Backward-Euler Matrix Lyapunov Equation Solver - Theory and User Manual

A research MATLAB package for time-domain covariance analysis of linear time-invariant and time-varying structural systems subjected to non-stationary stochastic excitation, including systems written in augmented state space with input-generating filters. Response second-moment statistics are obtained by integrating the non-stationary differential Lyapunov equation directly in matrix form, using a variable-step backward-Euler method in which every implicit update is recast as an algebraic Lyapunov equation and solved by a reusable complex-Schur Bartels-Stewart factorization. The solver is implemented in base MATLAB and does not require the Control System Toolbox. Temporal accuracy is controlled by step doubling: one full step is compared with two consecutive half steps, yielding a Richardson local-error estimate that drives a proportional or proportional-integral step-size controller. The accepted covariance is symmetric and, under standard conditions, provably positive semidefinite. The package returns covariance, variance, interstorey-drift, velocity, and acceleration statistics, and is validated against an analytical scalar solution, the stationary algebraic Lyapunov limit, symmetry/positive-semidefiniteness checks, and tolerance-monotonicity tests, with single-degree-of-freedom and six-storey shear-building examples. A full theory and user manual is included with the release.