We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In the present paper we describe a direct and an iterative method ba...

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...

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations. In the case of Toeplitz inputs, we decrease the solution time from quadratic to nearly linear, and our tests show dramatic decrease of the CPU time as well. We prove numerical stability of our randomized algorithms and extend our approach to solving nonsingular line...

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diago...

We consider the problem of solving linear systems of equations with limited-memory members of the restricted Broyden class and symmetric rank-one matrices. In this paper, we present various methods for solving these linear systems, and propose a new approach based on a practical implementation of the compact representation for the inverse of these limited-memory matrices. Using the proposed app...

By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that when α is big enough, it has an eigenvalue at 1 with multiplicity at least n, and the remaining eigenvalues are all located in a unit c...

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX À XD À AX þ B ¼ 0 from transport theory (Juang 1995), with M ½D; ÀC; ÀB; AŠ 2 R 2nÂ2n being a nonsingular M-matrix. In addition, A; D are rank-1 updates of diagonal matrices , with the products A À1 u; A À> u; D À1 v and D À> v computable in OðnÞ complexity, for some vectors u and v, and B, C are rank 1. The...

X iv :1 30 3. 07 38 v1 [ cs .S C ] 4 M ar 2 01 3 New Symbolic Algorithms For Solving A General Bordered Tridiagonal Linear System A. A. KARAWIA∗ Computer science unit, Deanship of educational services, Qassim University, P.O.Box 6595, Buraidah 51452, Saudi Arabia. E-mail: [email protected] Abstract In this paper, the author present reliable symbolic algorithms for solving a general bordered tri...

The capacitance matrix method has been widely used as an e cient numerical tool for solving the boundary value problems on irregular regions. Initially, this method was based on the Sherman-Morrison-Woodbury formula, an expression for the inverse of the matrix (A+UVT ) with A 2 <n n and U;V 2 <n p. Extensions to the capacitance matrix method with restrictive assumptions on the matrices A and (A...

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations. In the case of Toeplitz inputs, we decrease the solution time from quadratic to nearly linear, and our tests show dramatic decrease of the CPU time as well. We prove numerical stability of our randomized algorithms and extend our approach to solving nonsingular line...