A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation

Abstract

Abstract. A special instance of the algebraic Riccati equation XCX − XE − AX + B = 0 encountered in transport theory is considered where the n × n matrix coefficients A, B, C, E are rank structured matrices. Relying on the structural properties of Cauchy-like matrices, an algorithm is designed for performing the customary Newton iteration in O(n 2 ) arithmetic operations (ops). The same technique is used to reduce the cost of the algorithm proposed by L.-Z. Lu in [Numer. Linear Algebra Appl., 12 (2005), pp. 191-200] from O(n 3 ) to O(n 2 ) ops still preserving quadratic convergence in the generic case. As a byproduct we show that the latter algorithm is closely related to the customary Newton method by simple formal relations.In critical cases where the Jacobian is singular and quadratic convergence turns to linear, we provide an adaptation of the shift technique in order to get rid of the singularity. The original equation is transformed into an equivalent Riccati equation where singularity is removed and the matrix coefficients maintain the same structure as in the original equation. This leads to a quadratically convergent algorithm with complexity O(n 2 ) which provides approximations with the full precision.Numerical experiments and comparisons which confirm the effectiveness of the new approach are reported.

A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices

“…Notice that a Trummer-like matrix can be fully recovered from G, B, s, and d = diag(T). Trummer-like matrices are related to interpolation problems [8], and may arise from the transformation of Toeplitz and similar displacement structure [15], or directly from the discretization of differential problems [3].…”

“…Then use Eqs. (3) to obtain the generators G (2) and B (2) of the Schur complement C (2) of C. Repeat the algorithm setting G ← G (2) , B ← B (2) , s ← s 2:n and t ← t 2:n to get the second row of U and the second column of L, and so on. A simple implementation is outlined in Algorithm 1.…”

“…A special class of Cauchy-like matrices which may arise in application [3,4,8,15] is that of Trummer-like matrices, i.e., those for which the two node vectors coincide (t = s) and are injective. The partial reconstructibility of these matrices requires special care to be taken in the implementation of the solution algorithms.…”

A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

“…For the further physical meaning of these parameters, please see [13] and the references therein. In this paper, NARE (1) is always referred to the nonsymmetric algebraic Riccati equation (1) associated with the special structure given by (3) and (4). In applications from transport theory, the minimal nonnegative solution of NARE (1) is of interest.…”

General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

“…The Popov function based approach for nonsymmetric algebraic Riccati equation has been developed in [24,37]. Iterative numerical methods based on Newton's method and various refinements to solve nonsymmetric algebraic Riccati equation can be found in [4,15,16]. The invariant subspace method has been extended under various restrictive assumptions on the matrix coefficients in [1,9,10].…”