Contents

function out = LQR_LTV_smallscale_BDF(k, istest)
% Computes the optimal feedback via a BDF method for a small-scale LTV
% system, where the matrices are time-varying in the sense that
% A(t) = a(t) \hat{A}.
%
% Inputs:
%
% k           k-step BDF method
%             possible values: 1, ..., 4
%             (optinal, defaults to 2)
%
% istest      flag to determine whether this demo runs as a CI test or
%             interactive demo
%             (optional, defaults to 0, i.e. interactive demo)
%
%
%
% This program is free software; you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation; either version 2 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; if not, see <http://www.gnu.org/licenses/>.
%
% Copyright (C) Jens Saak, Martin Koehler, Peter Benner and others
%               2009-2020
%
if nargin < 1
    k = 2;
end
if nargin < 2
    istest = 0;
end

Nx = 10;
Nx2 = Nx^2;

% Finite-difference approximation of Laplacian on [0,1]^2 with Dirichlet
% boundary conditions
xx = linspace(0,1, Nx+2);
x = xx(2:end-1);
dx = 1/(Nx+1);
e = ones(Nx, 1);
A_1D = 1/dx^2*spdiags([e, -2*e, e], -1:1, Nx, Nx);

[X, Y] = meshgrid(x,x);
I_1D = eye(Nx, Nx);
A_2D = kron(A_1D, I_1D) + kron(I_1D, A_1D);
M = speye(Nx2, Nx2); % mass matrix = I in this case

% 3 inputs, control on the squares [j/4, j/4 + 1/8] x [j/4, j/4 + 1/8]
Nb = 3;
B = zeros(Nx2, Nb);
for j = 1:Nb
    B(:, j) = reshape((X > j/4) & (X < j/4 + 1/8) & (Y > j/4) & (Y < j/4 + 1/8), Nx2, 1);
end

% 1 output, average of all states
C = 1/Nx2 * ones(1, Nx2);

% Time dependency through factors
alpha = @(t) 1 + 10*sin(2*pi*t); % A
mu = @(t) 2 + 7.1*sin(2*pi*t);   % M
dmu = @(t) 7.1*2*pi*cos(2*pi*t); % dM/dt
beta = @(t) 3 + cos(t);          % B
gamma = @(t) 1 - min(t, 1);      % C

% Alternative, autonomous case:
% alpha = @(t) 1;        % A
% mu = @(t) 1;           % M
% dmu = @(t) 0;          % dM/dt
% beta = @(t) 1;         % B
% gamma = @(t) 1;        % C

At = @(t) alpha(t) * A_2D;
Mt = @(t) mu(t) * M;
dMt = @(t) dmu(t) * M;
Bt = @(t) beta(t) * B;
Ct = @(t) gamma(t) * C;

eqn.A_time = At;
eqn.E_time = Mt;
eqn.dt_E_time = dMt;
eqn.B_time = Bt;
eqn.C_time = Ct;

eqn.haveE = 1;

eqn.type = 'T';

% LDLT-factorization of initial condition P(0) = 0
L0 = zeros(Nx2, 1);
eqn.L0 = L0;
eqn.D0 = eye(size(L0, 2));

eqn.LTV = 1; % Specify that this is a time-varying problem

% Time interval [0, 0.1] and 100 time steps
t0 = 0;
tend = 0.1;
Nt = 1000;


% Set up and initialize operator
oper = operatormanager('default');

opts = struct;
[eqn, opts, oper] = oper.eval_matrix_functions(eqn, opts, oper, tend);

General BDF parameters

opts.norm = 'fro';

% ADI tolerances and maximum iteration number
opts.adi.maxiter = 100;
opts.adi.res_tol = 1e-14;
opts.adi.rel_diff_tol = 1e-16;
opts.adi.info = 0;
opts.adi.compute_sol_fac = 1;
opts.cc_info = 0;
%Heuristic shift parameters via basic Arnoldi
n=oper.size(eqn, opts);
opts.shifts.num_desired=7;
opts.shifts.num_Ritz=50;
opts.shifts.num_hRitz=25;
opts.shifts.method = 'heur';

% opts.shifts.b0=ones(n,1);

Newton tolerances and maximum iteration number

opts.nm.maxiter = 8;
opts.nm.res_tol = 1e-10;
opts.nm.rel_diff_tol = 1e-16;
opts.nm.info = 0;
opts.norm = 'fro';
opts.nm.accumulateRes = 1;
opts.nm.linesearch = 1;

BDF parameters

opts.bdf.time_steps = linspace(t0, tend, Nt+1);
opts.bdf.step = k;
opts.bdf.info = 1;
opts.bdf.save_solution = 1;
opts.bdf.startup_iter = 7;

Compute the approximation

tic;
[out, ~,opts, ~] = mess_bdf_dre(eqn,opts,oper);
toc;
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BDF step 7.500000e-02 s
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BDF step 7.490000e-02 s
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BDF step 7.480000e-02 s
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BDF step 7.470000e-02 s
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BDF step 7.460000e-02 s
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BDF step 7.450000e-02 s
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BDF step 7.440000e-02 s
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	 Rank 8
BDF step 7.430000e-02 s
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	 Rank 8
BDF step 7.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.410000e-02 s
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	 Rank 8
BDF step 7.400000e-02 s
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	 Rank 8
BDF step 7.390000e-02 s
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	 Rank 8
BDF step 7.380000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.360000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.350000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.340000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.330000e-02 s
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	 Rank 8
BDF step 7.320000e-02 s
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	 Rank 8
BDF step 7.310000e-02 s
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	 Rank 8
BDF step 7.300000e-02 s
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	 Rank 8
BDF step 7.290000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.260000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.250000e-02 s
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	 Rank 8
BDF step 7.240000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.230000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.220000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.210000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.200000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.190000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.180000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.170000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.160000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.150000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.140000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.130000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.120000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.110000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.100000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.080000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.070000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.000000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.990000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.980000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.970000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.960000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.950000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.940000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.930000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.920000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.910000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.900000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.890000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.880000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.870000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.860000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.850000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.840000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.830000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.820000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.810000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.800000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.790000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.780000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.770000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.760000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.750000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.740000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.730000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.720000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.710000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.700000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.690000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.680000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.670000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.660000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.650000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.640000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.630000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.620000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.610000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.600000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.590000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.580000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.570000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.560000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.550000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.540000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.530000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.520000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.510000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.500000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.490000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.480000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.470000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.460000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.450000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.440000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.430000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.410000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.400000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.390000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.380000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.360000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.350000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.340000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.330000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.320000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.310000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.300000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.290000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.260000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.250000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.240000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.230000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.220000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.210000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.200000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.190000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.180000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.170000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.160000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.150000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.140000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.130000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.120000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.110000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.100000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.080000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.070000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.000000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.990000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.980000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.970000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.960000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.950000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.940000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.930000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.920000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.910000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.900000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.890000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.880000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.870000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.860000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.850000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.840000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.830000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.820000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.810000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.800000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.790000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.780000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.770000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.760000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.750000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.740000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.730000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.720000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.710000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.700000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.690000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.680000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.670000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.660000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.650000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.640000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.630000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.620000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.610000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.600000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.590000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.580000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.570000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.560000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.550000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.540000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.530000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.520000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.510000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.500000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.490000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.480000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.470000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.460000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.450000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.440000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.430000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.410000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.400000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.390000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.380000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.360000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.350000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.340000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.330000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.320000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.310000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.300000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.290000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.260000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.250000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.240000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.230000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.220000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.210000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.200000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.190000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.180000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.170000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.160000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.150000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.140000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.130000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.120000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.110000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.100000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.080000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.070000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.000000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.990000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.980000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.970000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.960000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.950000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.940000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.930000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.920000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.910000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.900000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.890000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.880000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.870000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.860000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.850000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.840000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.830000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.820000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.810000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.800000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.790000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.780000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.770000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.760000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.750000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.740000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.730000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.720000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.710000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.700000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.690000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.680000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.670000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.660000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.650000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.640000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.630000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.620000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.610000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.600000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.590000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.580000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.570000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.560000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.550000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.540000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.530000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.520000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.510000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.500000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.490000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.480000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.470000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.460000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.450000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.440000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.430000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.410000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.400000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.390000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.380000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.360000e-02 s
	 Newton steps:  2 
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BDF step 4.350000e-02 s
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BDF step 4.340000e-02 s
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BDF step 4.330000e-02 s
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BDF step 4.320000e-02 s
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BDF step 4.310000e-02 s
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BDF step 4.300000e-02 s
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BDF step 4.290000e-02 s
	 Newton steps:  2 
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BDF step 4.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.260000e-02 s
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	 Rank 8
BDF step 4.250000e-02 s
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BDF step 4.240000e-02 s
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BDF step 4.230000e-02 s
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BDF step 4.220000e-02 s
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BDF step 4.210000e-02 s
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BDF step 4.200000e-02 s
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BDF step 4.190000e-02 s
	 Newton steps:  2 
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BDF step 4.180000e-02 s
	 Newton steps:  2 
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BDF step 4.170000e-02 s
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BDF step 4.160000e-02 s
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BDF step 4.150000e-02 s
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BDF step 4.140000e-02 s
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BDF step 4.130000e-02 s
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BDF step 4.120000e-02 s
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	 Rank 8
BDF step 4.110000e-02 s
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BDF step 4.100000e-02 s
	 Newton steps:  2 
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BDF step 4.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.080000e-02 s
	 Newton steps:  2 
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BDF step 4.070000e-02 s
	 Newton steps:  2 
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BDF step 4.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.000000e-02 s
	 Newton steps:  2 
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BDF step 3.990000e-02 s
	 Newton steps:  2 
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BDF step 3.980000e-02 s
	 Newton steps:  2 
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BDF step 3.970000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.960000e-02 s
	 Newton steps:  2 
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BDF step 3.950000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.940000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.930000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.920000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.910000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.900000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.890000e-02 s
	 Newton steps:  2 
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BDF step 3.880000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.870000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.860000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.850000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.840000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.830000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.820000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.810000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.800000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.790000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.780000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.770000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.760000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.750000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.740000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.730000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.720000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.710000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.700000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.690000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.680000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.670000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.660000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.650000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.640000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.630000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.620000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.610000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.600000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.590000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.580000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.570000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.560000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.550000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.540000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.530000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.520000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.510000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.500000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.490000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.480000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.470000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.460000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.450000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.440000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.430000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.410000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.400000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.390000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.380000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.360000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.350000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.340000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.330000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.320000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.310000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.300000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.290000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.260000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.250000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.240000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.230000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.220000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.210000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.200000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.190000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.180000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.170000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.160000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.150000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.140000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.130000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.120000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.110000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.100000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.080000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.070000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.000000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.990000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.980000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.970000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.960000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.950000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.940000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.930000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.920000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.910000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.900000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.890000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.880000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.870000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.860000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.850000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.840000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.830000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.820000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.810000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.800000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.790000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.780000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.770000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.760000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.750000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.740000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.730000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.720000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.710000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.700000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.690000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.680000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.670000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.660000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.650000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.640000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.630000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.620000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.610000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.600000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.590000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.580000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.570000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.560000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.550000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.540000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.530000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.520000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.510000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.500000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.490000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.480000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.470000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.460000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.450000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.440000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.430000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.410000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.400000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.390000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.380000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.360000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.350000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.340000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.330000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.320000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.310000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.300000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.290000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.260000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.250000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.240000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.230000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.220000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.210000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.200000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.190000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.180000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.170000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.160000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.150000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.140000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.130000e-02 s
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	 Rank 8
BDF step 2.120000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.110000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.100000e-02 s
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	 Rank 8
BDF step 2.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.080000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.070000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.000000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.990000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.980000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.970000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.960000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.950000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.940000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.930000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.920000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.910000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.900000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.890000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.880000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.870000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.860000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.850000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.840000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.830000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.820000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.810000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.800000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.790000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.780000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.770000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.760000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.750000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.740000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.730000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.720000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.710000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.700000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.690000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.680000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.670000e-02 s
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	 Rank 8
BDF step 1.660000e-02 s
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	 Rank 8
BDF step 1.650000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.640000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.630000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.620000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.610000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.600000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.590000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.580000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.570000e-02 s
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	 Rank 8
BDF step 1.560000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.550000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.540000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.530000e-02 s
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	 Rank 8
BDF step 1.520000e-02 s
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	 Rank 8
BDF step 1.510000e-02 s
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BDF step 1.500000e-02 s
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BDF step 1.490000e-02 s
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	 Rank 8
BDF step 1.480000e-02 s
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	 Rank 8
BDF step 1.470000e-02 s
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	 Rank 8
BDF step 1.460000e-02 s
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	 Rank 8
BDF step 1.450000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.440000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.430000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.420000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.410000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.400000e-02 s
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	 Rank 8
BDF step 1.390000e-02 s
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	 Rank 8
BDF step 1.380000e-02 s
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	 Rank 8
BDF step 1.370000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.360000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.350000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.340000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.330000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.320000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.310000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.300000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.290000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.280000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.270000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.260000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.250000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.240000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.230000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.220000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.210000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.200000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.190000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.180000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.170000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.160000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.150000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.140000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.130000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.120000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.110000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.100000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.090000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.080000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.070000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.060000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.050000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.040000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.030000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.020000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.010000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.000000e-02 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.900000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.800000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.700000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.600000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.500000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.400000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.300000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.200000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.100000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.000000e-03 s
	 Newton steps:  2 
	 Rank 8
BDF step 9.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 8.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 7.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 6.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 5.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 4.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 3.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 2.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step 1.000000e-04 s
	 Newton steps:  2 
	 Rank 8
BDF step    0 s
	 Newton steps:  2 
	 Rank 8
Elapsed time is 25.131421 seconds.
if not(istest)
    t = opts.bdf.time_steps;

    y = zeros(1,length(out.Ks));
    for i=1:length(out.Ks)
        y(i) = out.Ks{i}(1,1);
    end

    figure;
    plot(t, y);
    title('evolution of component (1,1) of the optimal feedback');
else

    if abs(norm(out.Ds{1}) / 6.079051242083189e-05 - 1) >= 1e-10
       error('MESS:TEST:accuracy','unexpectedly inaccurate result');
   end
end
ans = 

  struct with fields:

    res: [1×1000 double]
     rc: [1×1000 double]
     Ls: {1×1001 cell}
     Ds: {1×1001 cell}
     Ks: {1×1001 cell}