MEPACK: Matrix Equation PACKage

MEPACK is a FORTRAN software library for the solution of dense Sylvester-like matrix equations. This includes the following equations:

• Standard Sylvester Equation (SYLV) \(AX \pm XB=Y\)
• Standard Lyapunov Equation (LYAP) \(AX+XA^T=Y\)
• Standard Stein (Discrete-Time Lyapunov) Equation (STEIN) \(AXA^T−X=Y\)
• Discrete-Time Sylvester Equation (SYLV2) \(AXB−X=Y\)
• Generalized Sylvester Equation (GSYLV) \(AXB \pm CXD=Y\)
• Generalized Lyapunov Equation (GLYAP) \( AXB^T+BXA^T=Y\)
• Generalized Stein (Discrete-Time Lyapunov) Equation (GSTEIN) \(AXA^T−EXE^T=Y\)
• Generalized Coupled Sylvester Equation (CSYLV) (\(AR \pm LB=E\), \(CR\pm LD=F\)) and its dual counterpart (CSYLV_DUAL) (\(AR+CL=E\),\(\pm RB \pm LD=F\)).

The implemented solvers use a forward/backward substitution scheme on matrix equations with (quasi) triangular coefficient matrices. The algorithms are implemented in different ways, including the following features:

• Level-3 BLAS enabled solvers with adjustable block/tile sizes.
• OpenMP 4 accelerated Directed-Acyclic-Graph scheduled solvers with focus on current multicore CPUs
• Optimized BLAS level-2 solvers for problems with right-hand sides smaller than 128x128.
• Iterative refinement solvers to improve the accuracy if the (generalized) Schur decomposition is perturbed.

Furthermore, the package includes interfaces to GNU Octave and MATLAB.