Contents

function DEMO_RI_GE_T_N(istest)
% Computes the solution of the Hinf Riccati equation for a random generated
% generalized system. The computations are done with the RADI method for
% the control and filter Hinf Riccati equations. Afterwards, the real
% residual norms are shown and compared to the set tolerance.
%
% Input:
% istest  decides whether the function runs as an interactive demo or a
%         continuous integration test. (optional; defaults to 0, i.e.
%         interactive demo)
%

%
% This file is part of the M-M.E.S.S. project
% (http://www.mpi-magdeburg.mpg.de/projects/mess).
% Copyright (c) 2009-2025 Jens Saak, Martin Koehler, Peter Benner and others.
% All rights reserved.
% License: BSD 2-Clause License (see COPYING)
%
if nargin < 1
    istest = false;
end

Construction of system data.

if exist('OCTAVE_VERSION', 'builtin')
    rand('seed', 1.0); %#ok<RAND>
    eqn.A_ = rand(500) - 250 * eye(500);
    rand('seed', 2.0); %#ok<RAND>
    eqn.E_ = rand(500);
    eqn.E_ = eqn.E_' * eqn.E_;

    rand('seed', 3.0); %#ok<RAND>
    eqn.B2 = rand(500, 2);
    rand('seed', 4.0); %#ok<RAND>
    B1 = rand(500, 2);

    rand('seed', 5.0); %#ok<RAND>
    eqn.C2 = rand(3, 500);
    rand('seed', 6.0); %#ok<RAND>
    C1 = rand(3, 500);
else
    rng(1.0);
    eqn.A_ = rand(500) - 250 * eye(500);
    rng(2.0);
    eqn.E_ = rand(500);
    eqn.E_ = eqn.E_' * eqn.E_;

    rng(3.0);
    eqn.B2 = rand(500, 2);
    rng(4.0);
    B1 = rand(500, 2);

    rng(5.0);
    eqn.C2 = rand(3, 500);
    rng(6.0);
    C1 = rand(3, 500);
end

eqn.haveE = true;

gam = 5; % Scaling term for disturbances.

Set operator.

opts = struct();
[oper, opts] = operatormanager(opts, 'default');

Construction of options struct.

RADI settings.

opts.radi.maxiter      = 100;
opts.radi.res_tol      = 1.0e-12;
opts.radi.rel_diff_tol = 1.0e-16;
opts.radi.info         = 1;
opts.radi.trunc_tol    = eps;

% Shift options.
opts.shifts.num_desired     = 5;
opts.shifts.method = 'projection';

% RI settings.
opts.ri.riccati_solver = 'radi';
opts.ri.maxiter        = 10;
opts.ri.res_tol        = 1.0e-09;
opts.ri.rel_diff_tol   = 1.0e-16;
opts.ri.compres_tol    = 1.0e-16;
opts.ri.info           = 1;
opts.ri.trunc_tol      = eps;

% global options
opts.norm           = 2;

Solve the control equation.

t_solve_eqn = tic;
eqn.type = 'T';
eqn.B1   = 1 / gam * B1;
eqn.C1   = C1;
[outControl, eqn, opts, oper] = mess_lrri(eqn, opts, oper);
t_elapsed1 = toc(t_solve_eqn);
mess_fprintf(opts, ...
             'solving the control equation took %6.2f seconds \n', t_elapsed1);
RADI step:    1 pc: -6.595647e+00 + 0.000000e+00i normalized residual: 9.236423e-01 relative change in Z: 1.000000e+00  
RADI step:    2 pc: -5.503746e+00 + 0.000000e+00i normalized residual: 9.138310e-01 relative change in Z: 5.497570e-01  
RADI step:    3 pc: -4.751833e-04 + 0.000000e+00i normalized residual: 8.805950e-01 relative change in Z: 1.497923e-01  
RADI step:    4 pc: -2.103132e+00 + 0.000000e+00i normalized residual: 8.752092e-01 relative change in Z: 2.918227e-01  
RADI step:    5 pc: -9.586933e-04 + 0.000000e+00i normalized residual: 6.573983e-01 relative change in Z: 5.055678e-02  
RADI step:    6 pc: -8.245415e+01 + 0.000000e+00i normalized residual: 6.513484e-01 relative change in Z: 7.981100e-01  
RADI step:    7 pc: -8.192174e+00 + 0.000000e+00i normalized residual: 6.469973e-01 relative change in Z: 2.040896e-01  
RADI step:    8 pc: -5.428094e+01 + 0.000000e+00i normalized residual: 6.479712e-01 relative change in Z: 4.332318e-01  
RADI step:    9 pc: -1.948282e+00 + 0.000000e+00i normalized residual: 6.420417e-01 relative change in Z: 7.723426e-02  
RADI step:   10 pc: -2.641502e-03 + 0.000000e+00i normalized residual: 1.004718e-01 relative change in Z: 1.427654e-02  
RADI step:   11 pc: -1.399131e+03 + 0.000000e+00i normalized residual: 9.760732e-02 relative change in Z: 8.108794e-01  
RADI step:   12 pc: -3.081852e+01 + 0.000000e+00i normalized residual: 9.786744e-02 relative change in Z: 1.039725e-01  
RADI step:   13 pc: -1.266450e+02 + 0.000000e+00i normalized residual: 9.722493e-02 relative change in Z: 1.954600e-01  
RADI step:   14 pc: -5.328434e+00 + 0.000000e+00i normalized residual: 9.721685e-02 relative change in Z: 3.879684e-02  
RADI step:   15 pc: -1.643385e+04 + 0.000000e+00i normalized residual: 9.593976e-02 relative change in Z: 8.343566e-01  
RADI step:   16 pc: -4.700072e-03 + 0.000000e+00i normalized residual: 1.014750e-03 relative change in Z: 1.347891e-03  
RADI step:   17 pc: -2.847214e+02 + 0.000000e+00i normalized residual: 9.363992e-04 relative change in Z: 9.714434e-02  
RADI step:   18 pc: -3.895369e+01 + 0.000000e+00i normalized residual: 9.299439e-04 relative change in Z: 3.494885e-02  
RADI step:   19 pc: -1.770581e+01 + 0.000000e+00i normalized residual: 9.297762e-04 relative change in Z: 2.349740e-02  
RADI step:   20 pc: -2.062053e+05 + 0.000000e+00i normalized residual: 3.897790e-04 relative change in Z: 8.204956e-01  
RADI step:   21 pc: -4.973095e-03 + 0.000000e+00i normalized residual: 3.615338e-04 relative change in Z: 1.321272e-04  
RADI step:   22 pc: -3.571063e+03 + 0.000000e+00i normalized residual: 2.990567e-04 relative change in Z: 9.651357e-02  
RADI step:   23 pc: -1.855638e+00 + 0.000000e+00i normalized residual: 2.990224e-04 relative change in Z: 2.076200e-03  
RADI step:   24 pc: -1.509223e+02 + 0.000000e+00i normalized residual: 2.972224e-04 relative change in Z: 1.865532e-02  
RADI step:   25 pc: -4.972659e-03 + 0.000000e+00i normalized residual: 2.972223e-04 relative change in Z: 1.067691e-04  
RADI step:   26 pc: -3.896030e+06 + 0.000000e+00i normalized residual: 6.819848e-05 relative change in Z: 8.524739e-01  
RADI step:   27 pc: -2.032899e+04 + 0.000000e+00i normalized residual: 4.213357e-05 relative change in Z: 5.229367e-02  
RADI step:   28 pc: -1.616135e+00 + 0.000000e+00i normalized residual: 4.212647e-05 relative change in Z: 4.208969e-04  
RADI step:   29 pc: -3.038673e+03 + 0.000000e+00i normalized residual: 4.035805e-05 relative change in Z: 1.765306e-02  
RADI step:   30 pc: -7.297810e+06 + 0.000000e+00i normalized residual: 1.676264e-05 relative change in Z: 2.615684e-01  
RADI step:   31 pc: -1.643177e+00 + 0.000000e+00i normalized residual: 1.675957e-05 relative change in Z: 2.820280e-04  
RADI step:   32 pc: -1.233952e+05 + 0.000000e+00i normalized residual: 4.932777e-06 relative change in Z: 4.867860e-02  
RADI step:   33 pc: -3.777727e+02 + 0.000000e+00i normalized residual: 4.667946e-06 relative change in Z: 1.795552e-03  
RADI step:   34 pc: -4.633498e+03 + 0.000000e+00i normalized residual: 4.347147e-06 relative change in Z: 5.958112e-03  
RADI step:   35 pc: -1.997170e+00 + 0.000000e+00i normalized residual: 4.346960e-06 relative change in Z: 1.215089e-04  
RADI step:   36 pc: -7.300033e+06 + 0.000000e+00i normalized residual: 3.389639e-06 relative change in Z: 1.342975e-02  
RADI step:   37 pc: -1.521493e+05 + 0.000000e+00i normalized residual: 8.825533e-07 relative change in Z: 2.170168e-02  
RADI step:   38 pc: -4.604189e+02 + 0.000000e+00i normalized residual: 8.696676e-07 relative change in Z: 8.243518e-04  
RADI step:   39 pc: -2.623980e+04 + 0.000000e+00i normalized residual: 6.717361e-07 relative change in Z: 5.725098e-03  
RADI step:   40 pc: -7.490807e+01 + 0.000000e+00i normalized residual: 6.711510e-07 relative change in Z: 2.869247e-04  
RADI step:   41 pc: -4.416645e+06 + 0.000000e+00i normalized residual: 4.416699e-07 relative change in Z: 6.568614e-03  
RADI step:   42 pc: -9.701277e+02 + 0.000000e+00i normalized residual: 4.372664e-07 relative change in Z: 8.351353e-04  
RADI step:   43 pc: -4.619811e+05 + 0.000000e+00i normalized residual: 1.243000e-09 relative change in Z: 9.055297e-03  
RADI step:   44 pc: -7.538829e+03 + 0.000000e+00i normalized residual: 2.861340e-10 relative change in Z: 1.106760e-04  
RADI step:   45 pc: -1.133607e+02 + 0.000000e+00i normalized residual: 2.820544e-10 relative change in Z: 9.592405e-06  
RADI step:   46 pc: -4.619818e+05 + 0.000000e+00i normalized residual: 1.559032e-10 relative change in Z: 8.668422e-05  
RADI step:   47 pc: -9.717227e+02 + 0.000000e+00i normalized residual: 1.376846e-10 relative change in Z: 2.028000e-05  
RADI step:   48 pc: -3.939646e+04 + 0.000000e+00i normalized residual: 2.236632e-11 relative change in Z: 7.505788e-05  
RADI step:   49 pc: -5.173481e+03 + 0.000000e+00i normalized residual: 1.756753e-11 relative change in Z: 1.302221e-05  
RADI step:   50 pc: -5.228692e+01 + 0.000000e+00i normalized residual: 1.706861e-11 relative change in Z: 1.212844e-06  
RADI step:   51 pc: -4.619818e+05 + 0.000000e+00i normalized residual: 7.175107e-12 relative change in Z: 2.026675e-05  
RADI step:   52 pc: -3.939162e+04 + 0.000000e+00i normalized residual: 1.627667e-12 relative change in Z: 1.511270e-05  
RADI step:   53 pc: -8.050656e+02 + 0.000000e+00i normalized residual: 1.534713e-12 relative change in Z: 1.367708e-06  
RADI step:   54 pc: -1.705825e+03 + 0.000000e+00i normalized residual: 1.430744e-12 relative change in Z: 1.906269e-06  
RADI step:   55 pc: -1.450528e+01 + 0.000000e+00i normalized residual: 1.223698e-12 relative change in Z: 1.809569e-07  
RADI step:   56 pc: -1.530545e+05 + 0.000000e+00i normalized residual: 6.500179e-14 relative change in Z: 6.937971e-06  
RI step:    1 normalized residual: 3.560137e-02 relative change in Z: 1.000000e+00  
               number of RADI steps:   56  
  
RADI step:    1 pc: -3.403506e+00 + 0.000000e+00i normalized residual: 9.943999e-01 relative change in Z: 1.000000e+00  
RADI step:    2 pc: -4.771368e-03 + 0.000000e+00i normalized residual: 1.193780e-04 relative change in Z: 9.949786e-01  
RADI step:    3 pc: -1.196700e+01 + 0.000000e+00i normalized residual: 1.190079e-04 relative change in Z: 5.992934e-02  
RADI step:    4 pc: -6.831295e+00 + 0.000000e+00i normalized residual: 1.185588e-04 relative change in Z: 3.302110e-02  
RADI step:    5 pc: -2.168815e+00 + 0.000000e+00i normalized residual: 1.175102e-04 relative change in Z: 1.730776e-02  
RADI step:    6 pc: -4.875837e-03 + 0.000000e+00i normalized residual: 9.386139e-08 relative change in Z: 1.069987e-02  
RADI step:    7 pc: -1.711991e+00 + 0.000000e+00i normalized residual: 8.971046e-08 relative change in Z: 1.480520e-02  
RADI step:    8 pc: -4.875801e-03 + 0.000000e+00i normalized residual: 8.969924e-08 relative change in Z: 7.813892e-04  
RADI step:    9 pc: -2.710502e+02 + 0.000000e+00i normalized residual: 2.177494e-08 relative change in Z: 1.092012e-01  
RADI step:   10 pc: -2.307914e+01 + 0.000000e+00i normalized residual: 1.366111e-08 relative change in Z: 2.464808e-02  
RADI step:   11 pc: -5.814548e+00 + 0.000000e+00i normalized residual: 1.295964e-08 relative change in Z: 1.119859e-02  
RADI step:   12 pc: -2.298447e+00 + 0.000000e+00i normalized residual: 1.272513e-08 relative change in Z: 6.939127e-03  
RADI step:   13 pc: -2.040699e+03 + 0.000000e+00i normalized residual: 2.907191e-09 relative change in Z: 1.160054e-01  
RADI step:   14 pc: -4.875801e-03 + 0.000000e+00i normalized residual: 2.906884e-09 relative change in Z: 1.643910e-04  
RADI step:   15 pc: -5.423219e+01 + 0.000000e+00i normalized residual: 1.870928e-09 relative change in Z: 1.551389e-02  
RADI step:   16 pc: -2.334785e+02 + 0.000000e+00i normalized residual: 1.687032e-09 relative change in Z: 2.762474e-02  
RADI step:   17 pc: -6.750872e+00 + 0.000000e+00i normalized residual: 1.683103e-09 relative change in Z: 4.465060e-03  
RADI step:   18 pc: -1.511730e+04 + 0.000000e+00i normalized residual: 4.694473e-10 relative change in Z: 1.344507e-01  
RADI step:   19 pc: -4.875776e-03 + 0.000000e+00i normalized residual: 4.694457e-10 relative change in Z: 6.789777e-05  
RADI step:   20 pc: -3.686070e+02 + 0.000000e+00i normalized residual: 4.313667e-10 relative change in Z: 1.761330e-02  
RADI step:   21 pc: -5.319727e+01 + 0.000000e+00i normalized residual: 4.281389e-10 relative change in Z: 6.414951e-03  
RADI step:   22 pc: -5.816955e+02 + 0.000000e+00i normalized residual: 4.077584e-10 relative change in Z: 2.063975e-02  
RADI step:   23 pc: -2.564565e+00 + 0.000000e+00i normalized residual: 4.076828e-10 relative change in Z: 1.344832e-03  
RADI step:   24 pc: -9.235864e+04 + 0.000000e+00i normalized residual: 1.603660e-10 relative change in Z: 1.673551e-01  
RADI step:   25 pc: -3.601629e+01 + 0.000000e+00i normalized residual: 1.597409e-10 relative change in Z: 2.747174e-03  
RADI step:   26 pc: -9.122426e+03 + 0.000000e+00i normalized residual: 1.273375e-10 relative change in Z: 3.868424e-02  
RADI step:   27 pc: -9.625940e+02 + 0.000000e+00i normalized residual: 1.252362e-10 relative change in Z: 1.200583e-02  
RADI step:   28 pc: -6.202076e+05 + 0.000000e+00i normalized residual: 4.061536e-11 relative change in Z: 1.915115e-01  
RADI step:   29 pc: -1.580679e+00 + 0.000000e+00i normalized residual: 4.061416e-11 relative change in Z: 2.693313e-04  
RADI step:   30 pc: -2.649398e+04 + 0.000000e+00i normalized residual: 3.689326e-11 relative change in Z: 3.316074e-02  
RADI step:   31 pc: -3.219121e+01 + 0.000000e+00i normalized residual: 3.688569e-11 relative change in Z: 1.136712e-03  
RADI step:   32 pc: -4.038845e+03 + 0.000000e+00i normalized residual: 3.659692e-11 relative change in Z: 1.264380e-02  
RADI step:   33 pc: -1.576024e+00 + 0.000000e+00i normalized residual: 3.659682e-11 relative change in Z: 2.490798e-04  
RADI step:   34 pc: -7.161213e+06 + 0.000000e+00i normalized residual: 1.147303e-12 relative change in Z: 2.509567e-01  
RADI step:   35 pc: -7.706048e+01 + 0.000000e+00i normalized residual: 1.144103e-12 relative change in Z: 3.920669e-04  
RADI step:   36 pc: -1.503327e+05 + 0.000000e+00i normalized residual: 2.551177e-13 relative change in Z: 1.090171e-02  
RI step:    2 normalized residual: 1.522697e-05 relative change in Z: 5.864875e-08  
               number of RADI steps:   36  
  
RADI step:    1 pc: -2.634325e+00 + 0.000000e+00i normalized residual: 9.926300e-01 relative change in Z: 1.000000e+00  
RADI step:    2 pc: -4.869153e-03 + 0.000000e+00i normalized residual: 6.879507e-08 relative change in Z: 9.963082e-01  
RADI step:    3 pc: -4.398236e+00 + 0.000000e+00i normalized residual: 6.848595e-08 relative change in Z: 2.928788e-05  
RADI step:    4 pc: -2.525668e+00 + 0.000000e+00i normalized residual: 6.795855e-08 relative change in Z: 2.435723e-05  
RADI step:    5 pc: -4.871718e-03 + 0.000000e+00i normalized residual: 1.194577e-14 relative change in Z: 2.606869e-04  
RI step:    3 normalized residual: 2.731859e-12 relative change in Z: 2.050871e-11  
               number of RADI steps:    5  
  
solving the control equation took   0.70 seconds   

Solve the filter equation.

t_solve_eqn = tic;
eqn.type = 'N';
eqn.B1   = B1;
eqn.C1   = 1 / gam * C1;
[outFilter, eqn, opts, ~] = mess_lrri(eqn, opts, oper);
t_elapsed2 = toc(t_solve_eqn);
mess_fprintf(opts, ...
             'solving the filter equation took %6.2f seconds \n', t_elapsed2);
RADI step:    1 pc: -6.733023e+00 + 0.000000e+00i normalized residual: 9.045840e-01 relative change in Z: 1.000000e+00  
RADI step:    2 pc: -6.101093e-04 + 0.000000e+00i normalized residual: 8.537235e-01 relative change in Z: 1.610629e-01  
RADI step:    3 pc: -3.305161e+01 + 0.000000e+00i normalized residual: 8.370617e-01 relative change in Z: 7.791775e-01  
RADI step:    4 pc: -1.388380e+01 + 0.000000e+00i normalized residual: 8.325987e-01 relative change in Z: 3.781421e-01  
RADI step:    5 pc: -3.178887e+00 + 0.000000e+00i normalized residual: 8.289544e-01 relative change in Z: 1.730686e-01  
RADI step:    6 pc: -1.228092e-03 + 0.000000e+00i normalized residual: 5.357085e-01 relative change in Z: 3.301825e-02  
RADI step:    7 pc: -1.746772e+00 + 0.000000e+00i normalized residual: 5.318790e-01 relative change in Z: 1.235617e-01  
RADI step:    8 pc: -3.186932e-03 + 0.000000e+00i normalized residual: 4.938200e-02 relative change in Z: 1.686849e-02  
RADI step:    9 pc: -5.233183e+02 + 0.000000e+00i normalized residual: 4.311709e-02 relative change in Z: 8.291444e-01  
RADI step:   10 pc: -3.773325e+01 + 0.000000e+00i normalized residual: 4.368398e-02 relative change in Z: 1.806956e-01  
RADI step:   11 pc: -1.073615e+01 + 0.000000e+00i normalized residual: 4.267700e-02 relative change in Z: 9.272202e-02  
RADI step:   12 pc: -4.624748e+00 + 0.000000e+00i normalized residual: 4.334177e-02 relative change in Z: 6.031714e-02  
RADI step:   13 pc: -4.955884e-03 + 0.000000e+00i normalized residual: 4.224669e-03 relative change in Z: 2.766628e-03  
RADI step:   14 pc: -4.148474e+03 + 0.000000e+00i normalized residual: 1.893079e-03 relative change in Z: 8.057708e-01  
RADI step:   15 pc: -9.544956e+01 + 0.000000e+00i normalized residual: 1.709084e-03 relative change in Z: 1.124349e-01  
RADI step:   16 pc: -3.929615e+02 + 0.000000e+00i normalized residual: 1.580603e-03 relative change in Z: 2.147091e-01  
RADI step:   17 pc: -4.402123e+01 + 0.000000e+00i normalized residual: 1.573369e-03 relative change in Z: 7.063560e-02  
RADI step:   18 pc: -5.072252e-03 + 0.000000e+00i normalized residual: 1.572828e-03 relative change in Z: 7.575539e-04  
RADI step:   19 pc: -1.244760e+05 + 0.000000e+00i normalized residual: 1.288151e-03 relative change in Z: 9.476746e-01  
RADI step:   20 pc: -2.550884e+03 + 0.000000e+00i normalized residual: 1.178627e-03 relative change in Z: 1.236269e-01  
RADI step:   21 pc: -1.989884e+00 + 0.000000e+00i normalized residual: 1.178542e-03 relative change in Z: 3.393657e-03  
RADI step:   22 pc: -1.365710e+02 + 0.000000e+00i normalized residual: 1.175046e-03 relative change in Z: 2.807692e-02  
RADI step:   23 pc: -3.694524e+06 + 0.000000e+00i normalized residual: 2.311225e-04 relative change in Z: 9.353425e-01  
RADI step:   24 pc: -5.072256e-03 + 0.000000e+00i normalized residual: 2.311224e-04 relative change in Z: 3.093179e-05  
RADI step:   25 pc: -1.667683e+04 + 0.000000e+00i normalized residual: 1.581918e-04 relative change in Z: 5.001227e-02  
RADI step:   26 pc: -1.732932e+00 + 0.000000e+00i normalized residual: 1.581834e-04 relative change in Z: 4.829152e-04  
RADI step:   27 pc: -1.837718e+02 + 0.000000e+00i normalized residual: 1.574734e-04 relative change in Z: 4.966974e-03  
RADI step:   28 pc: -7.299948e+06 + 0.000000e+00i normalized residual: 1.229622e-04 relative change in Z: 3.086848e-01  
RADI step:   29 pc: -5.071265e-03 + 0.000000e+00i normalized residual: 1.229622e-04 relative change in Z: 1.669139e-05  
RADI step:   30 pc: -3.231355e+04 + 0.000000e+00i normalized residual: 9.054101e-05 relative change in Z: 3.849275e-02  
RADI step:   31 pc: -1.785471e+00 + 0.000000e+00i normalized residual: 9.053733e-05 relative change in Z: 2.663204e-04  
RADI step:   32 pc: -1.532272e+03 + 0.000000e+00i normalized residual: 8.888265e-05 relative change in Z: 7.742925e-03  
RADI step:   33 pc: -7.300040e+06 + 0.000000e+00i normalized residual: 6.910066e-05 relative change in Z: 3.145313e-02  
RADI step:   34 pc: -1.732368e+00 + 0.000000e+00i normalized residual: 6.909949e-05 relative change in Z: 2.284200e-04  
RADI step:   35 pc: -9.937564e+04 + 0.000000e+00i normalized residual: 2.863383e-05 relative change in Z: 4.476690e-02  
RADI step:   36 pc: -2.093663e+02 + 0.000000e+00i normalized residual: 2.857128e-05 relative change in Z: 1.613629e-03  
RADI step:   37 pc: -3.746753e+03 + 0.000000e+00i normalized residual: 2.760528e-05 relative change in Z: 6.754995e-03  
RADI step:   38 pc: -7.300040e+06 + 0.000000e+00i normalized residual: 2.142997e-05 relative change in Z: 1.756580e-02  
RADI step:   39 pc: -3.537326e+00 + 0.000000e+00i normalized residual: 2.142927e-05 relative change in Z: 1.812062e-04  
RADI step:   40 pc: -1.532049e+05 + 0.000000e+00i normalized residual: 5.401103e-06 relative change in Z: 2.827969e-02  
RADI step:   41 pc: -1.219376e+02 + 0.000000e+00i normalized residual: 5.394630e-06 relative change in Z: 5.339948e-04  
RADI step:   42 pc: -6.378059e+03 + 0.000000e+00i normalized residual: 5.097280e-06 relative change in Z: 3.803608e-03  
RADI step:   43 pc: -7.299889e+06 + 0.000000e+00i normalized residual: 3.956165e-06 relative change in Z: 7.548035e-03  
RADI step:   44 pc: -6.374481e+00 + 0.000000e+00i normalized residual: 3.955938e-06 relative change in Z: 1.044249e-04  
RADI step:   45 pc: -1.532062e+05 + 0.000000e+00i normalized residual: 9.968264e-07 relative change in Z: 1.215312e-02  
RADI step:   46 pc: -3.109059e+02 + 0.000000e+00i normalized residual: 9.939270e-07 relative change in Z: 3.658705e-04  
RADI step:   47 pc: -1.041231e+03 + 0.000000e+00i normalized residual: 9.848651e-07 relative change in Z: 6.674490e-04  
RADI step:   48 pc: -4.619818e+05 + 0.000000e+00i normalized residual: 3.953228e-10 relative change in Z: 7.011233e-03  
RADI step:   49 pc: -1.055809e+01 + 0.000000e+00i normalized residual: 3.936007e-10 relative change in Z: 1.439466e-06  
RADI step:   50 pc: -2.766855e+04 + 0.000000e+00i normalized residual: 5.106401e-11 relative change in Z: 2.857172e-05  
RADI step:   51 pc: -6.481527e+03 + 0.000000e+00i normalized residual: 4.055691e-12 relative change in Z: 7.839430e-06  
RADI step:   52 pc: -4.569029e+03 + 0.000000e+00i normalized residual: 1.087769e-12 relative change in Z: 2.239775e-06  
RADI step:   53 pc: -1.704332e+01 + 0.000000e+00i normalized residual: 9.973275e-13 relative change in Z: 1.088868e-07  
RI step:    1 normalized residual: 3.356225e-02 relative change in Z: 1.000000e+00  
               number of RADI steps:   53  
  
RADI step:    1 pc: -7.574421e+00 + 0.000000e+00i normalized residual: 9.974283e-01 relative change in Z: 1.000000e+00  
RADI step:    2 pc: -3.220744e+00 + 0.000000e+00i normalized residual: 9.913783e-01 relative change in Z: 6.216922e-01  
RADI step:    3 pc: -4.872673e-03 + 0.000000e+00i normalized residual: 1.123939e-04 relative change in Z: 9.903921e-01  
RADI step:    4 pc: -1.975461e+00 + 0.000000e+00i normalized residual: 1.113818e-04 relative change in Z: 2.264970e-02  
RADI step:    5 pc: -4.976483e-03 + 0.000000e+00i normalized residual: 1.610903e-07 relative change in Z: 1.044024e-02  
RADI step:    6 pc: -8.610186e+01 + 0.000000e+00i normalized residual: 4.827575e-08 relative change in Z: 9.088001e-02  
RADI step:    7 pc: -1.324074e+01 + 0.000000e+00i normalized residual: 3.435078e-08 relative change in Z: 2.870125e-02  
RADI step:    8 pc: -4.713681e+00 + 0.000000e+00i normalized residual: 3.290065e-08 relative change in Z: 1.586653e-02  
RADI step:    9 pc: -2.928941e+00 + 0.000000e+00i normalized residual: 3.218537e-08 relative change in Z: 1.231829e-02  
RADI step:   10 pc: -1.835636e+03 + 0.000000e+00i normalized residual: 6.682073e-09 relative change in Z: 1.410303e-01  
RADI step:   11 pc: -4.976497e-03 + 0.000000e+00i normalized residual: 6.681494e-09 relative change in Z: 2.314884e-04  
RADI step:   12 pc: -8.100719e+01 + 0.000000e+00i normalized residual: 3.752070e-09 relative change in Z: 2.522862e-02  
RADI step:   13 pc: -1.336724e+01 + 0.000000e+00i normalized residual: 3.498875e-09 relative change in Z: 8.943885e-03  
RADI step:   14 pc: -1.147559e+02 + 0.000000e+00i normalized residual: 2.562014e-09 relative change in Z: 2.354616e-02  
RADI step:   15 pc: -1.307414e-01 + 0.000000e+00i normalized residual: 2.561364e-09 relative change in Z: 7.328725e-04  
RADI step:   16 pc: -7.606031e+04 + 0.000000e+00i normalized residual: 1.715988e-09 relative change in Z: 2.041502e-01  
RADI step:   17 pc: -4.250217e+00 + 0.000000e+00i normalized residual: 1.696639e-09 relative change in Z: 3.355057e-03  
RADI step:   18 pc: -2.952054e+03 + 0.000000e+00i normalized residual: 7.117716e-10 relative change in Z: 5.563952e-02  
RADI step:   19 pc: -1.241621e+03 + 0.000000e+00i normalized residual: 4.657995e-10 relative change in Z: 2.812855e-02  
RADI step:   20 pc: -1.574560e+00 + 0.000000e+00i normalized residual: 4.650671e-10 relative change in Z: 9.768988e-04  
RADI step:   21 pc: -6.189848e+05 + 0.000000e+00i normalized residual: 3.106910e-10 relative change in Z: 3.557936e-01  
RADI step:   22 pc: -3.827683e+01 + 0.000000e+00i normalized residual: 3.008424e-10 relative change in Z: 3.625177e-03  
RADI step:   23 pc: -1.972188e+04 + 0.000000e+00i normalized residual: 1.865602e-10 relative change in Z: 6.043148e-02  
RADI step:   24 pc: -3.560785e+03 + 0.000000e+00i normalized residual: 1.717880e-10 relative change in Z: 2.435425e-02  
RADI step:   25 pc: -7.299434e+06 + 0.000000e+00i normalized residual: 1.968735e-11 relative change in Z: 4.690285e-01  
RADI step:   26 pc: -2.019993e+00 + 0.000000e+00i normalized residual: 1.937090e-11 relative change in Z: 2.002453e-04  
RADI step:   27 pc: -1.502418e+05 + 0.000000e+00i normalized residual: 1.588878e-11 relative change in Z: 1.270581e-02  
RADI step:   28 pc: -3.802545e+01 + 0.000000e+00i normalized residual: 1.122051e-11 relative change in Z: 7.196279e-04  
RADI step:   29 pc: -1.391610e+04 + 0.000000e+00i normalized residual: 9.011609e-12 relative change in Z: 2.704259e-03  
RADI step:   30 pc: -2.129751e+01 + 0.000000e+00i normalized residual: 7.351018e-12 relative change in Z: 4.155737e-04  
RADI step:   31 pc: -7.300039e+06 + 0.000000e+00i normalized residual: 7.294359e-12 relative change in Z: 2.346279e-03  
RADI step:   32 pc: -5.155106e+02 + 0.000000e+00i normalized residual: 5.103889e-13 relative change in Z: 9.417067e-04  
RI step:    2 normalized residual: 1.335744e-05 relative change in Z: 1.918792e-08  
               number of RADI steps:   32  
  
RADI step:    1 pc: -2.822653e+00 + 0.000000e+00i normalized residual: 9.929780e-01 relative change in Z: 1.000000e+00  
RADI step:    2 pc: -4.969368e-03 + 0.000000e+00i normalized residual: 1.059607e-07 relative change in Z: 9.964829e-01  
RADI step:    3 pc: -1.823158e+01 + 0.000000e+00i normalized residual: 1.058438e-07 relative change in Z: 2.068935e-05  
RADI step:    4 pc: -4.995697e+00 + 0.000000e+00i normalized residual: 1.054212e-07 relative change in Z: 2.260557e-05  
RADI step:    5 pc: -2.908775e+00 + 0.000000e+00i normalized residual: 1.047007e-07 relative change in Z: 2.699593e-05  
RADI step:    6 pc: -2.051294e+00 + 0.000000e+00i normalized residual: 1.036897e-07 relative change in Z: 3.180115e-05  
RADI step:    7 pc: -4.972615e-03 + 0.000000e+00i normalized residual: 1.444481e-16 relative change in Z: 3.220073e-04  
RI step:    3 normalized residual: 2.076327e-12 relative change in Z: 4.678203e-12  
               number of RADI steps:    7  
  
solving the filter equation took   0.58 seconds   

Compute real residuals.

absControl = norm(eqn.A_' * (outControl.Z * outControl.Z') * eqn.E_ + ...
                  eqn.E_' * (outControl.Z * outControl.Z') * eqn.A_ + ...
                  eqn.E_' * (outControl.Z * outControl.Z') * ...
                  (1 / gam^2 * (B1 * B1') - eqn.B2 * eqn.B2') * ...
                  (outControl.Z * outControl.Z') * eqn.E_ + ...
                  C1' * C1, 2);
relControl = absControl / norm(C1 * C1', 2);
mess_fprintf(opts, ...
             '\nControl -> set tolerance vs. real residual: %e | %e\n', ...
             opts.ri.res_tol, relControl);

absFilter = norm(eqn.A_ * (outFilter.Z * outFilter.Z') * eqn.E_' + ...
                 eqn.E_ * (outFilter.Z * outFilter.Z') * eqn.A_' + ...
                 eqn.E_ * (outFilter.Z * outFilter.Z') * ...
                 (1 / gam^2 * (C1' * C1) - eqn.C2' * eqn.C2) * ...
                 (outFilter.Z * outFilter.Z') * eqn.E_' + ...
                 B1 * B1', 2);
relFilter = absFilter / norm(B1' * B1, 2);
mess_fprintf(opts, 'Filter  -> set tolerance vs. real residual: %e | %e\n', ...
             opts.ri.res_tol, relFilter);

% safety factor mostly used for Octave
safety = 10;
if istest
    mess_assert(opts, relControl < opts.ri.res_tol * safety, ...
                'TEST:accuracy', 'unexpectedly inaccurate result');
    mess_assert(opts, relFilter < opts.ri.res_tol * safety, ...
                'TEST:accuracy', 'unexpectedly inaccurate result');
end
  
Control -> set tolerance vs. real residual: 1.000000e-09 | 3.608465e-11  
Filter  -> set tolerance vs. real residual: 1.000000e-09 | 3.248554e-11