Contents

function [Ar, Br, Cr] = bt_mor_rail_tol(k,tol,shifts,istest)
% bt_mor_FDM_tol computes a reduced order model via the standard Lyapunov
% balanced truncation (see e.g. [1]) for a finite difference discretized
% convection diffusion model on the unit square described in [2].
%
% Usage:
%    [Ar, Br, Cr] = bt_mor_rail_tol(k,tol,max_ord,n0,test)
%
% Inputs
%
% k           refinement level of the model to use
%             (1-4, i.e. 1357-79841Dofs)
%             (optinal, defaults to 1)
%
% tol         truncation tolerance for the Hankel singular values
%             (optional; defalts to 1e-6)
%
% shifts      shift selection used in ADI;  possible choices:
%               'heur'       :   Penzl heuristic shifts
%               'projection' :   projection shifts using the last columns
%                                of the solution factor
%             (optional, defaults to 'heur')
%
% istest      flag to determine whether this demo runs as a CI test or
%             interactive demo
%             (optional, defaults to 0, i.e. interactive demo)
%
% Outputs
%
% Ar, Br, Cr  the reduced orde system matrices.
%
% References
% [1] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Vol.
%     6 of Adv. Des. Control, SIAM Publications, Philadelphia, PA, 2005.
%     https://doi.org/10.1137/1.9780898718713.
%
% [2] J. Saak, Effiziente numerische Lösung eines
%     Optimalsteuerungsproblems für die Abkühlung von Stahlprofilen,
%     Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universität
%     Bremen, D-28334 Bremen (Sep. 2003).
%
% [3] P. Benner, J. Saak, A semi-discretized heat transfer model for
%     optimal cooling of steel profiles, in: P. Benner, V. Mehrmann, D.
%     Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45
%     of Lect. Notes Comput. Sci. Eng., Springer-Verlag, Berlin/Heidelberg,
%     Germany, 2005, pp. 353–356. https://doi.org/10.1007/3-540-27909-1_19.
%
% [4] J. Saak, Efficient numerical solution of large scale algebraic matrix
%     equations in PDE control and model order reduction, Dissertation,
%     Technische Universität Chemnitz, Chemnitz, Germany (Jul. 2009).
%     URL http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901642
%
%
% 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
%
narginchk(0,4);

% BT tolerance and maximum order for the ROM
if nargin<1, k=3; end
if nargin<2, tol=1e-5; end
if nargin<3, shifts='heur'; end
if nargin<4, istest=0; end


% ADI tolerance and maximum iteration number
opts.adi.maxiter = 100;              % maximum iteration number
opts.adi.res_tol = 1e-10;            % residual norm tolerance
opts.adi.rel_diff_tol = 1e-16;       % relative change norm tolerance
opts.adi.info = 1;                   % turn output on
opts.norm = 'fro';                   % Frobenius norm for stopping criteria

oper = operatormanager('default');

Problem data

eqn = getrail(k);                           % load system matrices
n = oper.size(eqn, opts);                   % number of equations

Shift Parameters

opts.shifts.num_desired=25;                 % number of parameters for
                                            % 'heur' and 'wachspress'
switch lower(shifts)
    case 'heur'
        opts.shifts.method = 'heur';
        opts.shifts.num_Ritz=50;            % number Arnoldi steps with F
        opts.shifts.num_hRitz=25;           % Arnoldi steps with inv(F)
        opts.shifts.b0=ones(n,1);           % initial guess for Arnoldi
    case 'wachspress'
        opts.shifts.method = 'wachspress';
        opts.shifts.wachspress = 'T';
    case 'projection'
        opts.shifts.method = 'projection';
end

opts.shifts.p=mess_para(eqn,opts,oper); % compute shift parameters

disp('(Initial) ADI shifts');
disp(opts.shifts.p);
(Initial) ADI shifts
 -152.2715
 -106.4367
  -66.0034
  -39.1359
  -24.3281
  -15.0964
   -7.9169
   -2.9888
   -1.8181
   -0.9599
   -0.3584
   -0.2108
   -0.0446
   -0.0193
   -0.0101
   -0.0065
   -0.0035
   -0.0021
   -0.0009
   -0.0004
   -0.0003
   -0.0002
   -0.0001
   -0.0000
   -0.0000

Compute low-rank factor of Controllability Gramian

eqn.type='N';                       % Lyapunov eq. for Controllability Gram.
tic;
outB = mess_lradi(eqn, opts, oper); % run ADI iteration
toc;
% residual norm plot
if istest
    if min(outB.res)>=opts.adi.res_tol
        error('MESS:TEST:accuracy','unexpectedly inaccurate result');
    end
else
    figure(1);
    semilogy(outB.res);
    title('AXM^T + MXA^T = -BB^T');
    xlabel('number of iterations');
    ylabel('normalized residual norm');
    pause(1);
end
disp('size outB.Z:');
disp(size(outB.Z));
ADI step:    1 normalized residual: 9.672701e-01 relative change in Z: 1.000000e+00
ADI step:    2 normalized residual: 9.296996e-01 relative change in Z: 7.065275e-01
ADI step:    3 normalized residual: 8.826245e-01 relative change in Z: 6.045051e-01
ADI step:    4 normalized residual: 8.241819e-01 relative change in Z: 5.458853e-01
ADI step:    5 normalized residual: 7.587891e-01 relative change in Z: 4.919914e-01
ADI step:    6 normalized residual: 6.898272e-01 relative change in Z: 4.526419e-01
ADI step:    7 normalized residual: 6.115317e-01 relative change in Z: 4.486345e-01
ADI step:    8 normalized residual: 5.083161e-01 relative change in Z: 4.875644e-01
ADI step:    9 normalized residual: 4.224550e-01 relative change in Z: 4.329871e-01
ADI step:   10 normalized residual: 3.428018e-01 relative change in Z: 4.322948e-01
ADI step:   11 normalized residual: 2.554907e-01 relative change in Z: 4.870484e-01
ADI step:   12 normalized residual: 1.930484e-01 relative change in Z: 4.515040e-01
ADI step:   13 normalized residual: 1.137732e-01 relative change in Z: 5.923103e-01
ADI step:   14 normalized residual: 7.090317e-02 relative change in Z: 5.483890e-01
ADI step:   15 normalized residual: 5.193073e-02 relative change in Z: 5.070545e-01
ADI step:   16 normalized residual: 4.392870e-02 relative change in Z: 4.589057e-01
ADI step:   17 normalized residual: 3.946526e-02 relative change in Z: 4.668622e-01
ADI step:   18 normalized residual: 3.465249e-02 relative change in Z: 4.524043e-01
ADI step:   19 normalized residual: 2.764332e-02 relative change in Z: 4.989289e-01
ADI step:   20 normalized residual: 1.856660e-02 relative change in Z: 5.125207e-01
ADI step:   21 normalized residual: 1.176669e-02 relative change in Z: 4.609735e-01
ADI step:   22 normalized residual: 6.769639e-03 relative change in Z: 3.917797e-01
ADI step:   23 normalized residual: 2.287358e-03 relative change in Z: 3.785449e-01
ADI step:   24 normalized residual: 1.313940e-04 relative change in Z: 2.883884e-01
ADI step:   25 normalized residual: 7.144918e-05 relative change in Z: 6.805685e-02
ADI step:   26 normalized residual: 7.107662e-05 relative change in Z: 6.824600e-05
ADI step:   27 normalized residual: 7.056731e-05 relative change in Z: 8.052261e-05
ADI step:   28 normalized residual: 6.979614e-05 relative change in Z: 1.005920e-04
ADI step:   29 normalized residual: 6.860685e-05 relative change in Z: 1.279988e-04
ADI step:   30 normalized residual: 6.690384e-05 relative change in Z: 1.587034e-04
ADI step:   31 normalized residual: 6.450525e-05 relative change in Z: 1.968148e-04
ADI step:   32 normalized residual: 6.058395e-05 relative change in Z: 2.638983e-04
ADI step:   33 normalized residual: 5.219462e-05 relative change in Z: 4.043165e-04
ADI step:   34 normalized residual: 4.172419e-05 relative change in Z: 4.660896e-04
ADI step:   35 normalized residual: 2.799693e-05 relative change in Z: 5.418764e-04
ADI step:   36 normalized residual: 1.017742e-05 relative change in Z: 6.255297e-04
ADI step:   37 normalized residual: 2.209369e-06 relative change in Z: 4.299523e-04
ADI step:   38 normalized residual: 1.954612e-07 relative change in Z: 2.420888e-04
ADI step:   39 normalized residual: 8.093605e-08 relative change in Z: 1.050616e-04
ADI step:   40 normalized residual: 5.230207e-08 relative change in Z: 1.025783e-04
ADI step:   41 normalized residual: 3.267056e-08 relative change in Z: 8.982669e-05
ADI step:   42 normalized residual: 2.513449e-08 relative change in Z: 7.577071e-05
ADI step:   43 normalized residual: 2.178487e-08 relative change in Z: 4.845362e-05
ADI step:   44 normalized residual: 2.077728e-08 relative change in Z: 2.956070e-05
ADI step:   45 normalized residual: 2.040621e-08 relative change in Z: 1.197793e-05
ADI step:   46 normalized residual: 2.018731e-08 relative change in Z: 2.770131e-06
ADI step:   47 normalized residual: 2.002868e-08 relative change in Z: 1.952378e-06
ADI step:   48 normalized residual: 1.995646e-08 relative change in Z: 1.297859e-06
ADI step:   49 normalized residual: 1.993238e-08 relative change in Z: 7.472417e-07
ADI step:   50 normalized residual: 1.991770e-08 relative change in Z: 5.829711e-07
ADI step:   51 normalized residual: 1.986731e-08 relative change in Z: 1.061608e-06
ADI step:   52 normalized residual: 1.979529e-08 relative change in Z: 1.266703e-06
ADI step:   53 normalized residual: 1.967936e-08 relative change in Z: 1.602856e-06
ADI step:   54 normalized residual: 1.948455e-08 relative change in Z: 2.069997e-06
ADI step:   55 normalized residual: 1.917350e-08 relative change in Z: 2.603105e-06
ADI step:   56 normalized residual: 1.867970e-08 relative change in Z: 3.262395e-06
ADI step:   57 normalized residual: 1.776859e-08 relative change in Z: 4.409074e-06
ADI step:   58 normalized residual: 1.556152e-08 relative change in Z: 6.836204e-06
ADI step:   59 normalized residual: 1.252311e-08 relative change in Z: 8.008842e-06
ADI step:   60 normalized residual: 8.321434e-09 relative change in Z: 9.427946e-06
ADI step:   61 normalized residual: 2.794217e-09 relative change in Z: 1.089724e-05
ADI step:   62 normalized residual: 4.507473e-10 relative change in Z: 7.253566e-06
ADI step:   63 normalized residual: 4.643685e-11 relative change in Z: 3.181829e-06
Elapsed time is 2.107552 seconds.
size outB.Z:
       20209         441

Compute low-rank factor of Observatibility Gramian

eqn.type = 'T';                     % Lyapunov eq. for Observability Gram.
tic;
outC = mess_lradi(eqn, opts, oper); % run ADI iteration
toc;
% residual norm plot
if istest
    if min(outC.res)>=opts.adi.res_tol
        error('MESS:TEST:accuracy','unexpectedly inaccurate result');
    end
else
    figure(2);
    semilogy(outC.res);
    title('A^TXM + M^TXA = -C^TC');
    xlabel('number of iterations');
    ylabel('normalized residual norm');
    pause(1);
end
disp('size outC.Z:');
disp(size(outC.Z));
ADI step:    1 normalized residual: 9.391027e-01 relative change in Z: 1.000000e+00
ADI step:    2 normalized residual: 8.692555e-01 relative change in Z: 7.220041e-01
ADI step:    3 normalized residual: 7.829058e-01 relative change in Z: 6.158664e-01
ADI step:    4 normalized residual: 6.783179e-01 relative change in Z: 5.516109e-01
ADI step:    5 normalized residual: 5.646896e-01 relative change in Z: 4.905379e-01
ADI step:    6 normalized residual: 4.473005e-01 relative change in Z: 4.400441e-01
ADI step:    7 normalized residual: 3.188496e-01 relative change in Z: 4.167621e-01
ADI step:    8 normalized residual: 1.786504e-01 relative change in Z: 4.140493e-01
ADI step:    9 normalized residual: 1.066603e-01 relative change in Z: 3.170179e-01
ADI step:   10 normalized residual: 6.224956e-02 relative change in Z: 2.783011e-01
ADI step:   11 normalized residual: 3.018132e-02 relative change in Z: 2.828326e-01
ADI step:   12 normalized residual: 1.628973e-02 relative change in Z: 2.369609e-01
ADI step:   13 normalized residual: 5.224343e-03 relative change in Z: 2.956199e-01
ADI step:   14 normalized residual: 1.500785e-03 relative change in Z: 2.401792e-01
ADI step:   15 normalized residual: 4.412941e-04 relative change in Z: 1.829698e-01
ADI step:   16 normalized residual: 1.948652e-04 relative change in Z: 1.343735e-01
ADI step:   17 normalized residual: 9.074454e-05 relative change in Z: 1.032315e-01
ADI step:   18 normalized residual: 5.231776e-05 relative change in Z: 6.404962e-02
ADI step:   19 normalized residual: 3.885750e-05 relative change in Z: 5.184031e-02
ADI step:   20 normalized residual: 3.279765e-05 relative change in Z: 5.548217e-02
ADI step:   21 normalized residual: 2.852168e-05 relative change in Z: 5.424737e-02
ADI step:   22 normalized residual: 2.552789e-05 relative change in Z: 4.880522e-02
ADI step:   23 normalized residual: 2.310104e-05 relative change in Z: 4.737782e-02
ADI step:   24 normalized residual: 2.231966e-05 relative change in Z: 3.095236e-02
ADI step:   25 normalized residual: 2.230059e-05 relative change in Z: 5.622151e-03
ADI step:   26 normalized residual: 2.214752e-05 relative change in Z: 3.467168e-04
ADI step:   27 normalized residual: 2.194029e-05 relative change in Z: 4.013048e-04
ADI step:   28 normalized residual: 2.162967e-05 relative change in Z: 4.878202e-04
ADI step:   29 normalized residual: 2.115476e-05 relative change in Z: 5.977384e-04
ADI step:   30 normalized residual: 2.047690e-05 relative change in Z: 7.083316e-04
ADI step:   31 normalized residual: 1.952103e-05 relative change in Z: 8.407251e-04
ADI step:   32 normalized residual: 1.797508e-05 relative change in Z: 1.086631e-03
ADI step:   33 normalized residual: 1.488645e-05 relative change in Z: 1.609753e-03
ADI step:   34 normalized residual: 1.147171e-05 relative change in Z: 1.802037e-03
ADI step:   35 normalized residual: 7.422971e-06 relative change in Z: 2.051176e-03
ADI step:   36 normalized residual: 2.491404e-06 relative change in Z: 2.326024e-03
ADI step:   37 normalized residual: 4.502215e-07 relative change in Z: 1.559597e-03
ADI step:   38 normalized residual: 3.925129e-08 relative change in Z: 7.952223e-04
ADI step:   39 normalized residual: 1.602659e-08 relative change in Z: 2.392763e-04
ADI step:   40 normalized residual: 1.039098e-08 relative change in Z: 1.790298e-04
ADI step:   41 normalized residual: 7.859436e-09 relative change in Z: 1.450926e-04
ADI step:   42 normalized residual: 6.778021e-09 relative change in Z: 1.109427e-04
ADI step:   43 normalized residual: 6.203145e-09 relative change in Z: 6.065464e-05
ADI step:   44 normalized residual: 5.973348e-09 relative change in Z: 2.877420e-05
ADI step:   45 normalized residual: 5.871757e-09 relative change in Z: 1.230214e-05
ADI step:   46 normalized residual: 5.809292e-09 relative change in Z: 8.401568e-06
ADI step:   47 normalized residual: 5.764135e-09 relative change in Z: 7.109816e-06
ADI step:   48 normalized residual: 5.743597e-09 relative change in Z: 4.790191e-06
ADI step:   49 normalized residual: 5.736751e-09 relative change in Z: 2.764968e-06
ADI step:   50 normalized residual: 5.732576e-09 relative change in Z: 2.158827e-06
ADI step:   51 normalized residual: 5.716271e-09 relative change in Z: 4.055486e-06
ADI step:   52 normalized residual: 5.693115e-09 relative change in Z: 4.837057e-06
ADI step:   53 normalized residual: 5.656166e-09 relative change in Z: 6.117509e-06
ADI step:   54 normalized residual: 5.594841e-09 relative change in Z: 7.895356e-06
ADI step:   55 normalized residual: 5.498467e-09 relative change in Z: 9.922104e-06
ADI step:   56 normalized residual: 5.348182e-09 relative change in Z: 1.242715e-05
ADI step:   57 normalized residual: 5.075973e-09 relative change in Z: 1.678060e-05
ADI step:   58 normalized residual: 4.428808e-09 relative change in Z: 2.596681e-05
ADI step:   59 normalized residual: 3.550571e-09 relative change in Z: 3.033124e-05
ADI step:   60 normalized residual: 2.347448e-09 relative change in Z: 3.559557e-05
ADI step:   61 normalized residual: 7.783330e-10 relative change in Z: 4.098519e-05
ADI step:   62 normalized residual: 1.212365e-10 relative change in Z: 2.715455e-05
ADI step:   63 normalized residual: 1.293672e-11 relative change in Z: 1.176613e-05
Elapsed time is 2.114844 seconds.
size outC.Z:
       20209         378

Compute reduced system matrices

Perform Square Root Method

opts.srm.tol=tol;
opts.srm.max_ord=n;
opts.srm.info=2;
[TL,TR,hsv] = mess_square_root_method(eqn,opts,oper,outB.Z,outC.Z);
reduced system order: 67  (max possible/allowed: 378/20209)

compute ROM matrices

Ar = TL'*oper.mul_A(eqn, opts, 'N', TR, 'N');
Br = TL'*eqn.B;
Cr = eqn.C*TR;
Er = eye(size(Ar,1));

Plots

ROM.A=Ar;
ROM.E=Er;
ROM.B=Br;
ROM.C=Cr;
if istest
    opts.sigma.info=0;
else
    opts.sigma.info=2;
end

opts.sigma.fmin=-3;
opts.sigma.fmax=4;

out = mess_sigma_plot(eqn, opts, oper, ROM); err = out.err;

if istest
    if max(err)>tol
        error('MESS:TEST:accuracy','unexpectedly inaccurate result');
    end
else
    figure;
    semilogy(hsv);
    title('Computed Hankel singular values');
    xlabel('index');
    ylabel('magnitude');
end
Computing TFMs of original and reduced order systems and MOR errors
 Step  10 / 100 Step  20 / 100 Step  30 / 100 Step  40 / 100 Step  50 / 100 Step  60 / 100 Step  70 / 100 Step  80 / 100 Step  90 / 100 Step 100 / 100


ans =

  Columns 1 through 7

   -0.0000    0.0000   -0.0000    0.0001   -0.0000    0.0000   -0.0002
   -0.0000   -0.0009   -0.0012   -0.0002   -0.0001    0.0012    0.0002
    0.0000   -0.0009   -0.0014    0.0001   -0.0003    0.0015   -0.0002
   -0.0002   -0.0001    0.0001   -0.0068    0.0041    0.0002    0.0018
    0.0001   -0.0001   -0.0001    0.0040   -0.0029    0.0002   -0.0002
    0.0000    0.0010    0.0013    0.0002    0.0002   -0.0024    0.0001
    0.0001   -0.0000    0.0001    0.0008   -0.0004    0.0001   -0.0066
   -0.0001   -0.0002    0.0005   -0.0012   -0.0018    0.0002   -0.0006
    0.0000    0.0017    0.0022    0.0016   -0.0005   -0.0045    0.0002
   -0.0001    0.0002    0.0005   -0.0102    0.0069   -0.0005   -0.0015
   -0.0000    0.0054    0.0069    0.0004    0.0004   -0.0094   -0.0001
    0.0002    0.0000   -0.0003    0.0140   -0.0099    0.0000    0.0012
   -0.0003   -0.0008   -0.0003   -0.0193    0.0126    0.0012    0.0007
    0.0001   -0.0022   -0.0028    0.0041   -0.0030    0.0032   -0.0004
    0.0000   -0.0026   -0.0043   -0.0014    0.0048    0.0055    0.0087
    0.0000   -0.0065   -0.0082    0.0008   -0.0030    0.0150   -0.0062
   -0.0002   -0.0011   -0.0010   -0.0022   -0.0023    0.0025    0.0180
    0.0001    0.0007    0.0005    0.0097   -0.0061   -0.0016    0.0031
   -0.0000   -0.0034   -0.0038    0.0003   -0.0000    0.0007    0.0002
   -0.0000    0.0002    0.0005   -0.0068    0.0054   -0.0007   -0.0082
   -0.0000    0.0016    0.0021   -0.0005    0.0005   -0.0038    0.0013
   -0.0000   -0.0031   -0.0041   -0.0000   -0.0020    0.0103   -0.0007
   -0.0001   -0.0019   -0.0027   -0.0101    0.0085    0.0055    0.0013
    0.0001   -0.0017   -0.0030    0.0094   -0.0049    0.0050    0.0006
    0.0000    0.0041    0.0052    0.0006    0.0018   -0.0110   -0.0004
   -0.0001    0.0002    0.0001   -0.0032    0.0024   -0.0006    0.0138
    0.0001    0.0000   -0.0005    0.0032    0.0009   -0.0007   -0.0044
    0.0000   -0.0001   -0.0009    0.0004   -0.0014    0.0080   -0.0012
    0.0000    0.0029    0.0042    0.0020   -0.0007   -0.0105   -0.0016
   -0.0001    0.0011    0.0015   -0.0055    0.0045   -0.0031    0.0022
   -0.0000   -0.0017   -0.0019   -0.0022    0.0001    0.0039   -0.0004
    0.0000   -0.0001   -0.0004    0.0013   -0.0002    0.0012    0.0007
    0.0000   -0.0021   -0.0030    0.0006   -0.0002    0.0056    0.0005
    0.0001    0.0002    0.0002    0.0066   -0.0049   -0.0006    0.0004
   -0.0000    0.0008    0.0012   -0.0006    0.0008   -0.0030    0.0003
   -0.0000   -0.0009   -0.0015   -0.0006   -0.0008    0.0068   -0.0001
   -0.0001   -0.0001    0.0000   -0.0054    0.0035    0.0001    0.0028
    0.0000    0.0005    0.0001    0.0005    0.0017   -0.0004    0.0011
   -0.0000    0.0008    0.0012   -0.0020    0.0009   -0.0013   -0.0001
   -0.0000   -0.0008   -0.0010   -0.0022    0.0019    0.0010    0.0007
    0.0000   -0.0000   -0.0001    0.0024   -0.0012   -0.0007   -0.0011
   -0.0000    0.0002    0.0005   -0.0003    0.0009   -0.0030    0.0005
   -0.0000   -0.0004    0.0001   -0.0005   -0.0023    0.0005   -0.0016
    0.0000   -0.0001   -0.0002    0.0007   -0.0003    0.0004   -0.0002
    0.0000   -0.0001   -0.0002    0.0016   -0.0006    0.0001   -0.0031
    0.0000   -0.0009   -0.0011    0.0004   -0.0000    0.0004    0.0002
    0.0000    0.0003    0.0003    0.0020   -0.0012   -0.0004   -0.0040
   -0.0000    0.0009    0.0011    0.0004   -0.0006   -0.0013    0.0012
   -0.0000   -0.0002   -0.0002   -0.0008    0.0005    0.0000   -0.0016
    0.0000   -0.0012   -0.0014    0.0006   -0.0007    0.0016    0.0010
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   -0.0000   -0.0015   -0.0019   -0.0002   -0.0002    0.0028    0.0014
   -0.0000   -0.0000   -0.0001   -0.0002   -0.0001    0.0004    0.0004
   -0.0000    0.0012    0.0018    0.0000    0.0000   -0.0030   -0.0001
   -0.0000    0.0002    0.0002   -0.0003    0.0004   -0.0005    0.0019
    0.0000    0.0002    0.0002    0.0004    0.0003   -0.0006   -0.0013
    0.0000   -0.0001   -0.0001    0.0002   -0.0001    0.0002   -0.0018
    0.0000   -0.0001   -0.0002   -0.0000   -0.0000    0.0005    0.0001
   -0.0000   -0.0000    0.0000   -0.0003    0.0001   -0.0000    0.0021
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    0.0000   -0.0002   -0.0003    0.0000    0.0001    0.0005    0.0001
    0.0000   -0.0000    0.0000   -0.0000   -0.0002    0.0001   -0.0014
    0.0000    0.0001   -0.0001    0.0004    0.0010   -0.0002    0.0003
    0.0000    0.0003    0.0003    0.0001    0.0003   -0.0006    0.0000
   -0.0000    0.0001    0.0002   -0.0000   -0.0000   -0.0005   -0.0001
   -0.0000   -0.0000   -0.0000   -0.0000   -0.0001    0.0002   -0.0002

  Columns 8 through 14

    0.0000    0.0000    0.0000    0.0000   -0.0000    0.0001   -0.0000
    0.0001    0.0019    0.0002    0.0052    0.0000   -0.0007   -0.0021
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   -0.0009    0.0017   -0.0106    0.0006    0.0153   -0.0209    0.0044
   -0.0011   -0.0004    0.0074    0.0008   -0.0111    0.0138   -0.0035
    0.0001   -0.0055   -0.0007   -0.0099    0.0000    0.0013    0.0038
   -0.0004    0.0001   -0.0006   -0.0001    0.0007    0.0013   -0.0006
   -0.0142    0.0004   -0.0011   -0.0009   -0.0011   -0.0058    0.0002
    0.0005   -0.0155    0.0016   -0.0242   -0.0058    0.0117    0.0072
   -0.0009    0.0016   -0.0291   -0.0034    0.0483   -0.0685    0.0166
   -0.0009   -0.0239   -0.0036   -0.1006    0.0012    0.0147    0.0581
   -0.0013   -0.0055    0.0466    0.0013   -0.0851    0.1262   -0.0296
   -0.0057    0.0115   -0.0670    0.0141    0.1267   -0.2082    0.0361
    0.0000    0.0077    0.0165    0.0558   -0.0296    0.0360   -0.0508
    0.0369    0.0201    0.0008    0.0750    0.0145   -0.0071   -0.0419
   -0.0110    0.0509    0.0086    0.1608   -0.0084   -0.0244   -0.1076
   -0.0175    0.0071    0.0145    0.0236   -0.0336    0.0181   -0.0222
    0.0063   -0.0101    0.0411   -0.0163   -0.0841    0.1492   -0.0204
   -0.0017   -0.0004    0.0022    0.1110   -0.0062   -0.0192   -0.1095
   -0.0003    0.0020   -0.0406   -0.0067    0.0882   -0.1408    0.0355
   -0.0005   -0.0129   -0.0026   -0.0449    0.0015    0.0051    0.0330
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    0.0154    0.0263   -0.0432    0.0564    0.1068   -0.1822    0.0062
    0.0188    0.0144    0.0435    0.0542   -0.0836    0.1609   -0.0711
    0.0087   -0.0423   -0.0081   -0.1181    0.0105    0.0198    0.0892
    0.0084   -0.0024    0.0047   -0.0059   -0.0145    0.0233    0.0047
    0.0219   -0.0026    0.0026    0.0071    0.0049    0.0252   -0.0093
    0.0003    0.0279    0.0053   -0.0143   -0.0032    0.0164    0.0478
   -0.0047   -0.0387    0.0008   -0.0840   -0.0139    0.0380    0.0426
    0.0033   -0.0082   -0.0259   -0.0334    0.0568   -0.0948    0.0498
   -0.0089    0.0150   -0.0063    0.0507    0.0146   -0.0520   -0.0354
    0.0073    0.0039    0.0065    0.0038   -0.0108    0.0300   -0.0029
    0.0067    0.0213    0.0066    0.0673   -0.0066    0.0059   -0.0549
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    0.0004   -0.0103   -0.0041   -0.0248    0.0059   -0.0056    0.0179
   -0.0022    0.0220    0.0023    0.0171   -0.0002   -0.0034    0.0121
   -0.0006    0.0021   -0.0149    0.0012    0.0256   -0.0537    0.0115
    0.0173   -0.0022   -0.0002   -0.0115    0.0072    0.0152    0.0169
   -0.0049   -0.0037   -0.0109   -0.0300    0.0234   -0.0434    0.0393
    0.0026    0.0054   -0.0114    0.0282    0.0305   -0.0617   -0.0149
    0.0027   -0.0034    0.0110    0.0035   -0.0268    0.0544   -0.0199
    0.0025   -0.0082   -0.0035    0.0003    0.0070   -0.0123   -0.0119
   -0.0221    0.0020    0.0009    0.0066   -0.0112   -0.0131   -0.0127
    0.0015    0.0008    0.0021    0.0048   -0.0033    0.0078   -0.0052
    0.0028   -0.0007    0.0082    0.0030   -0.0231    0.0505   -0.0144
    0.0013    0.0018    0.0016    0.0349   -0.0021   -0.0043   -0.0405
   -0.0010   -0.0023    0.0082   -0.0101   -0.0239    0.0509   -0.0023
   -0.0023   -0.0043    0.0006   -0.0308   -0.0008    0.0050    0.0302
   -0.0020   -0.0002   -0.0030    0.0079    0.0030   -0.0095   -0.0087
   -0.0005    0.0032    0.0037    0.0418   -0.0062   -0.0012   -0.0438
   -0.0004    0.0012   -0.0032    0.0087    0.0071   -0.0169   -0.0058
   -0.0003    0.0089    0.0012    0.0827   -0.0021   -0.0166   -0.0764
   -0.0004    0.0056    0.0014    0.0518   -0.0012   -0.0131   -0.0468
   -0.0011   -0.0016   -0.0005    0.0005    0.0004   -0.0026    0.0029
   -0.0024   -0.0030   -0.0007   -0.0447    0.0004    0.0062    0.0364
    0.0021   -0.0004   -0.0005   -0.0053    0.0029   -0.0032    0.0059
    0.0033   -0.0010    0.0001   -0.0071    0.0002    0.0079    0.0047
   -0.0007    0.0002   -0.0003    0.0017   -0.0006    0.0018   -0.0020
    0.0006   -0.0022   -0.0003    0.0047    0.0000   -0.0000   -0.0017
   -0.0002    0.0002   -0.0023    0.0000    0.0090   -0.0196    0.0048
   -0.0005    0.0018    0.0002   -0.0006    0.0002   -0.0013   -0.0009
    0.0015    0.0021    0.0003    0.0069    0.0006   -0.0006   -0.0058
   -0.0025    0.0005   -0.0005    0.0005   -0.0003   -0.0017   -0.0010
    0.0095    0.0000    0.0003    0.0008    0.0033    0.0080    0.0001
    0.0024   -0.0048   -0.0007   -0.0070    0.0012    0.0034    0.0068
   -0.0009   -0.0005   -0.0002   -0.0024   -0.0001   -0.0009   -0.0003
   -0.0012   -0.0008   -0.0003    0.0011   -0.0003   -0.0012   -0.0003

  Columns 15 through 21

    0.0001   -0.0001    0.0003    0.0000   -0.0000   -0.0001    0.0000
   -0.0031   -0.0064   -0.0013    0.0006   -0.0030    0.0004    0.0015
   -0.0040   -0.0085   -0.0009    0.0006   -0.0038    0.0004    0.0021
   -0.0016    0.0005   -0.0009    0.0111    0.0002   -0.0095   -0.0002
    0.0038   -0.0029   -0.0038   -0.0074   -0.0006    0.0080    0.0003
    0.0062    0.0161    0.0026   -0.0017    0.0025   -0.0007   -0.0040
    0.0090   -0.0065    0.0189    0.0036    0.0002   -0.0109    0.0017
    0.0368   -0.0107   -0.0173    0.0064   -0.0014   -0.0002   -0.0005
    0.0199    0.0513    0.0069   -0.0106   -0.0019    0.0027   -0.0139
    0.0016    0.0078    0.0170    0.0434    0.0021   -0.0475   -0.0019
    0.0748    0.1606    0.0236   -0.0161    0.1222   -0.0059   -0.0434
    0.0150   -0.0084   -0.0338   -0.0851   -0.0066    0.0936    0.0005
   -0.0076   -0.0237    0.0142    0.1504   -0.0197   -0.1495    0.0058
   -0.0422   -0.1083   -0.0213   -0.0210   -0.1124    0.0367    0.0313
   -0.3085   -0.0809    0.0224   -0.0203   -0.0972    0.0334    0.0491
   -0.0823   -0.4142   -0.0457    0.0559   -0.2227   -0.0067    0.1245
    0.0227   -0.0464   -0.2162   -0.0365   -0.0407    0.1468   -0.0005
   -0.0181    0.0552   -0.0333   -0.1335    0.0235    0.1499   -0.0163
   -0.0885   -0.1993   -0.0356    0.0195   -0.4440    0.0077    0.0638
    0.0267    0.0014    0.1179    0.1393    0.0111   -0.2132    0.0040
    0.0493    0.1241    0.0043   -0.0147    0.0764    0.0034   -0.0452
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   -0.2172   -0.1274    0.0726    0.1653   -0.0454   -0.1893    0.0679
   -0.2029   -0.1299    0.0044   -0.1523   -0.0598    0.1826    0.0663
    0.0914    0.3985    0.1072   -0.0463    0.1856   -0.0469   -0.1438
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   -0.1673    0.0294    0.1622   -0.0221   -0.0342   -0.0662    0.0157
   -0.0011   -0.0670   -0.0003    0.0035    0.4335    0.0134    0.0352
    0.1765    0.2929    0.0429   -0.0518   -0.0038   -0.0102   -0.1329
   -0.0045    0.1357    0.0338    0.0768    0.0623   -0.1132   -0.0526
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   -0.0774   -0.0018    0.0164   -0.0382    0.0506    0.0399    0.0111
   -0.1622   -0.2107   -0.0137    0.0000   -0.1221    0.0386    0.1025
    0.0326    0.0119   -0.0561   -0.1561   -0.0106    0.2049   -0.0111
    0.0301    0.0992    0.0137   -0.0051    0.0026   -0.0117   -0.0455
   -0.0137   -0.1211   -0.0340    0.0197    0.2472    0.0200    0.0563
   -0.0260    0.0176   -0.0554    0.0273   -0.0064    0.0352   -0.0146
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   -0.0201    0.0010    0.0080   -0.0628   -0.0631    0.0728    0.0006
   -0.0239    0.0370    0.0188    0.0042   -0.1660   -0.0248   -0.0157
    0.2146   -0.0975   -0.1050    0.0422   -0.0617    0.0100    0.0085
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   -0.0256   -0.0198    0.0035    0.0005   -0.0183    0.0008    0.0116
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  Columns 22 through 28

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   -0.0032   -0.0021   -0.0019    0.0042   -0.0001   -0.0000   -0.0002
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    0.0001   -0.0121    0.0108    0.0004   -0.0013    0.0024    0.0005
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    0.0080   -0.0472    0.0470   -0.0088    0.0100    0.0013    0.0058
    0.0743    0.0526    0.0504   -0.1152   -0.0067    0.0087   -0.0397
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   -0.0641    0.0908   -0.0086    0.1098   -0.1733    0.1680    0.0073
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    0.0023   -0.0232   -0.0379    0.1592    0.0214   -0.0375    0.4901
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    0.3987    0.1689    0.1913   -0.6110   -0.0458   -0.1153    0.1900
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    0.0767   -0.1850   -0.1790   -0.1126    0.0759   -0.3985    0.0746
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    0.3992    0.3579    0.2651   -0.5513    0.0466    0.0261    0.6006
    0.1487   -0.1486    0.2174   -0.2314   -0.0697   -0.0296    0.1109
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    0.0861   -0.1884   -0.1998   -0.2030   -0.1841   -0.4203   -0.1509
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   -0.0790   -0.0634   -0.0826    0.2318    0.0274   -0.0222    0.2644
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   -0.1999   -0.1677   -0.1803    0.4746    0.0660   -0.0423    0.2010
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    0.0272   -0.0315   -0.0040   -0.0448   -0.0842   -0.0353    0.0393
    0.0362   -0.0095   -0.0354   -0.0703    0.0109   -0.1138    0.0320
   -0.0108    0.0145   -0.0015    0.0137    0.0635   -0.0027   -0.0262
   -0.0013   -0.0104   -0.0105   -0.0127   -0.0095   -0.0077   -0.0919
    0.0026   -0.0716    0.0616   -0.0062    0.0339   -0.0037    0.0019
   -0.0062   -0.0002    0.0052    0.0233    0.0105    0.0070    0.0606
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    0.0563    0.0035    0.0005   -0.1116   -0.0287   -0.0649    0.0013
    0.0167    0.0228    0.0248   -0.0111    0.0141    0.0158    0.1088
   -0.0049    0.0150    0.0172    0.0031    0.0185    0.0339   -0.0344

  Columns 29 through 35

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   -0.0013    0.0007   -0.0009    0.0012    0.0009    0.0043    0.0008
   -0.0043    0.0029   -0.0070    0.0055    0.0051   -0.0009   -0.0000
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   -0.0688   -0.0306    0.0492    0.0016    0.0657   -0.0047   -0.0230
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    0.1638    0.0036   -0.0030   -0.0637   -0.1493    0.0125    0.0337
    0.2764    0.1270   -0.1865   -0.0025   -0.2125    0.0264    0.1015
    0.0357    0.0538   -0.0412    0.0032   -0.0253   -0.0972    0.0091
   -0.0524    0.0862    0.0691   -0.0340    0.0034   -0.1592   -0.0027
   -0.0623    0.0460   -0.1162    0.0507   -0.1227   -0.0101    0.0023
   -0.0030   -0.1318   -0.0383    0.0433    0.0414    0.2249   -0.0142
   -0.1238   -0.0468    0.0675    0.0124    0.1043   -0.0177   -0.0457
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    0.6258    0.3295   -0.6118   -0.0225   -1.0326    0.0536    0.5884
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    0.0820    0.0181   -0.0263   -0.0401   -0.1370    0.0234    0.0496
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    0.0127    0.0152   -0.0284    0.0200    0.0267   -0.0049    0.0273

  Columns 36 through 42

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   -0.0329   -0.0998    0.0465    0.0523    0.0564   -0.0534    0.0330
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    0.0204    0.0086   -0.0001   -0.0887   -0.1053    0.0971   -0.0369
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    0.2317   -0.2106   -0.0836   -0.1302   -0.1024    0.0641   -0.2359
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  Columns 43 through 49

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   -0.4303    0.0211   -1.2595    0.1278   -1.5870    0.1791   -0.5075
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   -0.5350   -0.0328   -0.5243   -0.4584   -0.3913    0.7587   -0.5644
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    3.3710   -0.2646   -0.5919   -0.4956    0.2807    0.6956    0.5110
    1.0078   -0.0681   -0.1152    0.5626   -0.0528   -0.3051    0.1657
   -0.4136    0.0981    0.0378   -0.3972   -0.0466   -0.0109   -0.1468
   -0.4121   -0.0076    0.0462    0.2402   -0.0705   -0.0611   -0.1141

  Columns 50 through 56

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  Columns 57 through 63

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  Columns 64 through 67

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