Contents
MORLAB Demo: Matrix Equation Solvers
This demo script contains the application of the MORLAB matrix equation solver routines.
See also ml_cabe_sgn, ml_care_nwt_fac, ml_caredl_sgn, ml_caredl_sgn_fac, ml_dare_nwt_fac, ml_daredl_sda, ml_daredl_sda_fac, ml_dlyap_smith, ml_dlyap_smith_fac, ml_dlyap_smith_ldl, ml_dlyapdl_smith, ml_dlyapdl_smith_fac, ml_dlyapdl_smith_ldl, ml_dsylv_smith, ml_dsylv_smith_fac, ml_gdlyapdl_smith_fac, ml_icare_ric_fac, ml_lyap_sgn, ml_lyap_sgn_fac, ml_lyap_sgn_ldl, ml_lyapdl_sgn, ml_lyapdl_sgn_fac, ml_lyapdl_sgn_ldl, ml_pcare_nwt_fac, ml_sylv_sgn, ml_sylv_sgn_fac.
% % This file is part of the MORLAB toolbox % (https://www.mpi-magdeburg.mpg.de/projects/morlab). % Copyright (C) 2006-2023 Peter Benner, Jens Saak, and Steffen W. R. Werner % All rights reserved. % License: BSD 2-Clause License (see COPYING) %
General Handling
The MORLAB toolbox contains a bunch of iterative matrix equation solvers, which are used for the implemented model reduction methods. Due to the modolarity of the toolbox, the user can use each of the matrix equation solver routines by itself. MORLAB has implemented solvers for the following types of matrix equations:
- Continuous-time Lyapunov equations A*X*E' + E*X*A' + Q = 0: ml_lyap_sgn, ml_lyap_sgn_fac, ml_lyap_sgn_ldl, ml_lyapdl_sgn, ml_lyapdl_sgn_fac, ml_lyapdl_sgn_ldl
- Continuous-time Riccati equations A'*X*E + E'*X*A + G - E'*X*Q*X*E = 0: ml_care_nwt_fac, ml_caredl_sgn, ml_caredl_sgn_fac, ml_icare_ric_fac, ml_pcare_nwt_fac
- Continuous-time Bernoulli equations A'*X*E + E'*X*A - E'*X*Q*X*E = 0: ml_cabe_sgn
- Continuous-time Sylvester equations A*X*F + E*X*B + C = 0: ml_sylv_sgn, ml_sylv_sgn_fac
- Discrete-time Lyapunov equations A*X*A' - E*X*E' + Q = 0: ml_dlyap_smith, ml_dlyap_smith_fac, ml_dlyap_smith_ldl, ml_dlyapdl_smith, ml_dlyapdl_smith_fac, ml_dlyapdl_smith_ldl, ml_gdlyapdl_smith_fac
- Discrete-time Riccati equations A'*X*A - E'*X*E + G - A'*X*inv(I - Q*X)*Q*X*A = 0: ml_dare_nwt_fac, ml_daredl_sda, ml_daredl_sda_fac
- Discrete-time Sylvester equations A*X*B - E*X*F + C = 0: ml_dsylv_smith, ml_dsylv_smith_fac
In contrast to the system-based routines in MORLAB, the matrix equation solvers just need the coefficient matrices of the equation that has to be solved. For demonstration, we load a prepared data file containing state-space matrices for a stable passive system:
if exist('OCTAVE_VERSION', 'builtin') orig_warn = warning('off', 'Octave:data-file-in-path'); load morlab_data_std_pr.mat; warning(orig_warn); else load morlab_data_std_pr.mat; end
Now we want to solve a Lyapunov equation with those coefficient matrices.
Q = B * B'; [X, info] = ml_lyap_sgn(A, Q);
The matrix X is the solution of the Lyapunov equation A*X + X*A' + Q = 0 as we will show by computing a normalized residual below. The structure info instead contains information about the underlying iteration like the number of iteration steps, the absolute and relative error of the method.
res = norm(A * X + X * A' + Q) / norm(X); disp(res);
5.7288e-14
Solver Types and Naming Scheme
There are different types of solvers implemented in MORLAB. The type of solver and the underlying method can be determined by the naming scheme of the MORLAB function:
- the ending _fac stands for the factorized version of a solver, i.e., certain coefficient matrices are factorized and the solution will also be factorized,
- the ending _ldl stands for the LDL^T version of a solver, i.e., it takes certain coefficient matrices in this format and returns an LDL^T factorized solution,
- if dl is attached to the equation name, two dual matrix equations can be solved by the function at the same time,
- _nwt stands for Newton-type solvers, _sgn for the matrix sign function, _smith for the Smith or squared Smith iteration, _sda for the structure-preserving doubling method and _ric for the Riccati iteration.
As example, we will use the solver ml_lyapdl_sgn_fac. By the naming scheme above, we will solve the dual Lyapunov equations
A*X + X*A' + B*B' = 0, A'*Y + Y*A + C'*C = 0,
where the solutions are factorized as X = R*R' and Y = L*L'.
[R, L] = ml_lyapdl_sgn_fac(A, B, C); X = R*R'; Y = L*L'; resX = norm(A * X + X * A' + B * B') / norm(X); resY = norm(A' * Y + Y * A + C' * C) / norm(Y); disp(resX); disp(resY);
7.3929e-14 6.0740e-14
Remarks
- All matrix equation solver routines are more or less equivalent to each other, which means that all usage examples here also apply to the other equations solvers.
- The dual Riccati equation solvers have the optional parameter to determine if either the primal, the dual or both equations should be solved.