], [17]. Algorithm 1 is potentially faster (and more accurate) than the widely used Schur vector method [20].…”

“…Rounding errors make it unlikely that any Riccati solver will produce much accuracy, but with its excellent structure-preserving rounding error properties, Newton's method is likely to squeeze out as much accuracy as possible [2], [16], [17].…”