Abstract. We propose a new matrix geometric mean satisfying the ten properties given by Ando, Li and Mathias [Linear Alg. Appl. 2004]. This mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by Ando, Li and Mathias is the limit of a sequence having order of convergence 1. This makes this new mean very easily computable. We provide a geometric interpretation and a generalization which includes as special cases our mean and the Ando-Li-Mathias mean.

The problem of reducing an algebraic Riccati equation XC X − AX − X D + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind P X 2 + Q X + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28 (2): [295][296][297][298][299][300][301][302][303][304][305][306] 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12: [95][96][97][98][99][100][101][102][103][104][105][106][107][108][109][110][111][112][113] 1965) and Buzbee et al. (SIAM J Numer Anal 7:627-656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.

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.

We consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is dual pencils, a pair of pencils with the same regular part and related singular structures. They were introduced by V. Kublanovskaya in the 1980s. The other is Wong chains, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T. Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials.We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in L 1 , L 2 linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials.

We derive a new representation of Lagrangian subspaces in the form ImΠ T I X , where Π is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and X satisfies |X ij | ≤ 1 if i = j, √ 2 if i = j. This representation allows us to limit element growth in the context of doubling algorithms for the computation of Lagrangian subspaces and the solution of Riccati equations. It is shown that a simple doubling algorithm using this representation can reach full machine accuracy on a wide range of problems, obtaining invariant subspaces of the same quality as those computed by the state-of-the-art algorithms based on orthogonal transformations. The same idea carries over to representations of arbitrary subspaces and can be used for other types of structured pencils.