Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

“…The following theorem generalizes the convergence results of the SDA given in . Theorem Suppose that Algorithm 3 can be applied with no breakdown to a weakly d‐split starting pencil, and that the normalization factors are chosen in any way such that and are bounded.…”

“…In the case of symplectic matrices arising in control theory, the assumptions of this theorem are typically satisfied, , so that every step of Algorithm 2 is well defined. However, the doubling algorithm is also applied in cases when − L 0 and M 0 are not symmetric positive semidefinite . In this case, the algorithm may break down when becomes singular or ill‐conditioned at some step of the algorithm.…”

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

“…The Euclidean norm of the ECT V (P, α) bounds the asymptotic convergence rates of many matrix splitting iteration methods such as (i) the alternating direction implicit (ADI) method [58,1,36,61], the Hermitian and skew-Hermitian splitting (HSS) method [13,10], the normal and skew-Hermitian splitting (NSS) method [14], the positive-definite and skew-Hermitian splitting (PSS) method [12], the shift-splitting preconditioning method [18] and the triangular skew-Hermitian splitting method [50,60,17,51] for solving large sparse and non-Hermitian positive-definite systems of linear equations, (ii) the preconditioned HSS (PHSS) method [15,11], the accelerated HSS (AHSS) method [9,4], the dimensional split preconditioning method [20] and the block alternating splitting implicit (BASI) method [7] for solving large sparse saddle-point linear systems, (iii) the modulus method [22,47,57,52], the modified modulus method [28], the extrapolated modulus method [38,40,37] and the modulus-based splitting methods [6] for solving large sparse linear complementarity problems, and (iv) the alternately linearized implicit (ALI) method [16], the structure-preserving doubling algorithm [35,54,43,23] and the inexact Newton methods based on doubling iteration scheme [32] for computing the minimal nonnegative solutions of large sparse nonsymmetric algebraic Riccati equations; see also [31,15,21,…”

A structure‐preserving doubling algorithm for Lur'e equations

“…We present a numerical method based on a modification of the SDA, an iterative scheme for continuous‐time and discrete‐time AREs . It is shown in that, unlike other iterative schemes, this algorithm has good convergence properties also when the pencil has eigenvalues (of even multiplicity) on the unit circle, as is the case in our problem. The algorithm is tailored to small‐scale, dense problems and requires O ( n 3 log ϵ −1 ) floating point operations to reach convergence up to an accuracy ϵ .…”