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A special instance of the algebraic Riccati equation XCX − XE − AX + B = 0 where the n × n matrix coefficients A, B, C, E are rank structured matrices is considered. 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(More)
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(More)
Consider a finite set of identical computational entities that can move freely in the Eu-clidean plane operating in Look-Compute-Move cycles. Let p(t) denote the location of entity p at time t; entity p can see entity q at time t if at that time no other entity lies on the line segment p(t)q(t). We consider the basic problem called Mutual Visibility:(More)
The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind P X 2 + QX + R = 0 is analyzed. New reductions are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm of B. a suitable(More)
We consider a special instance of the algebraic Riccati equation XCX − XE − AX + B = 0 encountered in transport theory, where the n × n matrix coefficients A, B, C, E are rank structured matrices. The equation is reduced to unilateral form A 1 X 2 + A 0 X + A −1 = 0 and solved by means of Cyclic Reduction (CR). It is shown that the matrices generated by CR(More)
We survey theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton's iteration is carried out in the cases of(More)
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(More)
The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both these eigenvalues are exactly zero, the problem is called critical or null(More)