On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term AXB − X + EFT = 0. These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the...

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B = 0, with M ≡ [D,−C;−B,A] ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix, and A,D being sparse-like (with the products A−1v, A−>v, D−1v and D−>v computable in O(n1) or O(n2) complexity, for some vector v) and B,C are low-ranked. The structure-preserving doubling algorithm by Guo, Lin and Xu (2...

On Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems

The Biorthogonal Lanczos and the Biconjugate Gradients methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems. Sonneveld [19] obtained the Conjugate Gradient Squared by squaring the matrix polynomials of the Biconjugate Gra­ dients method. Here we square the Biorthogonal Lanczos, the Biconjugate Residual and the Biconjugate Orth...

Krylov Subspace Methods for Large-Scale Constrained Sylvester Equations

We consider the numerical approximation to the solution of the matrix equation A1X+XA2 −Y C = 0 in the unknown matrices X, Y , under the constraint XB = 0, with A1, A2 of large dimensions. We propose a new formulation of the problem that entails the numerical solution of an unconstrained Sylvester equation. The spectral properties of the resulting coefficient matrices call for appropriately des...

Preconditioned Krylov Subspace Methods for Eigenvalue Problems