Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems ẋ(t) = A0x(t) +A1x(t− τ) +B0u(t). We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unkn...

Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning

We present a nested splitting conjugate gradient iteration method for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non-Hermitian) positive semi-definite, and at least one of them is positive definite. This method is actually inner/outer iterations, which employs the Sylvester conjugate gradient method as inner iteration to approximate each outer it...

The use of iterative methods for solving Black-Scholes equation

Iterative methods for solving non linear Fokker-Planck equation

Direct Methods for Matrix Sylvester and Lyapunov Equations

We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method for A1X +XA2 +D = 0 and Hammarling’s method for AX + XA + BB = 0 with A stable. We construct three schemes for solving the unitarily reduced quasitriangular systems. We also construct a new rank-1 updating scheme in Hammarling’s method. This new scheme is able to accommodate a B wit...