Contents

function [Ar, Br, Cr] = bt_mor_rail_tol(k, tol, shifts, istest)
% bt_mor_rail_tol computes a reduced order model via the standard Lyapunov
% balanced truncation (see e.g. [1]) for a finite element discretized
% heat conduction model on a rail profile cross-section described
% in [2, 3, 4].
%
% 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
%               'wachspress' :   optimal Wachspress parameters
%             (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 Lecture Notes in Computational Science and Engineering,
%     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 (c) 2009-2025 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 = 2;
end
if nargin < 2
    tol = 1e-5;
end
if nargin < 3
    shifts = 'heur';
end
if nargin < 4
    istest = false;
end

% ADI tolerance and maximum iteration number
opts.adi.maxiter = 150;              % 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, opts] = operatormanager(opts, '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.num_Ritz   = 50;        % number Arnoldi steps with F
        opts.shifts.num_hRitz  = 25;        % Arnoldi steps with inv(F)
        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);
mess_fprintf(opts, 'mess_lradi took %6.2f seconds \n', t_elapsed1);

% residual norm plot
if istest
    if min(outB.res) >= opts.adi.res_tol
        mess_err(opts, '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
[mZ, nZ] = size(outB.Z);
mess_fprintf(opts, 'size outB.Z: %d x %d\n\n', mZ, nZ);
ADI step:    1 normalized residual: 9.474594e-01 relative change in Z: 1.000000e+00  
ADI step:    2 normalized residual: 8.927351e-01 relative change in Z: 6.848676e-01  
ADI step:    3 normalized residual: 8.295058e-01 relative change in Z: 5.707160e-01  
ADI step:    4 normalized residual: 7.572184e-01 relative change in Z: 5.031271e-01  
ADI step:    5 normalized residual: 6.781431e-01 relative change in Z: 4.582740e-01  
ADI step:    6 normalized residual: 6.054060e-01 relative change in Z: 4.121435e-01  
ADI step:    7 normalized residual: 5.394436e-01 relative change in Z: 3.929989e-01  
ADI step:    8 normalized residual: 4.648237e-01 relative change in Z: 4.302504e-01  
ADI step:    9 normalized residual: 4.009874e-01 relative change in Z: 3.913635e-01  
ADI step:   10 normalized residual: 3.424997e-01 relative change in Z: 3.686088e-01  
ADI step:   11 normalized residual: 2.796104e-01 relative change in Z: 3.931841e-01  
ADI step:   12 normalized residual: 2.152434e-01 relative change in Z: 4.338421e-01  
ADI step:   13 normalized residual: 1.732821e-01 relative change in Z: 4.398806e-01  
ADI step:   14 normalized residual: 1.546354e-01 relative change in Z: 4.130482e-01  
ADI step:   15 normalized residual: 1.390549e-01 relative change in Z: 4.697030e-01  
ADI step:   16 normalized residual: 1.256237e-01 relative change in Z: 4.324842e-01  
ADI step:   17 normalized residual: 1.098831e-01 relative change in Z: 4.296118e-01  
ADI step:   18 normalized residual: 8.679367e-02 relative change in Z: 4.894507e-01  
ADI step:   19 normalized residual: 6.927757e-02 relative change in Z: 4.376648e-01  
ADI step:   20 normalized residual: 5.211548e-02 relative change in Z: 4.381249e-01  
ADI step:   21 normalized residual: 3.668472e-02 relative change in Z: 4.042950e-01  
ADI step:   22 normalized residual: 1.696367e-02 relative change in Z: 4.378921e-01  
ADI step:   23 normalized residual: 1.006346e-03 relative change in Z: 4.097701e-01  
ADI step:   24 normalized residual: 4.571904e-06 relative change in Z: 1.171674e-01  
ADI step:   25 normalized residual: 3.372406e-07 relative change in Z: 7.682132e-03  
ADI step:   26 normalized residual: 3.279503e-07 relative change in Z: 1.282941e-05  
ADI step:   27 normalized residual: 3.162034e-07 relative change in Z: 1.405301e-05  
ADI step:   28 normalized residual: 2.998289e-07 relative change in Z: 1.623522e-05  
ADI step:   29 normalized residual: 2.770748e-07 relative change in Z: 1.871936e-05  
ADI step:   30 normalized residual: 2.460390e-07 relative change in Z: 2.149050e-05  
ADI step:   31 normalized residual: 2.090601e-07 relative change in Z: 2.335330e-05  
ADI step:   32 normalized residual: 1.644663e-07 relative change in Z: 2.592568e-05  
ADI step:   33 normalized residual: 1.010301e-07 relative change in Z: 3.200632e-05  
ADI step:   34 normalized residual: 5.643748e-08 relative change in Z: 2.899845e-05  
ADI step:   35 normalized residual: 3.225024e-08 relative change in Z: 2.443140e-05  
ADI step:   36 normalized residual: 1.791635e-08 relative change in Z: 2.312635e-05  
ADI step:   37 normalized residual: 8.983323e-09 relative change in Z: 2.386534e-05  
ADI step:   38 normalized residual: 4.452219e-09 relative change in Z: 2.168001e-05  
ADI step:   39 normalized residual: 2.273559e-09 relative change in Z: 1.783116e-05  
ADI step:   40 normalized residual: 5.867649e-10 relative change in Z: 1.809708e-05  
ADI step:   41 normalized residual: 1.166301e-10 relative change in Z: 1.211339e-05  
ADI step:   42 normalized residual: 2.933250e-11 relative change in Z: 7.394417e-06  
mess_lradi took   0.26 seconds   
size outB.Z: 1357 x 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);
mess_fprintf(opts, 'mess_lradi took %6.2f seconds \n', t_elapsed2);

% residual norm plot
if istest
    if min(outC.res) >= opts.adi.res_tol
        mess_err(opts, '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
[mZ, nZ] = size(outC.Z);
mess_fprintf(opts, 'size outC.Z: %d x %d\n\n', mZ, nZ);
ADI step:    1 normalized residual: 8.798717e-01 relative change in Z: 1.000000e+00  
ADI step:    2 normalized residual: 7.681639e-01 relative change in Z: 6.801309e-01  
ADI step:    3 normalized residual: 6.541966e-01 relative change in Z: 5.513119e-01  
ADI step:    4 normalized residual: 5.392809e-01 relative change in Z: 4.694506e-01  
ADI step:    5 normalized residual: 4.252802e-01 relative change in Z: 4.095090e-01  
ADI step:    6 normalized residual: 3.254585e-01 relative change in Z: 3.491357e-01  
ADI step:    7 normalized residual: 2.370696e-01 relative change in Z: 3.125751e-01  
ADI step:    8 normalized residual: 1.480505e-01 relative change in Z: 3.139988e-01  
ADI step:    9 normalized residual: 9.724878e-02 relative change in Z: 2.544873e-01  
ADI step:   10 normalized residual: 6.397794e-02 relative change in Z: 2.195765e-01  
ADI step:   11 normalized residual: 3.488421e-02 relative change in Z: 2.167150e-01  
ADI step:   12 normalized residual: 1.318338e-02 relative change in Z: 2.111119e-01  
ADI step:   13 normalized residual: 4.708713e-03 relative change in Z: 1.765635e-01  
ADI step:   14 normalized residual: 2.234807e-03 relative change in Z: 1.346802e-01  
ADI step:   15 normalized residual: 7.754380e-04 relative change in Z: 1.128388e-01  
ADI step:   16 normalized residual: 3.933989e-04 relative change in Z: 6.253186e-02  
ADI step:   17 normalized residual: 2.776729e-04 relative change in Z: 4.768086e-02  
ADI step:   18 normalized residual: 2.090779e-04 relative change in Z: 5.549394e-02  
ADI step:   19 normalized residual: 1.679729e-04 relative change in Z: 5.325733e-02  
ADI step:   20 normalized residual: 1.237168e-04 relative change in Z: 5.909578e-02  
ADI step:   21 normalized residual: 8.109693e-05 relative change in Z: 5.893471e-02  
ADI step:   22 normalized residual: 3.186350e-05 relative change in Z: 6.728979e-02  
ADI step:   23 normalized residual: 5.963682e-07 relative change in Z: 5.911083e-02  
ADI step:   24 normalized residual: 1.431414e-07 relative change in Z: 1.531810e-02  
ADI step:   25 normalized residual: 1.409976e-07 relative change in Z: 1.005224e-03  
ADI step:   26 normalized residual: 1.345585e-07 relative change in Z: 6.139024e-05  
ADI step:   27 normalized residual: 1.270898e-07 relative change in Z: 6.560745e-05  
ADI step:   28 normalized residual: 1.176820e-07 relative change in Z: 7.288393e-05  
ADI step:   29 normalized residual: 1.060327e-07 relative change in Z: 7.991251e-05  
ADI step:   30 normalized residual: 9.186680e-08 relative change in Z: 8.678308e-05  
ADI step:   31 normalized residual: 7.649342e-08 relative change in Z: 9.003189e-05  
ADI step:   32 normalized residual: 5.919861e-08 relative change in Z: 9.679658e-05  
ADI step:   33 normalized residual: 3.604682e-08 relative change in Z: 1.162086e-04  
ADI step:   34 normalized residual: 2.017143e-08 relative change in Z: 1.027865e-04  
ADI step:   35 normalized residual: 1.113342e-08 relative change in Z: 8.492447e-05  
ADI step:   36 normalized residual: 5.497048e-09 relative change in Z: 7.650150e-05  
ADI step:   37 normalized residual: 1.989369e-09 relative change in Z: 7.124317e-05  
ADI step:   38 normalized residual: 5.271695e-10 relative change in Z: 5.598915e-05  
ADI step:   39 normalized residual: 2.114299e-10 relative change in Z: 3.657037e-05  
ADI step:   40 normalized residual: 5.357956e-11 relative change in Z: 3.061995e-05  
mess_lradi took   0.24 seconds   
size outC.Z: 1357 x 240  
  

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: 53  (max possible/allowed: 240/1357)  
  

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.tf_plot.info = 0;
else
    opts.tf_plot.info = 2;
end

opts.tf_plot.type = 'sigma';

opts.tf_plot.fmin = -6;
opts.tf_plot.fmax = 4;

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

if istest
    if max(err) > tol
        mess_err(opts, '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    0.0034    0.0002    0.0018
    0.0001    0.0001    0.0002    0.0033   -0.0027   -0.0000   -0.0005
    0.0000   -0.0000   -0.0014    0.0002   -0.0001   -0.0022    0.0000
    0.0001    0.0001    0.0000    0.0007   -0.0004   -0.0000   -0.0055
   -0.0001    0.0003    0.0002   -0.0010   -0.0013   -0.0005   -0.0004
   -0.0000    0.0000   -0.0004   -0.0047    0.0037   -0.0004   -0.0013
    0.0000   -0.0001   -0.0024    0.0010   -0.0005   -0.0039    0.0002
    0.0000   -0.0001   -0.0057    0.0001    0.0008   -0.0069    0.0000
    0.0000    0.0000   -0.0002    0.0038   -0.0032   -0.0005    0.0009
   -0.0000   -0.0000   -0.0015   -0.0005    0.0006   -0.0014   -0.0001
    0.0001   -0.0001   -0.0000    0.0062   -0.0047   -0.0002    0.0008
    0.0000   -0.0002   -0.0050    0.0002    0.0006   -0.0089    0.0007
    0.0001    0.0000   -0.0004    0.0026   -0.0015   -0.0007   -0.0092
   -0.0000    0.0003    0.0001   -0.0004   -0.0019   -0.0009   -0.0004
    0.0000   -0.0000    0.0002    0.0036   -0.0026    0.0004   -0.0006
    0.0000   -0.0000   -0.0005    0.0003   -0.0002   -0.0015   -0.0000
    0.0000   -0.0000   -0.0001    0.0022   -0.0019   -0.0003    0.0031
    0.0000    0.0000    0.0004    0.0001    0.0001    0.0021    0.0001
    0.0000    0.0000    0.0021    0.0001   -0.0002    0.0035    0.0001
    0.0000   -0.0001   -0.0009    0.0003    0.0004   -0.0017    0.0001
   -0.0000    0.0001    0.0022    0.0000   -0.0004    0.0042   -0.0001
   -0.0000    0.0000    0.0004   -0.0008    0.0006    0.0009    0.0002
   -0.0000   -0.0000   -0.0009   -0.0003    0.0002   -0.0024   -0.0000
   -0.0000    0.0001   -0.0002   -0.0001    0.0000    0.0010   -0.0001
   -0.0000    0.0000    0.0012   -0.0001   -0.0002    0.0021   -0.0001
   -0.0000    0.0000   -0.0002   -0.0001    0.0000   -0.0003    0.0002
    0.0000   -0.0000   -0.0002    0.0002   -0.0001   -0.0008   -0.0007
   -0.0000   -0.0000   -0.0000   -0.0006    0.0003   -0.0001    0.0024
    0.0000   -0.0000   -0.0000    0.0012   -0.0007    0.0000   -0.0028
   -0.0000    0.0000    0.0000    0.0000    0.0000    0.0014    0.0001
    0.0000   -0.0000    0.0000    0.0013   -0.0009   -0.0000   -0.0021
   -0.0000    0.0000    0.0006   -0.0001   -0.0000    0.0017    0.0001
   -0.0000    0.0000   -0.0000   -0.0010    0.0007   -0.0000    0.0004
   -0.0000    0.0001   -0.0001    0.0000   -0.0004   -0.0001   -0.0003
   -0.0000    0.0000    0.0004   -0.0004    0.0001   -0.0000    0.0003
    0.0000    0.0000    0.0003    0.0000   -0.0001    0.0002   -0.0003
    0.0000    0.0000    0.0004    0.0002   -0.0003    0.0005   -0.0001
   -0.0000    0.0000   -0.0001   -0.0004    0.0003    0.0002    0.0003
   -0.0000    0.0000    0.0001   -0.0003    0.0002   -0.0000    0.0001
   -0.0000   -0.0000    0.0001   -0.0001    0.0000   -0.0002    0.0001
    0.0000    0.0000   -0.0002    0.0002   -0.0002   -0.0002   -0.0001
    0.0000   -0.0000    0.0004   -0.0000    0.0000    0.0004    0.0000
   -0.0000    0.0000   -0.0001   -0.0001    0.0001   -0.0001   -0.0000
    0.0000   -0.0000   -0.0003    0.0000    0.0000   -0.0002    0.0000
    0.0000    0.0000   -0.0000   -0.0001    0.0001    0.0000   -0.0003
    0.0000    0.0000    0.0000    0.0000   -0.0000    0.0001    0.0001
    0.0000    0.0000    0.0003    0.0000   -0.0001    0.0005   -0.0000
    0.0000   -0.0000   -0.0001    0.0002   -0.0002   -0.0002    0.0001
    0.0000    0.0000    0.0002    0.0001   -0.0001    0.0004    0.0000
   -0.0000    0.0000   -0.0001    0.0000   -0.0001   -0.0002    0.0000

  Columns 8 through 14

    0.0000    0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000
   -0.0000    0.0000   -0.0000   -0.0001   -0.0000   -0.0000   -0.0001
    0.0002   -0.0004   -0.0026   -0.0056   -0.0002   -0.0015   -0.0000
   -0.0007   -0.0046    0.0010    0.0000    0.0039   -0.0005    0.0063
   -0.0008    0.0038   -0.0005    0.0007   -0.0034    0.0005   -0.0048
   -0.0003   -0.0006   -0.0047   -0.0071   -0.0006   -0.0016   -0.0002
   -0.0003   -0.0006    0.0001   -0.0000    0.0004   -0.0000   -0.0003
   -0.0111   -0.0005    0.0001    0.0007   -0.0013   -0.0005    0.0018
   -0.0004   -0.0087   -0.0007   -0.0031    0.0081   -0.0017    0.0134
    0.0000   -0.0007   -0.0133   -0.0171   -0.0023   -0.0029   -0.0024
    0.0008   -0.0025   -0.0141   -0.0482   -0.0018   -0.0182    0.0001
   -0.0013    0.0079   -0.0021   -0.0018   -0.0096    0.0004   -0.0158
   -0.0005   -0.0013   -0.0011   -0.0163    0.0005   -0.0087    0.0024
    0.0016    0.0136   -0.0023    0.0004   -0.0156    0.0025   -0.0307
    0.0020   -0.0048   -0.0303   -0.0577   -0.0026   -0.0152   -0.0003
   -0.0004    0.0001   -0.0023   -0.0041   -0.0015   -0.0010   -0.0029
   -0.0168    0.0006   -0.0012    0.0007   -0.0052   -0.0010    0.0010
    0.0015    0.0079    0.0003    0.0032   -0.0085    0.0023   -0.0196
   -0.0000   -0.0003   -0.0063   -0.0062   -0.0013    0.0006   -0.0018
    0.0002    0.0074   -0.0019   -0.0005   -0.0108    0.0016   -0.0210
    0.0012    0.0031    0.0155    0.0009   -0.0003   -0.0085   -0.0021
    0.0015    0.0009    0.0047    0.0315    0.0020    0.0160   -0.0007
    0.0040   -0.0007   -0.0060   -0.0123    0.0003   -0.0029   -0.0016
   -0.0009    0.0028    0.0149    0.0308    0.0011    0.0100   -0.0009
    0.0009   -0.0014    0.0031    0.0062    0.0034    0.0019    0.0058
   -0.0008   -0.0019   -0.0061   -0.0138    0.0002   -0.0041    0.0029
   -0.0014   -0.0006   -0.0018    0.0004    0.0003   -0.0015    0.0015
   -0.0011    0.0020    0.0124    0.0143    0.0005    0.0018    0.0002
   -0.0009   -0.0003   -0.0010   -0.0033   -0.0001   -0.0020    0.0008
   -0.0002   -0.0004   -0.0025   -0.0026   -0.0006    0.0004   -0.0003
   -0.0003   -0.0000   -0.0003   -0.0002   -0.0005    0.0001    0.0018
    0.0003    0.0016   -0.0002   -0.0003   -0.0047    0.0001   -0.0071
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   -0.0006    0.0001   -0.0004   -0.0015   -0.0005   -0.0008   -0.0002

  Columns 15 through 21

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   -0.0034   -0.0001   -0.0048    0.0002   -0.0009   -0.0006    0.0000

  Columns 22 through 28

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  Columns 29 through 35

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  Columns 36 through 42

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   -0.0206    0.0234   -0.0473   -0.0164   -0.0314    0.0182   -0.0352
   -0.0059    0.0729   -0.1400   -0.0584   -0.1119    0.1076   -0.0803
    0.0282   -0.0115    0.0274    0.0128    0.0212    0.0197    0.0090
    0.0180    0.0736   -0.1384   -0.0828   -0.1473    0.0814   -0.0359
   -0.0100   -0.0165    0.0285    0.0194    0.0427   -0.0254    0.0085
   -0.0062   -0.0640    0.1617    0.0886    0.1636   -0.1194    0.0877
   -0.0050   -0.0000    0.0112   -0.0027    0.0084   -0.0006    0.0022
    0.0044    0.0091    0.0011   -0.0013   -0.0111    0.0021   -0.0060
    0.0103    0.0335   -0.0613   -0.0555   -0.1212    0.0316   -0.0483
    0.0159   -0.0016    0.0192    0.0167    0.0264    0.0120    0.0272
    0.0097    0.0019   -0.0161   -0.0319   -0.0759   -0.0047   -0.0175
   -0.0019   -0.0574   -0.0067    0.0004    0.0192    0.0057    0.0034

  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   -0.0005   -0.0001    0.0000   -0.0000    0.0002   -0.0001
   -0.0002    0.0003    0.0002   -0.0001   -0.0000   -0.0002    0.0000
   -0.0008    0.0001   -0.0009    0.0002    0.0008    0.0000   -0.0001
   -0.0003    0.0006    0.0003    0.0002   -0.0000    0.0000   -0.0001
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   -0.0009    0.0013    0.0002   -0.0003    0.0002   -0.0010    0.0001
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   -0.0009    0.0040    0.0006   -0.0005   -0.0000   -0.0023    0.0006
   -0.0210    0.0032   -0.0230    0.0055    0.0213    0.0005   -0.0036
   -0.0040    0.0054    0.0011    0.0023    0.0014   -0.0000   -0.0007
   -0.0043   -0.0035    0.0046   -0.0006    0.0005   -0.0004   -0.0010
    0.0003    0.0041    0.0025   -0.0006   -0.0015   -0.0021    0.0008
   -0.0073    0.0010   -0.0060    0.0016    0.0076   -0.0000   -0.0008
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    0.0119   -0.0032    0.0121   -0.0018   -0.0114    0.0018    0.0016
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    0.0127   -0.0164    0.0470   -0.0094   -0.0085    0.0014    0.0085
    0.0588    0.0061    0.0268   -0.0088   -0.0657   -0.0026    0.0026
   -0.0069    0.0031   -0.0009    0.0021    0.0058    0.0003   -0.0007
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    0.0038   -0.0358    0.0717   -0.0247   -0.0579    0.0124    0.0235

  Columns 50 through 53

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