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
%             (0 - 5, i.e. 109 - 79841 Dofs)
%             (optional, defaults to 3, i.e. 5177 Dofs)
%
% tol         truncation tolerance for the Hankel singular values
%             (optional; defaults 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 order 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).
%     https://doi.org/10.5281/zenodo.1187040
%
% [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 file is part of the M-M.E.S.S. project
% (http://www.mpi-magdeburg.mpg.de/projects/mess).
% Copyright © 2009-2022 Jens Saak, Martin Koehler, Peter Benner and others.
% All rights reserved.
% License: BSD 2-Clause License (see COPYING)
%

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 = mess_get_linear_rail(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

Compute low-rank factor of Controllability Gramian

eqn.type='N';                       % Lyapunov eq. for Controllability Gram.
t_mess_lradi = tic;
outB = mess_lradi(eqn, opts, oper); % run ADI iteration
t_elapsed1 = toc(t_mess_lradi);
fprintf(1,'mess_lradi took %6.2f seconds \n',t_elapsed1);

% 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,'LineWidth',3);
    title('A X E^T + E X A^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: 9.540736e-01 relative change in Z: 1.000000e+00
ADI step:    2 normalized residual: 9.022193e-01 relative change in Z: 7.017286e-01
ADI step:    3 normalized residual: 8.376322e-01 relative change in Z: 5.940307e-01
ADI step:    4 normalized residual: 7.288196e-01 relative change in Z: 5.901053e-01
ADI step:    5 normalized residual: 6.307259e-01 relative change in Z: 4.849749e-01
ADI step:    6 normalized residual: 5.436156e-01 relative change in Z: 4.476573e-01
ADI step:    7 normalized residual: 4.550657e-01 relative change in Z: 4.706104e-01
ADI step:    8 normalized residual: 3.869504e-01 relative change in Z: 4.017214e-01
ADI step:    9 normalized residual: 3.235731e-01 relative change in Z: 3.896615e-01
ADI step:   10 normalized residual: 2.589330e-01 relative change in Z: 4.125858e-01
ADI step:   11 normalized residual: 1.856869e-01 relative change in Z: 4.801609e-01
ADI step:   12 normalized residual: 1.484909e-01 relative change in Z: 3.927720e-01
ADI step:   13 normalized residual: 1.132085e-01 relative change in Z: 4.410151e-01
ADI step:   14 normalized residual: 8.773470e-02 relative change in Z: 4.895876e-01
ADI step:   15 normalized residual: 7.751995e-02 relative change in Z: 4.676556e-01
ADI step:   16 normalized residual: 7.076337e-02 relative change in Z: 4.495181e-01
ADI step:   17 normalized residual: 5.919815e-02 relative change in Z: 5.280133e-01
ADI step:   18 normalized residual: 4.842990e-02 relative change in Z: 4.722096e-01
ADI step:   19 normalized residual: 3.774125e-02 relative change in Z: 4.728068e-01
ADI step:   20 normalized residual: 2.798475e-02 relative change in Z: 4.621671e-01
ADI step:   21 normalized residual: 1.961923e-02 relative change in Z: 4.213141e-01
ADI step:   22 normalized residual: 8.997049e-03 relative change in Z: 4.519668e-01
ADI step:   23 normalized residual: 5.275588e-04 relative change in Z: 4.202030e-01
ADI step:   24 normalized residual: 7.333388e-06 relative change in Z: 1.195977e-01
ADI step:   25 normalized residual: 9.877299e-07 relative change in Z: 1.393323e-02
ADI step:   26 normalized residual: 9.578007e-07 relative change in Z: 2.842105e-05
ADI step:   27 normalized residual: 9.281348e-07 relative change in Z: 2.404943e-05
ADI step:   28 normalized residual: 8.928960e-07 relative change in Z: 2.349151e-05
ADI step:   29 normalized residual: 8.299175e-07 relative change in Z: 2.997828e-05
ADI step:   30 normalized residual: 7.611381e-07 relative change in Z: 3.066412e-05
ADI step:   31 normalized residual: 6.738629e-07 relative change in Z: 3.481919e-05
ADI step:   32 normalized residual: 5.377852e-07 relative change in Z: 4.638747e-05
ADI step:   33 normalized residual: 4.322907e-07 relative change in Z: 4.584316e-05
ADI step:   34 normalized residual: 3.444620e-07 relative change in Z: 4.777012e-05
ADI step:   35 normalized residual: 2.601868e-07 relative change in Z: 5.362071e-05
ADI step:   36 normalized residual: 1.610205e-07 relative change in Z: 6.658857e-05
ADI step:   37 normalized residual: 1.137514e-07 relative change in Z: 5.390905e-05
ADI step:   38 normalized residual: 7.023969e-08 relative change in Z: 6.161001e-05
ADI step:   39 normalized residual: 2.841649e-08 relative change in Z: 6.794112e-05
ADI step:   40 normalized residual: 6.238977e-09 relative change in Z: 5.166790e-05
ADI step:   41 normalized residual: 5.548212e-10 relative change in Z: 2.904573e-05
ADI step:   42 normalized residual: 6.583571e-11 relative change in Z: 1.616275e-05
mess_lradi took   1.13 seconds 
size outB.Z:
        5177         294

Compute low-rank factor of Observability Gramian

eqn.type = 'T';                     % Lyapunov eq. for Observability Gram.
t_mess_lradi = tic;
outC = mess_lradi(eqn, opts, oper); % run ADI iteration
t_elapsed2 = toc(t_mess_lradi);
fprintf(1,'mess_lradi took %6.2f seconds \n',t_elapsed2);

% 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,'LineWidth',3);
    title('A^T X E + E^T X A = -C^T C');
    xlabel('number of iterations');
    ylabel('normalized residual norm');
    pause(1);
end
disp('size outC.Z:');
disp(size(outC.Z));
ADI step:    1 normalized residual: 8.947589e-01 relative change in Z: 1.000000e+00
ADI step:    2 normalized residual: 7.869684e-01 relative change in Z: 6.992007e-01
ADI step:    3 normalized residual: 6.682720e-01 relative change in Z: 5.768862e-01
ADI step:    4 normalized residual: 4.969198e-01 relative change in Z: 5.506452e-01
ADI step:    5 normalized residual: 3.594852e-01 relative change in Z: 4.232765e-01
ADI step:    6 normalized residual: 2.415124e-01 relative change in Z: 3.614895e-01
ADI step:    7 normalized residual: 1.359892e-01 relative change in Z: 3.409900e-01
ADI step:    8 normalized residual: 8.687926e-02 relative change in Z: 2.530866e-01
ADI step:    9 normalized residual: 5.598895e-02 relative change in Z: 2.241878e-01
ADI step:   10 normalized residual: 3.347451e-02 relative change in Z: 2.213724e-01
ADI step:   11 normalized residual: 1.558178e-02 relative change in Z: 2.415891e-01
ADI step:   12 normalized residual: 8.570039e-03 relative change in Z: 1.781169e-01
ADI step:   13 normalized residual: 3.293371e-03 relative change in Z: 1.809690e-01
ADI step:   14 normalized residual: 9.946641e-04 relative change in Z: 1.695158e-01
ADI step:   15 normalized residual: 3.995725e-04 relative change in Z: 1.259353e-01
ADI step:   16 normalized residual: 1.640396e-04 relative change in Z: 8.252576e-02
ADI step:   17 normalized residual: 7.651346e-05 relative change in Z: 6.107904e-02
ADI step:   18 normalized residual: 5.866263e-05 relative change in Z: 4.823852e-02
ADI step:   19 normalized residual: 4.640613e-05 relative change in Z: 5.171545e-02
ADI step:   20 normalized residual: 3.395949e-05 relative change in Z: 5.643950e-02
ADI step:   21 normalized residual: 2.243128e-05 relative change in Z: 5.605705e-02
ADI step:   22 normalized residual: 8.819108e-06 relative change in Z: 6.394259e-02
ADI step:   23 normalized residual: 5.508455e-07 relative change in Z: 5.602514e-02
ADI step:   24 normalized residual: 4.772787e-07 relative change in Z: 1.453525e-02
ADI step:   25 normalized residual: 4.760426e-07 relative change in Z: 1.692512e-03
ADI step:   26 normalized residual: 4.159229e-07 relative change in Z: 2.342674e-04
ADI step:   27 normalized residual: 3.695560e-07 relative change in Z: 1.862429e-04
ADI step:   28 normalized residual: 3.259978e-07 relative change in Z: 1.603995e-04
ADI step:   29 normalized residual: 2.677560e-07 relative change in Z: 1.779658e-04
ADI step:   30 normalized residual: 2.229204e-07 relative change in Z: 1.562560e-04
ADI step:   31 normalized residual: 1.782333e-07 relative change in Z: 1.578359e-04
ADI step:   32 normalized residual: 1.177680e-07 relative change in Z: 1.973753e-04
ADI step:   33 normalized residual: 7.738125e-08 relative change in Z: 1.851612e-04
ADI step:   34 normalized residual: 5.040355e-08 relative change in Z: 1.827358e-04
ADI step:   35 normalized residual: 3.110822e-08 relative change in Z: 1.944998e-04
ADI step:   36 normalized residual: 1.304615e-08 relative change in Z: 2.265926e-04
ADI step:   37 normalized residual: 6.283452e-09 relative change in Z: 1.642206e-04
ADI step:   38 normalized residual: 2.575653e-09 relative change in Z: 1.637633e-04
ADI step:   39 normalized residual: 9.134495e-10 relative change in Z: 1.638016e-04
ADI step:   40 normalized residual: 1.833310e-10 relative change in Z: 1.222578e-04
ADI step:   41 normalized residual: 1.380298e-11 relative change in Z: 6.394106e-05
mess_lradi took   1.13 seconds 
size outC.Z:
        5177         246

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: 59  (max possible/allowed: 246/5177)

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=-6;
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,'LineWidth',3);
    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    0.0001
   -0.0000   -0.0000    0.0000    0.0002   -0.0001   -0.0000   -0.0003
   -0.0000   -0.0001   -0.0020    0.0000    0.0001   -0.0018    0.0000
    0.0001    0.0000    0.0001   -0.0049    0.0039    0.0003    0.0021
   -0.0001    0.0001    0.0002    0.0038   -0.0035   -0.0001   -0.0009
   -0.0000   -0.0000   -0.0015    0.0002   -0.0002   -0.0022   -0.0000
   -0.0001    0.0001    0.0000    0.0011   -0.0008   -0.0001   -0.0063
    0.0001    0.0004    0.0002   -0.0012   -0.0013   -0.0006   -0.0004
   -0.0001   -0.0000    0.0005    0.0062   -0.0055    0.0006    0.0006
    0.0000    0.0001    0.0024   -0.0017    0.0012    0.0039   -0.0001
    0.0000    0.0002    0.0067   -0.0001   -0.0011    0.0083   -0.0001
    0.0001    0.0000    0.0003   -0.0074    0.0068    0.0009    0.0006
   -0.0001   -0.0000    0.0023    0.0044   -0.0041    0.0023   -0.0012
    0.0001    0.0001    0.0009   -0.0108    0.0091    0.0015    0.0032
   -0.0001    0.0001   -0.0000    0.0028   -0.0024   -0.0004   -0.0156
    0.0001    0.0005    0.0036   -0.0015   -0.0028    0.0044    0.0015
   -0.0000    0.0000    0.0069    0.0001    0.0004    0.0122   -0.0011
    0.0000    0.0000   -0.0005   -0.0049    0.0044   -0.0006   -0.0020
    0.0000    0.0000    0.0009   -0.0009    0.0008    0.0022   -0.0008
   -0.0000   -0.0000   -0.0001    0.0033   -0.0034   -0.0003    0.0063
   -0.0000   -0.0001   -0.0039    0.0001    0.0005   -0.0044   -0.0002
    0.0000   -0.0000   -0.0031   -0.0002    0.0001   -0.0068   -0.0004
   -0.0000   -0.0002   -0.0029    0.0008    0.0009   -0.0051    0.0002
    0.0000   -0.0001   -0.0040   -0.0006    0.0009   -0.0075    0.0004
    0.0000    0.0000    0.0005   -0.0050    0.0047    0.0014    0.0016
   -0.0000   -0.0001    0.0007    0.0019   -0.0008    0.0017    0.0011
   -0.0000   -0.0002   -0.0024    0.0004    0.0007   -0.0050   -0.0001
   -0.0000    0.0000    0.0001    0.0006   -0.0000    0.0004   -0.0068
    0.0000    0.0001   -0.0004    0.0002   -0.0006   -0.0015   -0.0002
   -0.0000    0.0000   -0.0003    0.0024   -0.0025   -0.0009   -0.0001
   -0.0000   -0.0000    0.0010    0.0007   -0.0002    0.0025    0.0000
    0.0000   -0.0000   -0.0008   -0.0003   -0.0001   -0.0028   -0.0003
    0.0000    0.0001    0.0005   -0.0003   -0.0003    0.0009   -0.0004
    0.0000    0.0001    0.0014   -0.0002   -0.0004    0.0038   -0.0001
   -0.0000    0.0000   -0.0019    0.0001    0.0002   -0.0023   -0.0000
    0.0000    0.0000   -0.0001   -0.0014    0.0012   -0.0001    0.0007
   -0.0000   -0.0001   -0.0003    0.0002   -0.0002   -0.0024   -0.0004
    0.0000    0.0000    0.0000   -0.0016    0.0011   -0.0002    0.0034
   -0.0000   -0.0000    0.0000    0.0007   -0.0003   -0.0000   -0.0034
    0.0000    0.0000    0.0005   -0.0003    0.0001    0.0012    0.0005
   -0.0000    0.0000    0.0003    0.0005   -0.0004    0.0013   -0.0010
    0.0000    0.0000   -0.0001   -0.0005    0.0002    0.0002    0.0016
   -0.0000   -0.0000    0.0000    0.0001   -0.0003   -0.0011   -0.0004
    0.0000    0.0001    0.0001   -0.0002   -0.0003    0.0001   -0.0005
    0.0000    0.0000    0.0001   -0.0006    0.0002    0.0000    0.0018
    0.0000   -0.0000    0.0002   -0.0003    0.0004    0.0003    0.0012
   -0.0000   -0.0000    0.0001    0.0002   -0.0000    0.0005   -0.0004
   -0.0000    0.0000    0.0005    0.0003   -0.0003    0.0006   -0.0006
   -0.0000   -0.0000   -0.0007    0.0001    0.0001   -0.0010   -0.0000
   -0.0000   -0.0000    0.0005    0.0000   -0.0000    0.0000   -0.0005
    0.0000   -0.0000    0.0004   -0.0000   -0.0000    0.0002    0.0006
    0.0000    0.0000   -0.0001   -0.0001    0.0001   -0.0000    0.0005
    0.0000    0.0000    0.0003   -0.0001   -0.0001   -0.0000   -0.0000
   -0.0000   -0.0000    0.0001    0.0001    0.0002    0.0001    0.0002
    0.0000   -0.0000   -0.0002   -0.0000    0.0000   -0.0002   -0.0000
    0.0000   -0.0000   -0.0001   -0.0004    0.0004   -0.0001    0.0002
    0.0000    0.0000    0.0006   -0.0001   -0.0000    0.0009   -0.0000
    0.0000   -0.0000   -0.0004   -0.0000    0.0000   -0.0007   -0.0000
    0.0000   -0.0000   -0.0000   -0.0000    0.0000   -0.0002    0.0000

  Columns 8 through 14

   -0.0000    0.0000   -0.0000   -0.0000   -0.0000    0.0000   -0.0001
   -0.0000   -0.0000    0.0000    0.0001    0.0001   -0.0001    0.0002
    0.0002    0.0006    0.0026    0.0067    0.0003    0.0023    0.0008
   -0.0009    0.0061   -0.0017   -0.0000   -0.0074    0.0045   -0.0108
   -0.0007   -0.0056    0.0012   -0.0009    0.0070   -0.0041    0.0091
   -0.0004    0.0008    0.0049    0.0085    0.0009    0.0026    0.0015
   -0.0003   -0.0002    0.0001   -0.0000    0.0016   -0.0018    0.0048
   -0.0128    0.0008   -0.0000   -0.0009    0.0015    0.0017   -0.0029
    0.0007   -0.0141    0.0003   -0.0052    0.0183   -0.0126    0.0260
    0.0001    0.0004   -0.0140   -0.0202   -0.0059   -0.0032   -0.0090
   -0.0010   -0.0044   -0.0178   -0.0686   -0.0025   -0.0320   -0.0106
    0.0016    0.0184   -0.0058   -0.0025   -0.0301    0.0170   -0.0453
    0.0017   -0.0124   -0.0016   -0.0305    0.0171   -0.0300    0.0254
   -0.0027    0.0268   -0.0087   -0.0099   -0.0454    0.0255   -0.0775
   -0.0059    0.0002    0.0007    0.0002    0.0070   -0.0064    0.0195
   -0.0271   -0.0047   -0.0168   -0.0496    0.0063   -0.0162   -0.0140
    0.0104   -0.0078   -0.0392   -0.0970   -0.0100   -0.0443   -0.0147
   -0.0020    0.0162   -0.0012    0.0075   -0.0253    0.0215   -0.0424
   -0.0000    0.0012   -0.0096   -0.0124   -0.0066    0.0006   -0.0101
    0.0002   -0.0144    0.0034    0.0007    0.0253   -0.0143    0.0363
   -0.0008    0.0003    0.0017    0.0690    0.0040    0.0479    0.0171
   -0.0037    0.0072    0.0275    0.0473    0.0029    0.0189    0.0033
    0.0101    0.0038    0.0200    0.0462    0.0004    0.0169    0.0129
   -0.0013    0.0083    0.0269    0.0673    0.0003    0.0364    0.0034
    0.0020    0.0149   -0.0077   -0.0086   -0.0331    0.0187   -0.0598
    0.0090   -0.0068   -0.0029   -0.0137    0.0071   -0.0181    0.0187
    0.0065    0.0043    0.0186    0.0413   -0.0001    0.0158    0.0087
    0.0007    0.0027   -0.0019   -0.0023   -0.0044   -0.0019    0.0029
   -0.0035    0.0035    0.0183   -0.0006    0.0030   -0.0102   -0.0016
   -0.0027   -0.0085    0.0018    0.0062    0.0188   -0.0071    0.0317
    0.0040   -0.0030   -0.0051   -0.0194    0.0007   -0.0139    0.0038
   -0.0041    0.0015    0.0034    0.0182    0.0017    0.0114    0.0003
   -0.0039   -0.0003   -0.0036   -0.0095    0.0000   -0.0030   -0.0039
   -0.0035   -0.0024   -0.0106   -0.0264   -0.0005   -0.0097   -0.0052
   -0.0005    0.0031    0.0141    0.0306    0.0020    0.0171    0.0062
   -0.0002    0.0048   -0.0003    0.0023   -0.0110    0.0087   -0.0180
   -0.0002    0.0016    0.0090    0.0053    0.0020   -0.0049    0.0009
   -0.0003    0.0035   -0.0002   -0.0004   -0.0105    0.0065   -0.0198
    0.0003   -0.0001    0.0001   -0.0010    0.0031   -0.0036    0.0068
   -0.0003   -0.0007   -0.0051   -0.0087   -0.0018   -0.0019   -0.0042
   -0.0007   -0.0024   -0.0054   -0.0048    0.0026   -0.0013    0.0061
   -0.0013    0.0002   -0.0015    0.0014   -0.0015    0.0050   -0.0050
   -0.0014    0.0008    0.0047   -0.0012    0.0014   -0.0054   -0.0004
   -0.0042    0.0001   -0.0015   -0.0007    0.0010    0.0024   -0.0019
   -0.0011    0.0006   -0.0001   -0.0026   -0.0020    0.0004   -0.0071
    0.0019    0.0004   -0.0011   -0.0034   -0.0027   -0.0012   -0.0051
    0.0009   -0.0007   -0.0024   -0.0006   -0.0003    0.0005    0.0023
    0.0001   -0.0011   -0.0019   -0.0083    0.0012   -0.0068    0.0016
    0.0008    0.0004    0.0028    0.0130    0.0009    0.0079    0.0042
    0.0007    0.0008    0.0020   -0.0097   -0.0012   -0.0100   -0.0032
    0.0008    0.0000    0.0011   -0.0081   -0.0004   -0.0078   -0.0029
   -0.0003    0.0000   -0.0003    0.0014   -0.0002    0.0020   -0.0010
   -0.0010    0.0006    0.0018   -0.0073   -0.0003   -0.0068   -0.0036
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   -0.0001   -0.0001   -0.0005    0.0042    0.0002    0.0038    0.0012
    0.0001    0.0015   -0.0001    0.0022   -0.0029    0.0038   -0.0052
   -0.0005    0.0000   -0.0016   -0.0121   -0.0012   -0.0078   -0.0045
   -0.0004    0.0002    0.0007    0.0083    0.0005    0.0061    0.0018
   -0.0000    0.0004    0.0017   -0.0003   -0.0001   -0.0013   -0.0007

  Columns 15 through 21

    0.0001   -0.0000    0.0000    0.0000    0.0000   -0.0000    0.0000
   -0.0004    0.0002    0.0000   -0.0000    0.0000    0.0001   -0.0000
   -0.0000    0.0036    0.0071   -0.0005    0.0009   -0.0001   -0.0043
    0.0028   -0.0012   -0.0000   -0.0049   -0.0009    0.0032    0.0001
   -0.0016   -0.0026    0.0004    0.0046    0.0008   -0.0037    0.0005
   -0.0003    0.0048    0.0126   -0.0006    0.0022   -0.0003   -0.0047
   -0.0156    0.0015   -0.0010   -0.0012   -0.0007    0.0062   -0.0002
   -0.0059   -0.0271    0.0102   -0.0021   -0.0000    0.0002   -0.0007
    0.0023   -0.0054   -0.0082    0.0165    0.0012   -0.0148    0.0018
    0.0002   -0.0178   -0.0418   -0.0011   -0.0098    0.0037    0.0083
    0.0004   -0.0462   -0.0914    0.0072   -0.0103    0.0010    0.0791
    0.0040    0.0078   -0.0103   -0.0271   -0.0063    0.0245    0.0043
   -0.0058   -0.0148   -0.0429    0.0220    0.0011   -0.0133    0.0532
    0.0185   -0.0142   -0.0140   -0.0436   -0.0099    0.0341    0.0192
   -0.1010   -0.0123    0.0050   -0.0114   -0.0058    0.0540   -0.0032
   -0.0125   -0.1811   -0.0506    0.0032   -0.0189   -0.0130    0.0582
    0.0048   -0.0513   -0.2462    0.0169   -0.0470    0.0100    0.1275
   -0.0148    0.0053    0.0160   -0.0362   -0.0049    0.0421   -0.0103
   -0.0041   -0.0183   -0.0491   -0.0040   -0.0172    0.0112   -0.0005
    0.0424   -0.0097    0.0072    0.0363    0.0107   -0.0663   -0.0053
   -0.0029    0.0405    0.0951   -0.0080   -0.0010   -0.0042   -0.2331
   -0.0096    0.0521    0.1781   -0.0166    0.0463    0.0038   -0.0407
    0.0122    0.1321    0.1171   -0.0056    0.0378   -0.0007   -0.0674
    0.0002    0.0874    0.2148   -0.0231    0.0526    0.0030   -0.1248
    0.0206    0.0060   -0.0383   -0.0403   -0.0185    0.0485    0.0214
    0.0219    0.0452   -0.0655    0.0249   -0.0048   -0.0264    0.0450
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   -0.0648    0.0126   -0.0177   -0.0139   -0.0108    0.0607    0.0032
   -0.0081    0.0063    0.0838   -0.0047    0.0377    0.0013    0.1389
   -0.0070   -0.0196    0.0216    0.0225    0.0087   -0.0367   -0.0422
    0.0061    0.0048   -0.0727    0.0122   -0.0167   -0.0049    0.0674
   -0.0100   -0.0100    0.0675   -0.0098    0.0179    0.0024   -0.0846
   -0.0087   -0.0450   -0.0197   -0.0006   -0.0097    0.0039    0.0200
   -0.0047   -0.0707   -0.0897    0.0071   -0.0364   -0.0004    0.0661
   -0.0020    0.0475    0.1086   -0.0082    0.0253   -0.0005   -0.0417
    0.0094    0.0018    0.0084   -0.0111   -0.0024    0.0250   -0.0040
   -0.0059    0.0213    0.0589   -0.0053    0.0340    0.0009    0.0331
    0.0398   -0.0070    0.0082   -0.0012    0.0020    0.0042    0.0084
   -0.0387    0.0083   -0.0060   -0.0085   -0.0012    0.0152    0.0049
    0.0063   -0.0207   -0.0375    0.0027   -0.0146   -0.0014    0.0088
   -0.0117   -0.0197   -0.0360    0.0047   -0.0185    0.0002   -0.0188
    0.0164   -0.0157    0.0006    0.0010   -0.0050   -0.0096   -0.0192
   -0.0074   -0.0028    0.0288   -0.0026    0.0181    0.0032    0.0376
   -0.0110   -0.0378    0.0061   -0.0036   -0.0047    0.0024   -0.0146
    0.0183   -0.0143    0.0020    0.0011    0.0019   -0.0083    0.0142
    0.0164    0.0099   -0.0148    0.0011   -0.0008   -0.0058    0.0125
   -0.0030    0.0029   -0.0165    0.0020   -0.0082    0.0045   -0.0155
   -0.0064   -0.0090   -0.0241    0.0028   -0.0041    0.0011    0.0291
    0.0003    0.0229    0.0322   -0.0019    0.0064   -0.0013   -0.0505
   -0.0047    0.0030   -0.0078   -0.0008    0.0086    0.0084    0.0722
    0.0073    0.0007   -0.0097    0.0023    0.0062   -0.0045    0.0562
    0.0047   -0.0031    0.0022   -0.0000   -0.0010   -0.0040   -0.0109
   -0.0016   -0.0107    0.0005   -0.0008    0.0073    0.0018    0.0571
    0.0050    0.0170   -0.0107    0.0017    0.0002   -0.0003    0.0094
   -0.0003    0.0022    0.0055   -0.0007   -0.0018   -0.0004   -0.0293
    0.0025    0.0037    0.0044   -0.0049   -0.0006    0.0049   -0.0102
   -0.0008   -0.0173   -0.0272    0.0010   -0.0042    0.0028    0.0596
   -0.0013    0.0053    0.0205   -0.0023    0.0025   -0.0003   -0.0462
   -0.0001    0.0029    0.0072   -0.0007    0.0048    0.0008    0.0180

  Columns 22 through 28

    0.0000    0.0000   -0.0000   -0.0000   -0.0000    0.0000    0.0001
   -0.0001   -0.0001   -0.0000    0.0001    0.0000   -0.0001   -0.0002
   -0.0030   -0.0029   -0.0039    0.0005    0.0008   -0.0026    0.0002
   -0.0001    0.0007   -0.0006   -0.0051    0.0018    0.0004    0.0007
    0.0002    0.0008    0.0008    0.0047   -0.0010    0.0007    0.0002
   -0.0068   -0.0051   -0.0076    0.0014    0.0017   -0.0049    0.0004
   -0.0004    0.0002    0.0006    0.0023    0.0011   -0.0001   -0.0072
   -0.0037    0.0101   -0.0015    0.0020    0.0090    0.0066    0.0007
    0.0072    0.0042    0.0088    0.0147   -0.0073    0.0049    0.0030
    0.0276    0.0212    0.0290   -0.0081   -0.0041    0.0207   -0.0021
    0.0382    0.0405    0.0582   -0.0076   -0.0144    0.0393   -0.0024
    0.0022   -0.0000   -0.0001   -0.0329    0.0073   -0.0003   -0.0034
    0.0137    0.0137    0.0300    0.0196   -0.0186    0.0155   -0.0027
    0.0018    0.0114    0.0010   -0.0597    0.0189    0.0086    0.0045
   -0.0094    0.0113    0.0004    0.0206    0.0242    0.0054   -0.0717
    0.0477    0.1301    0.0816    0.0058    0.0408    0.1196    0.0138
    0.1639    0.1140    0.2105   -0.0382   -0.0689    0.1336   -0.0196
   -0.0161   -0.0055   -0.0222   -0.0416    0.0276   -0.0102   -0.0162
    0.0511    0.0396    0.0579   -0.0191   -0.0062    0.0464   -0.0108
    0.0052    0.0013    0.0037    0.0427   -0.0263    0.0047    0.0606
   -0.0005   -0.0368   -0.0743    0.0149    0.0406   -0.0523    0.0032
   -0.1740   -0.1272   -0.2168    0.0324    0.0621   -0.1660    0.0090
   -0.1226   -0.1736   -0.1869    0.0171   -0.0098   -0.2121    0.0048
   -0.2057   -0.1886   -0.3117    0.0337    0.0900   -0.2738    0.0247
    0.0292    0.0162    0.0323   -0.1192    0.0093    0.0281    0.0123
    0.0546   -0.0152    0.0758    0.0127   -0.1106    0.0138    0.0052
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    0.0084    0.0038    0.0201   -0.0146    0.0264    0.0042   -0.1461
   -0.1530   -0.0780   -0.1685    0.0362    0.0387   -0.1088    0.0065
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    0.0688    0.0385    0.1112   -0.0121   -0.0845    0.1023   -0.0150
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    0.0267    0.0674    0.0591   -0.0021    0.0333    0.1076   -0.0086
    0.1154    0.1559    0.2016   -0.0221   -0.0224    0.3132   -0.0141
   -0.1263   -0.1204   -0.2323    0.0369    0.0554   -0.1908    0.0110
   -0.0106   -0.0091   -0.0223   -0.0625    0.0404   -0.0207   -0.0079
   -0.1073   -0.0909   -0.1393    0.0308    0.0356   -0.2241    0.0091
   -0.0101   -0.0069   -0.0174   -0.0756    0.0352   -0.0207    0.0743
   -0.0020   -0.0012    0.0053    0.0338   -0.0116   -0.0060   -0.0884
    0.0542    0.0568    0.0904   -0.0235   -0.0175    0.1238    0.0072
    0.0635    0.0638    0.0948    0.0092   -0.0172    0.1456   -0.0349
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   -0.0288   -0.0475   -0.0763    0.0133    0.0148   -0.0958    0.0032
   -0.0228   -0.0137    0.0021   -0.0049   -0.0168   -0.0408   -0.0233
   -0.0102   -0.0062    0.0108   -0.0058   -0.0267   -0.0174    0.0234
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   -0.0175   -0.0124   -0.0194    0.0017    0.0013   -0.0269    0.0014

  Columns 29 through 35

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   -0.0007   -0.0025   -0.0004    0.0000   -0.0002   -0.0005   -0.0002
   -0.0005   -0.0011    0.0028   -0.0036    0.0009    0.0044   -0.0001
   -0.0002    0.0004    0.0001   -0.0004   -0.0005   -0.0001    0.0001
   -0.0035   -0.0027    0.0039   -0.0038   -0.0035   -0.0032   -0.0004
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   -0.0337   -0.0015   -0.0216    0.0314   -0.0019   -0.0247   -0.0244
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   -0.2864    0.0226    0.0440    0.0619    0.0791    0.1271   -0.7097
   -0.0026    0.0809    0.0293   -0.0109   -0.0078    0.0260   -0.0198
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   -0.0243    0.0186    0.0138   -0.0100   -0.0196   -0.0138    0.0362
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  Columns 36 through 42

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    0.0585   -0.0097    0.0636   -0.0323    0.0023   -0.0061    0.0005
    0.0039   -0.1044   -0.0204   -0.0070   -0.0192    0.0083    0.0167
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   -0.0094   -0.1528   -0.0262    0.0021    0.0467    0.0825    0.0386
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   -0.0364   -0.0586    0.0239    0.0162    0.2656    0.2299    0.0202
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    0.0288    0.0660    0.0822   -0.0661   -0.0449   -0.1587   -0.0819
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   -0.0110   -0.0282   -0.0734    0.0462   -0.0522    0.0434   -0.0036
   -0.0769    0.2227   -0.1251    0.1365   -0.1187   -0.1298   -0.1161
    0.0960    0.0367    0.1812   -0.1585   -0.0800   -0.1837    0.0758
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   -0.0296   -0.3445   -0.0879   -0.0218    0.0485    0.2765    0.2204
    0.0292   -0.2207   -0.0216   -0.0193    0.0135    0.1855    0.1290
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   -0.0043   -0.2698   -0.0501   -0.0181    0.0590    0.2447    0.1286
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   -0.0472   -0.0064   -0.0675    0.0381    0.0122    0.0563   -0.0338
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   -0.0078   -0.1146   -0.0173   -0.0044    0.0886    0.1625    0.0689

  Columns 43 through 49

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    0.0000    0.0000   -0.0000   -0.0000    0.0000    0.0000    0.0000
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   -0.0002   -0.0003    0.0014    0.0007   -0.0001   -0.0005   -0.0001
    0.0001   -0.0001   -0.0013   -0.0003    0.0001    0.0005    0.0000
   -0.0020    0.0003   -0.0004   -0.0002    0.0007   -0.0003    0.0007
    0.0003   -0.0002   -0.0007   -0.0008    0.0009    0.0001    0.0000
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    0.0023    0.0005   -0.0040   -0.0015   -0.0010    0.0018   -0.0001
    0.0081   -0.0018    0.0023    0.0011   -0.0028    0.0002   -0.0017
    0.0152   -0.0024    0.0036    0.0040   -0.0035    0.0075   -0.0121
    0.0005   -0.0002    0.0096    0.0030   -0.0004   -0.0032   -0.0014
    0.0081    0.0013   -0.0047   -0.0012   -0.0009    0.0081   -0.0085
    0.0003   -0.0040    0.0184    0.0098   -0.0012   -0.0044   -0.0041
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    0.0151    0.0112   -0.0205   -0.0401    0.0121    0.0756   -0.1011
    0.0305   -0.0348   -0.0955   -0.0309    0.0076   -0.2764    0.6175
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   -0.2179    0.0702   -0.0431    0.0030    0.1396   -0.0128    0.0776

  Columns 50 through 56

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    0.0003   -0.0001   -0.0003    0.0000    0.0000   -0.0000   -0.0004
   -0.0017   -0.0013    0.0002   -0.0011   -0.0002    0.0005    0.0002
    0.0006   -0.0005   -0.0003   -0.0000   -0.0001    0.0000    0.0003
    0.0002    0.0002   -0.0000    0.0004   -0.0008   -0.0000   -0.0001
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  Columns 57 through 59

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