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 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-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 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-0.0028 -0.0001 -0.0012 -0.0009 -0.0008 0.0062 0.0009 0.0051 0.0033 -0.0007 -0.0000 0.0004 0.0042 0.0002 0.0027 0.0003 -0.0005 0.0006 0.0008 0.0073 -0.0005 0.0047 -0.0016 0.0001 -0.0004 0.0017 -0.0021 0.0008 -0.0031 0.0026 -0.0003 -0.0008 -0.0005 0.0012 0.0008 0.0012 0.0024 -0.0002 -0.0005 -0.0022 0.0022 0.0001 0.0032 0.0007 -0.0003 0.0002 -0.0010 -0.0021 -0.0004 -0.0009 -0.0012 0.0006 0.0000 0.0007 0.0069 0.0008 0.0048 0.0001 -0.0001 -0.0004 -0.0000 -0.0017 0.0005 -0.0013 0.0013 0.0001 0.0003 0.0009 -0.0050 -0.0004 -0.0042 -0.0005 0.0000 -0.0004 0.0001 -0.0002 0.0006 -0.0003 0.0013 0.0001 0.0001 -0.0001 0.0005 -0.0002 0.0004 -0.0005 0.0001 0.0002 0.0005 0.0053 0.0002 0.0032 -0.0002 0.0001 0.0006 -0.0004 -0.0015 -0.0010 -0.0007 -0.0022 -0.0001 0.0003 0.0009 0.0038 -0.0001 0.0019 -0.0006 -0.0006 0.0001 -0.0004 -0.0015 -0.0005 -0.0008 -0.0002 Columns 15 through 21 0.0000 -0.0001 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0001 -0.0003 0.0000 -0.0001 -0.0000 0.0001 0.0000 -0.0050 -0.0004 0.0000 0.0003 -0.0006 -0.0001 -0.0002 0.0002 0.0025 -0.0002 0.0036 0.0004 0.0017 0.0001 0.0005 -0.0010 -0.0018 -0.0027 -0.0002 -0.0018 0.0001 -0.0091 -0.0007 -0.0009 0.0004 -0.0015 -0.0003 0.0017 0.0007 -0.0093 -0.0005 -0.0012 -0.0001 0.0040 0.0002 0.0022 -0.0004 -0.0167 0.0016 -0.0001 0.0001 0.0012 -0.0052 -0.0011 0.0006 0.0080 -0.0003 0.0069 0.0024 -0.0323 -0.0022 -0.0014 0.0005 -0.0065 -0.0019 0.0114 -0.0506 -0.0037 -0.0001 0.0030 -0.0048 -0.0006 -0.0111 -0.0023 0.0000 -0.0060 -0.0098 -0.0012 -0.0087 -0.0009 -0.0136 -0.0009 -0.0017 0.0026 0.0005 0.0011 -0.0142 -0.0002 -0.0050 0.0020 -0.0214 -0.0021 -0.0152 -0.0019 -0.1301 -0.0064 0.0031 0.0077 -0.0266 -0.0047 0.0360 -0.0064 -0.0442 -0.0020 -0.0087 -0.0026 0.0288 0.0039 0.0033 -0.0029 -0.0673 0.0044 -0.0008 -0.0030 0.0073 0.0075 -0.0061 0.0031 -0.0165 0.0003 -0.0078 -0.0043 -0.0267 -0.0028 -0.0024 0.0003 -0.0083 -0.0016 0.0242 -0.0031 0.0052 -0.0011 -0.0124 -0.0021 -0.0258 -0.0004 0.0564 0.0046 0.0077 -0.0047 0.0177 0.0002 -0.0839 0.0460 0.0040 0.0110 -0.0045 0.0075 0.0002 0.0348 -0.0373 -0.0020 0.0200 0.0001 -0.0095 -0.0007 0.0109 0.0918 0.0062 -0.0026 -0.0069 0.0221 0.0022 -0.0331 0.0178 0.0056 0.0073 0.0034 0.0051 0.0053 -0.0052 -0.0445 -0.0027 -0.0083 0.0060 -0.0143 0.0019 0.0147 0.0020 -0.0004 -0.0057 -0.0003 0.0070 0.0022 0.0183 0.0655 0.0039 -0.0055 -0.0033 0.0181 0.0022 -0.0681 -0.0072 0.0021 -0.0061 0.0012 -0.0006 0.0002 0.0001 -0.0150 -0.0078 -0.0029 0.0008 -0.0068 0.0008 0.0134 -0.0031 0.0197 -0.0016 0.0069 -0.0003 -0.0126 0.0017 0.0027 -0.0276 0.0020 -0.0036 0.0006 -0.0093 -0.0023 0.0268 0.0032 0.0033 -0.0031 0.0159 0.0005 -0.0372 0.0023 -0.0206 0.0006 -0.0079 -0.0011 -0.0060 0.0002 0.0436 0.0039 0.0004 -0.0028 0.0157 0.0015 -0.0367 -0.0014 0.0049 -0.0020 0.0090 0.0009 0.0052 -0.0000 0.0057 -0.0024 -0.0263 0.0005 0.0029 -0.0002 -0.0121 0.0016 0.0032 -0.0088 0.0041 -0.0047 0.0007 0.0249 0.0066 -0.0023 -0.0051 -0.0004 -0.0005 0.0022 0.0079 0.0122 -0.0005 -0.0036 -0.0023 -0.0002 -0.0011 0.0117 0.0057 0.0039 0.0019 0.0026 0.0054 0.0005 -0.0201 -0.0013 0.0010 -0.0017 0.0027 -0.0014 0.0012 0.0102 -0.0076 0.0001 -0.0018 0.0017 -0.0056 -0.0011 0.0270 -0.0058 -0.0017 -0.0025 -0.0018 -0.0014 0.0016 0.0039 0.0093 0.0011 0.0039 -0.0009 -0.0010 0.0014 0.0108 -0.0022 0.0001 -0.0005 0.0012 0.0004 0.0019 -0.0034 -0.0016 0.0000 0.0006 -0.0003 0.0030 -0.0005 -0.0227 0.0000 -0.0024 0.0005 0.0009 0.0002 0.0028 -0.0006 0.0005 0.0004 0.0005 -0.0005 -0.0000 -0.0008 0.0026 0.0094 0.0006 0.0008 -0.0010 0.0011 -0.0000 0.0107 -0.0036 -0.0001 0.0001 -0.0016 -0.0010 -0.0029 -0.0027 0.0095 0.0008 -0.0001 -0.0013 0.0021 -0.0006 0.0025 -0.0034 -0.0001 -0.0048 0.0002 -0.0009 -0.0006 0.0000 Columns 22 through 28 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0027 -0.0010 0.0021 0.0004 -0.0016 0.0019 -0.0000 -0.0000 0.0003 0.0001 -0.0006 -0.0002 -0.0002 -0.0001 -0.0002 0.0003 -0.0003 0.0005 0.0004 -0.0002 -0.0000 0.0037 -0.0017 0.0042 0.0009 -0.0027 0.0023 0.0011 0.0002 0.0000 -0.0001 -0.0005 -0.0004 -0.0002 -0.0001 0.0015 0.0040 -0.0003 0.0009 -0.0001 -0.0015 -0.0013 0.0017 -0.0009 0.0032 -0.0013 -0.0024 0.0004 0.0014 0.0102 -0.0071 0.0167 0.0037 -0.0095 0.0055 0.0082 0.0395 -0.0124 0.0254 0.0056 -0.0222 0.0316 -0.0063 0.0022 0.0002 0.0009 0.0030 0.0000 0.0021 -0.0006 0.0208 -0.0034 0.0069 0.0015 -0.0098 0.0192 -0.0103 -0.0001 -0.0018 -0.0015 0.0051 0.0023 0.0020 -0.0002 0.0657 -0.0408 0.0924 0.0192 -0.0612 0.0510 0.0361 0.0064 -0.0032 0.0058 -0.0028 -0.0077 0.0023 0.0024 0.0075 0.0208 0.0033 0.0063 -0.0012 -0.0069 -0.0054 -0.0046 0.0000 -0.0072 0.0019 0.0058 -0.0010 -0.0019 0.0048 -0.0086 0.0247 0.0060 -0.0127 -0.0010 0.0202 0.0008 -0.0008 0.0017 0.0089 0.0029 0.0030 0.0010 0.0180 0.0147 -0.0559 -0.0108 0.0111 0.0576 -0.0870 -0.0763 0.0180 -0.0452 -0.0130 0.0582 -0.0976 0.0363 0.0227 -0.0253 0.0385 0.0066 -0.0317 0.0320 0.0219 -0.0649 0.0461 -0.1135 -0.0250 0.0927 -0.0849 -0.0568 -0.0170 0.0066 -0.0247 -0.0108 0.0214 -0.0238 -0.0061 0.0468 -0.0242 0.0619 0.0160 -0.0753 0.0950 0.0067 -0.0339 0.0177 0.0007 -0.0006 0.0573 -0.1620 0.0702 -0.0044 0.0298 -0.0952 -0.0203 0.0359 0.0730 -0.1542 0.0088 -0.0003 0.0128 0.0036 -0.0104 0.0005 0.0055 0.0113 -0.0093 0.0248 0.0091 -0.0295 0.0391 0.0110 0.0013 0.0011 0.0053 -0.0075 -0.0064 0.0076 -0.0005 -0.0008 0.0032 0.0010 0.0138 0.0174 -0.0054 0.0004 -0.0123 0.0219 -0.0470 -0.0118 0.0722 -0.1134 -0.0238 -0.0015 0.0003 -0.0020 0.0137 0.0091 0.0077 -0.0032 -0.0257 0.0297 -0.0839 -0.0220 0.0823 -0.0663 -0.0835 0.0032 0.0019 0.0043 -0.0065 -0.0068 -0.0018 -0.0009 0.0138 0.0188 -0.0086 0.0094 -0.0049 -0.0166 -0.0370 -0.0162 0.0048 -0.0047 -0.0024 -0.0099 0.0167 0.0075 -0.0113 0.0064 -0.0139 -0.0023 0.0064 -0.0025 -0.0077 -0.0227 0.0094 -0.0268 -0.0050 0.0211 -0.0115 -0.0087 0.0075 0.0043 -0.0096 -0.0050 0.0133 -0.0138 -0.0258 -0.0066 0.0012 0.0013 -0.0018 0.0008 -0.0104 0.0166 -0.0128 -0.0030 0.0121 0.0020 -0.0075 -0.0078 0.0474 0.0034 -0.0019 0.0097 0.0035 -0.0050 -0.0158 0.0216 -0.0194 0.0017 -0.0190 -0.0076 0.0077 0.0192 -0.0130 0.0057 -0.0010 0.0049 0.0003 -0.0049 0.0004 -0.0003 0.0213 -0.0022 0.0066 0.0035 -0.0096 0.0188 -0.0304 0.0008 -0.0002 0.0005 -0.0008 -0.0006 -0.0002 -0.0008 -0.0040 0.0007 -0.0023 -0.0007 0.0060 -0.0113 0.0074 -0.0234 0.0077 -0.0232 -0.0075 0.0311 -0.0404 0.0108 0.0073 -0.0035 0.0084 0.0046 -0.0108 0.0175 -0.0024 -0.0158 0.0077 -0.0219 -0.0059 0.0268 -0.0310 -0.0034 0.0065 0.0002 0.0076 0.0041 -0.0103 0.0056 0.0015 Columns 29 through 35 -0.0000 0.0000 -0.0000 0.0001 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0002 0.0000 0.0001 0.0000 0.0000 -0.0006 -0.0001 0.0001 0.0015 -0.0001 0.0005 0.0001 -0.0004 0.0018 -0.0015 0.0000 -0.0017 0.0001 -0.0001 0.0003 -0.0014 0.0005 -0.0002 0.0011 -0.0001 -0.0001 -0.0010 -0.0003 0.0002 0.0026 -0.0000 0.0014 -0.0006 0.0008 0.0001 0.0063 -0.0001 0.0029 -0.0000 -0.0008 0.0000 0.0002 -0.0005 0.0002 -0.0004 -0.0005 -0.0002 -0.0011 0.0047 0.0009 0.0016 -0.0024 0.0015 -0.0006 -0.0038 -0.0016 0.0004 0.0093 0.0003 0.0070 0.0012 -0.0081 -0.0014 0.0019 0.0213 -0.0011 0.0043 -0.0000 0.0003 -0.0071 -0.0006 0.0010 0.0037 -0.0004 0.0013 -0.0034 0.0006 0.0009 0.0094 -0.0015 -0.0016 -0.0001 0.0022 -0.0166 0.0032 -0.0005 0.0111 -0.0012 -0.0009 -0.0257 -0.0049 0.0011 0.0627 -0.0029 0.0400 -0.0055 0.0057 -0.0024 0.0542 0.0035 0.0263 0.0027 -0.0043 0.0001 -0.0019 -0.0032 0.0035 -0.0010 -0.0017 -0.0003 0.0048 -0.0150 0.0100 -0.0051 0.0138 -0.0036 -0.0025 -0.0056 -0.0031 0.0016 0.0141 0.0017 0.0153 0.0034 -0.0039 -0.0167 -0.0374 0.0021 -0.0078 0.0000 0.0112 0.0071 -0.0000 -0.0000 -0.0117 -0.0022 -0.0510 -0.0046 0.0231 0.0045 -0.0075 -0.0666 0.0032 -0.0051 0.0018 -0.0145 -0.0042 0.0063 0.0364 0.0017 0.0257 0.0003 0.0441 0.0057 -0.0055 -0.1144 0.0053 -0.0794 0.0006 0.0108 0.0201 0.0097 -0.0324 -0.0026 -0.0170 0.0031 -0.0390 0.0004 0.0187 0.1158 -0.0044 0.0500 -0.0251 0.0338 0.0120 -0.0088 -0.1152 0.0079 0.0231 0.0170 0.0207 0.0045 0.0005 -0.0357 -0.0059 -0.1404 -0.0057 -0.0012 -0.0006 0.0220 0.0098 0.0117 0.0125 0.0048 -0.0247 -0.0007 -0.0301 0.0668 -0.0264 0.0356 0.0005 0.0036 -0.0518 0.0208 0.0120 0.0570 0.0004 0.0192 -0.0328 -0.0268 -0.3283 -0.0056 -0.1656 -0.0037 -0.0150 0.0553 0.0097 -0.0078 -0.2011 0.0151 -0.0864 0.0137 -0.0235 -0.0004 -0.2164 0.0102 -0.1425 -0.0022 0.0032 0.0581 0.0091 -0.0086 -0.2029 0.0097 -0.2391 -0.0025 -0.0013 -0.0146 0.0087 0.0032 0.0329 0.0041 -0.0182 0.0007 0.0015 -0.0077 -0.0287 -0.0029 -0.0345 0.0050 -0.0174 -0.0176 0.0133 0.0937 0.0083 0.0018 0.0024 -0.0020 0.0106 0.0108 0.0173 0.0007 -0.0374 0.0070 0.0035 0.0034 -0.0039 0.0073 -0.0139 -0.0768 -0.0031 0.0182 -0.0122 -0.0184 -0.0815 0.0081 -0.0620 0.0008 -0.0039 -0.0078 -0.0129 0.0166 -0.0022 0.0103 0.0033 -0.0175 -0.0110 -0.0228 0.0750 -0.0191 0.0763 -0.0082 0.0041 0.0178 0.0714 -0.0097 0.0460 0.0316 0.0108 -0.0031 0.0032 0.0384 0.0580 0.0155 -0.0274 -0.0033 0.0015 0.0020 0.0186 -0.0094 0.0189 0.0077 -0.0050 0.0054 0.0007 -0.0040 -0.0468 0.0029 -0.0192 0.0007 -0.0024 0.0090 -0.0154 -0.0014 -0.0130 -0.0003 -0.0000 0.0035 -0.0007 0.0008 -0.0105 0.0011 -0.0100 0.0044 0.0135 0.0027 -0.0040 -0.0315 -0.0037 -0.0811 -0.0005 -0.0062 -0.0023 -0.0035 0.0191 -0.0096 0.0368 0.0025 0.0147 0.0022 -0.0036 -0.0433 -0.0026 -0.0902 -0.0041 -0.0046 -0.0016 0.0085 0.0119 0.0052 0.0295 Columns 36 through 42 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0003 -0.0007 -0.0003 -0.0005 0.0004 -0.0002 0.0008 -0.0002 0.0004 -0.0002 -0.0002 0.0005 0.0003 -0.0008 -0.0001 -0.0003 0.0001 0.0002 -0.0004 -0.0003 -0.0000 0.0006 -0.0012 -0.0005 -0.0007 0.0007 -0.0003 0.0005 -0.0001 -0.0001 0.0002 -0.0001 -0.0002 -0.0001 -0.0002 -0.0009 -0.0001 -0.0001 -0.0002 -0.0001 -0.0001 0.0026 -0.0002 0.0002 -0.0006 -0.0010 0.0017 0.0005 -0.0005 0.0024 -0.0045 -0.0014 -0.0021 0.0030 -0.0014 0.0007 0.0044 -0.0097 -0.0047 -0.0080 0.0045 -0.0020 -0.0039 0.0001 -0.0022 0.0001 0.0003 -0.0015 -0.0013 0.0011 0.0013 -0.0040 -0.0026 -0.0045 0.0014 -0.0005 -0.0086 0.0021 -0.0032 0.0014 0.0020 -0.0043 -0.0025 0.0001 0.0148 -0.0299 -0.0111 -0.0172 0.0200 -0.0082 0.0029 0.0002 -0.0038 0.0012 -0.0016 -0.0004 -0.0016 -0.0029 -0.0079 -0.0035 -0.0012 -0.0019 -0.0002 -0.0019 -0.0074 0.0001 -0.0013 0.0021 0.0029 -0.0054 -0.0019 -0.0011 0.0059 -0.0069 -0.0012 -0.0020 0.0056 -0.0025 -0.0124 0.0027 -0.0035 -0.0004 0.0020 -0.0028 -0.0025 0.0015 -0.0075 0.0064 -0.0033 -0.0069 -0.0149 0.0046 -0.0028 -0.0159 0.0302 0.0151 0.0270 -0.0122 0.0058 0.0008 0.0127 -0.0186 -0.0067 -0.0104 0.0127 -0.0048 -0.0012 -0.0255 0.0589 0.0217 0.0331 -0.0411 0.0166 0.0094 -0.0135 0.0177 0.0046 0.0076 -0.0068 0.0062 0.0090 0.0370 -0.0597 -0.0254 -0.0452 0.0373 -0.0141 -0.0029 -0.0168 0.0753 0.0437 0.0724 -0.0211 0.0149 0.0075 -0.0047 0.0280 -0.0031 -0.0155 -0.0441 0.0155 0.0029 0.0134 0.0009 0.0049 0.0030 0.0023 -0.0006 0.0032 0.0261 -0.0382 -0.0199 -0.0301 0.0300 -0.0085 -0.0496 0.0147 -0.0262 0.0016 0.0052 -0.0179 -0.0174 -0.0501 0.0038 0.0236 -0.0211 0.0153 0.0162 0.0140 -0.0088 -0.0873 0.1526 0.0694 0.1214 -0.0907 0.0414 0.0059 -0.0034 0.0203 -0.0303 -0.0134 0.0402 0.0225 -0.0047 -0.0773 0.1808 0.0612 0.0890 -0.1540 0.0596 -0.0514 -0.0127 -0.0237 -0.0008 0.0137 -0.0201 -0.0212 -0.0039 -0.1757 0.0127 -0.0161 0.0238 -0.0281 -0.0167 -0.0181 0.0127 -0.1725 -0.0873 -0.1419 0.0687 -0.0777 0.0119 -0.0075 -0.0494 -0.0431 -0.0668 0.0267 -0.0219 0.0206 0.0024 -0.0627 -0.0558 -0.1041 0.0295 -0.0239 -0.0363 -0.0384 0.1106 0.0634 0.1045 -0.1100 0.0303 -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 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0003 0.0001 -0.0005 0.0001 0.0003 0.0000 -0.0001 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 -0.0004 -0.0005 0.0007 -0.0001 -0.0001 0.0000 -0.0001 -0.0003 -0.0009 -0.0007 0.0004 0.0006 0.0007 -0.0003 -0.0036 0.0005 -0.0033 0.0008 0.0036 -0.0000 -0.0005 -0.0037 0.0018 -0.0069 0.0015 0.0034 0.0000 -0.0012 -0.0009 0.0013 0.0002 -0.0003 0.0002 -0.0010 0.0001 -0.0003 0.0007 -0.0027 0.0006 0.0001 0.0001 -0.0006 -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 -0.0002 0.0002 -0.0017 -0.0020 0.0008 -0.0025 0.0008 0.0206 0.0029 0.0081 -0.0030 -0.0222 -0.0011 0.0011 0.0099 -0.0062 0.0210 -0.0047 -0.0082 0.0000 0.0040 -0.0117 0.0048 -0.0173 0.0041 0.0137 0.0003 -0.0018 0.0443 -0.0046 0.0442 -0.0112 -0.0447 -0.0015 0.0073 0.0119 -0.0032 0.0121 -0.0018 -0.0114 0.0018 0.0016 -0.0362 0.0096 -0.0479 0.0124 0.0361 0.0018 -0.0084 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 -0.0238 -0.0018 -0.0382 0.0065 0.0286 0.0021 -0.0056 -0.0116 0.0213 -0.0031 -0.0025 0.0062 -0.0145 0.0022 0.0347 -0.0743 -0.0250 -0.0282 -0.0029 -0.0001 0.0060 0.0961 -0.0223 0.1302 -0.0314 -0.1019 -0.0024 0.0236 0.0285 -0.0749 -0.0326 -0.0144 0.0007 0.0153 -0.0016 0.1856 -0.0081 0.1633 -0.0448 -0.2176 -0.0100 0.0237 -0.0133 0.0084 0.0092 -0.0106 0.0057 -0.0230 0.0060 -0.0008 -0.0656 0.1293 -0.0240 -0.0578 0.0016 0.0071 -0.1310 0.0317 -0.1207 0.0373 0.1327 -0.0128 -0.0342 -0.0282 -0.0161 -0.0420 0.0097 0.0286 0.0058 -0.0173 -0.0263 0.0076 -0.0775 0.0234 0.0327 0.0078 -0.0304 0.1072 0.0160 0.1455 -0.0393 -0.1580 -0.0285 0.0334 -0.0539 0.0235 -0.0403 0.0144 0.0631 -0.0173 -0.0116 -0.1841 0.0226 -0.1485 0.0551 0.2404 -0.0042 -0.0422 0.0041 -0.0742 0.0284 -0.0204 -0.0065 0.0329 0.0053 -0.0964 0.0428 -0.2393 0.0592 0.2010 0.0121 -0.0596 0.0327 -0.0137 0.0533 -0.0257 -0.0603 -0.0082 0.0241 0.2084 -0.0169 0.2347 -0.0817 -0.3724 -0.0118 0.0658 0.0115 -0.0060 0.0095 -0.0083 -0.0121 -0.0278 0.0070 -0.0148 -0.0097 0.0027 0.0038 0.0388 0.0097 -0.0188 -0.0874 0.0073 -0.1247 0.0542 0.2423 0.0129 -0.0704 0.0201 -0.0102 0.0210 -0.0004 -0.0710 0.0225 0.0144 -0.0142 0.0092 -0.0691 0.0374 0.1282 0.0145 -0.0493 0.0038 -0.0358 0.0717 -0.0247 -0.0579 0.0124 0.0235 Columns 50 through 53 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0004 0.0001 -0.0003 0.0001 0.0000 -0.0001 -0.0000 -0.0000 0.0001 0.0001 0.0000 0.0000 -0.0007 0.0003 -0.0005 0.0002 -0.0000 -0.0003 -0.0001 -0.0001 0.0001 0.0001 -0.0001 -0.0000 -0.0006 -0.0003 -0.0005 0.0000 -0.0031 0.0011 -0.0021 0.0007 -0.0049 0.0017 -0.0038 0.0010 -0.0001 0.0007 -0.0000 0.0003 -0.0015 0.0005 -0.0014 0.0003 0.0002 0.0011 0.0006 0.0002 -0.0198 0.0067 -0.0138 0.0046 -0.0018 -0.0017 -0.0017 -0.0003 -0.0001 0.0015 -0.0011 0.0007 0.0015 0.0002 0.0013 -0.0000 -0.0059 0.0020 -0.0037 0.0011 -0.0000 0.0033 0.0007 0.0010 0.0126 -0.0037 0.0068 -0.0028 0.0143 -0.0054 0.0115 -0.0022 -0.0130 0.0035 -0.0085 0.0028 0.0400 -0.0132 0.0277 -0.0097 0.0104 -0.0058 0.0067 -0.0020 -0.0390 0.0122 -0.0281 0.0067 0.0290 -0.0111 0.0256 -0.0083 0.0380 -0.0103 0.0213 -0.0123 -0.0034 -0.0001 -0.0015 -0.0019 -0.0288 0.0117 -0.0201 0.0063 -0.0048 0.0103 -0.0011 0.0034 0.0119 0.0267 0.0132 0.0066 0.1122 -0.0395 0.0811 -0.0223 -0.0001 0.0134 -0.0002 0.0042 0.1693 -0.0508 0.1074 -0.0405 0.0035 0.0171 0.0055 0.0104 0.0743 -0.0282 0.0329 0.0385 -0.1616 0.0622 -0.1280 0.0612 -0.0564 0.0228 -0.0534 0.0305 -0.0950 0.0322 -0.0890 0.0303 0.1634 -0.0473 0.1194 -0.0410 -0.0720 0.0320 -0.0590 0.0374 -0.2441 0.0824 -0.1841 0.0806 0.0404 -0.0038 0.0203 0.0217 -0.2972 0.1120 -0.2082 0.0303 0.0996 -0.0194 0.0773 -0.0091 0.4432 -0.1594 0.3372 -0.1278 0.0152 0.0130 0.0166 0.0016 -0.0836 0.0301 -0.0866 0.0220 -0.4493 0.1754 -0.4013 0.1162 0.1497 -0.1103 0.1469 -0.0690 -0.3491 0.1311 -0.3651 0.1880 0.1654 -0.0595 0.0982 -0.1699


