Solve Save + Borr Dynamic Programming Problem (Loop)

back to Fan's Dynamic Assets Repository Table of Content.

Contents

function result_map = ff_abz_vf(varargin)

FF_ABZ_VF solve infinite horizon exo shock + endo asset problem

This program solves the infinite horizon dynamic single asset and single shock problem with loops. This file contains codes that processes borrowing.

The borrowing problem is very similar to the savings problem. The code could be identical if one does not have to deal with default. The main addition here in comparison to the savings only code ff_az_vf is the ability to deal with default.

@param param_map container parameter container

@param support_map container support container

@param armt_map container container with states, choices and shocks grids that are inputs for grid based solution algorithm

@param func_map container container with function handles for consumption cash-on-hand etc.

@return result_map container contains policy function matrix, value function matrix, iteration results, and policy function, value function and iteration results tables.

keys included in result_map:

@example

  % Get Default Parameters
  it_param_set = 2;
  [param_map, support_map] = ffs_abz_set_default_param(it_param_set);
  % Chnage param_map keys for borrowing
  param_map('fl_b_bd') = -20; % borrow bound
  param_map('bl_default') = false; % true if allow for default
  param_map('fl_c_min') = 0.0001; % u(c_min) when default
  % Change Keys in param_map
  param_map('it_a_n') = 500;
  param_map('fl_z_r_borr_n') = 5;
  param_map('it_z_wage_n') = 15;
  param_map('it_z_n') = param_map('it_z_wage_n') * param_map('fl_z_r_borr_n');
  param_map('fl_a_max') = 100;
  param_map('fl_w') = 1.3;
  % Change Keys support_map
  support_map('bl_display') = false;
  support_map('bl_post') = true;
  support_map('bl_display_final') = false;
  % Call Program with external parameters that override defaults.
  ff_abz_vf(param_map, support_map);

@include

@seealso

Default

it_param_set = 4;
bl_input_override = true;
[param_map, support_map] = ffs_abz_set_default_param(it_param_set);

% Note: param_map and support_map can be adjusted here or outside to override defaults
param_map('it_a_n') = 75;
param_map('fl_z_r_borr_n') = 3;
param_map('it_z_wage_n') = 5;
param_map('it_z_n') = param_map('it_z_wage_n') * param_map('fl_z_r_borr_n');
% param_map('fl_r_save') = 0.025;
% param_map('fl_r_borr') = 0.035;

[armt_map, func_map] = ffs_abz_get_funcgrid(param_map, support_map, bl_input_override); % 1 for override
default_params = {param_map support_map armt_map func_map};

Parse Parameters 1

% if varargin only has param_map and support_map,
params_len = length(varargin);
[default_params{1:params_len}] = varargin{:};
param_map = [param_map; default_params{1}];
support_map = [support_map; default_params{2}];
if params_len >= 1 && params_len <= 2
    % If override param_map, re-generate armt and func if they are not
    % provided
    bl_input_override = true;
    [armt_map, func_map] = ffs_abz_get_funcgrid(param_map, support_map, bl_input_override);
else
    % Override all
    armt_map = [armt_map; default_params{3}];
    func_map = [func_map; default_params{4}];
end

% append function name
st_func_name = 'ff_abz_vf';
support_map('st_profile_name_main') = [st_func_name support_map('st_profile_name_main')];
support_map('st_mat_name_main') = [st_func_name support_map('st_mat_name_main')];
support_map('st_img_name_main') = [st_func_name support_map('st_img_name_main')];

Parse Parameters 2

% armt_map
params_group = values(armt_map, {'ar_a', 'mt_z_trans', 'ar_z_r_borr_mesh_wage', 'ar_z_wage_mesh_r_borr'});
[ar_a, mt_z_trans, ar_z_r_borr_mesh_wage, ar_z_wage_mesh_r_borr] = params_group{:};

% func_map
params_group = values(func_map, {'f_util_log', 'f_util_crra', 'f_cons_checkcmin', 'f_awithr_to_anor', 'f_coh', 'f_cons_coh'});
[f_util_log, f_util_crra, f_cons_checkcmin, f_awithr_to_anor, f_coh, f_cons_coh] = params_group{:};

% param_map
params_group = values(param_map, {'it_a_n', 'it_z_n', 'fl_crra', 'fl_beta', 'fl_c_min',...
    'fl_nan_replace', 'bl_default', 'fl_default_aprime'});
[it_a_n, it_z_n, fl_crra, fl_beta, fl_c_min, ...
    fl_nan_replace, bl_default, fl_default_aprime] = params_group{:};
params_group = values(param_map, {'it_maxiter_val', 'fl_tol_val', 'fl_tol_pol', 'it_tol_pol_nochange'});
[it_maxiter_val, fl_tol_val, fl_tol_pol, it_tol_pol_nochange] = params_group{:};

% support_map
params_group = values(support_map, {'bl_profile', 'st_profile_path', ...
    'st_profile_prefix', 'st_profile_name_main', 'st_profile_suffix',...
    'bl_time', 'bl_display_defparam', 'bl_display', 'it_display_every', 'bl_post'});
[bl_profile, st_profile_path, ...
    st_profile_prefix, st_profile_name_main, st_profile_suffix, ...
    bl_time, bl_display_defparam, bl_display, it_display_every, bl_post] = params_group{:};

Display Parameters

if(bl_display_defparam)
    fft_container_map_display(param_map);
    fft_container_map_display(support_map);
end
----------------------------------------
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Begin: Show all key and value pairs from container
CONTAINER NAME: PARAM_MAP
----------------------------------------
  Map with properties:

        Count: 35
      KeyType: char
    ValueType: any

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
----------------------------------------
----------------------------------------
pos = 32 ; key = st_analytical_stationary_type ; val = eigenvector
pos = 33 ; key = st_model ; val = abz
pos = 34 ; key = st_z_r_borr_drv_ele_type ; val = unif
pos = 35 ; key = st_z_r_borr_drv_prb_type ; val = poiss
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Scalars in Container and Sizes and Basic Statistics
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
                              i     idx    value 
                              __    ___    ______

    bl_b_is_principle          1     1          1
    bl_default                 2     2          1
    bl_loglin                  3     3          0
    fl_a_max                   4     4         50
    fl_a_min                   5     5          0
    fl_b_bd                    6     6        -20
    fl_beta                    7     7       0.94
    fl_c_min                   8     8       0.01
    fl_crra                    9     9        1.5
    fl_default_aprime         10    10          0
    fl_loglin_threshold       11    11          1
    fl_nan_replace            12    12     -99999
    fl_r_save                 13    13      0.025
    fl_tol_dist               14    14      1e-05
    fl_tol_pol                15    15      1e-05
    fl_tol_val                16    16      1e-05
    fl_w                      17    17       1.28
    fl_z_r_borr_max           18    18      0.095
    fl_z_r_borr_min           19    19      0.025
    fl_z_r_borr_n             20    20          3
    fl_z_r_borr_poiss_mean    21    21         10
    fl_z_wage_mu              22    22          0
    fl_z_wage_rho             23    23        0.8
    fl_z_wage_sig             24    24        0.2
    it_a_n                    25    25         75
    it_maxiter_dist           26    26       1000
    it_maxiter_val            27    27       1000
    it_tol_pol_nochange       28    28         25
    it_trans_power_dist       29    29       1000
    it_z_n                    30    30         15
    it_z_wage_n               31    31          5

----------------------------------------
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Begin: Show all key and value pairs from container
CONTAINER NAME: SUPPORT_MAP
----------------------------------------
  Map with properties:

        Count: 40
      KeyType: char
    ValueType: any

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
----------------------------------------
----------------------------------------
pos = 26 ; key = st_img_name_main ; val = ff_abz_vf_default
pos = 27 ; key = st_img_path ; val = C:/Users/fan/CodeDynaAsset//m_abz//solve/img/
pos = 28 ; key = st_img_prefix ; val = 
pos = 29 ; key = st_img_suffix ; val = _p4.png
pos = 30 ; key = st_mat_name_main ; val = ff_abz_vf_default
pos = 31 ; key = st_mat_path ; val = C:/Users/fan/CodeDynaAsset//m_abz//solve/mat/
pos = 32 ; key = st_mat_prefix ; val = 
pos = 33 ; key = st_mat_suffix ; val = _p4
pos = 34 ; key = st_mat_test_path ; val = C:/Users/fan/CodeDynaAsset//m_abz//test/ff_az_ds_vecsv/mat/
pos = 35 ; key = st_matimg_path_root ; val = C:/Users/fan/CodeDynaAsset//m_abz/
pos = 36 ; key = st_profile_name_main ; val = ff_abz_vf_default
pos = 37 ; key = st_profile_path ; val = C:/Users/fan/CodeDynaAsset//m_abz//solve/profile/
pos = 38 ; key = st_profile_prefix ; val = 
pos = 39 ; key = st_profile_suffix ; val = _p4
pos = 40 ; key = st_title_prefix ; val = 
----------------------------------------
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Scalars in Container and Sizes and Basic Statistics
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
                                    i     idx    value
                                    __    ___    _____

    bl_display                       1     1        0 
    bl_display_defparam              2     2        1 
    bl_display_dist                  3     3        0 
    bl_display_final                 4     4        1 
    bl_display_final_dist            5     5        0 
    bl_display_final_dist_detail     6     6        0 
    bl_display_funcgrids             7     7        0 
    bl_graph                         8     8        1 
    bl_graph_coh_t_coh               9     9        1 
    bl_graph_funcgrids              10    10        0 
    bl_graph_onebyones              11    11        1 
    bl_graph_pol_lvl                12    12        1 
    bl_graph_pol_pct                13    13        1 
    bl_graph_val                    14    14        1 
    bl_img_save                     15    15        0 
    bl_mat                          16    16        0 
    bl_post                         17    17        1 
    bl_profile                      18    18        0 
    bl_profile_dist                 19    19        0 
    bl_time                         20    20        1 
    it_display_every                21    21        5 
    it_display_final_colmax         22    22       15 
    it_display_final_rowmax         23    23      100 
    it_display_summmat_colmax       24    24        5 
    it_display_summmat_rowmax       25    25        5 

Initialize Output Matrixes

mt_val_cur = zeros(length(ar_a),it_z_n);
mt_val = mt_val_cur - 1;
mt_pol_a = zeros(length(ar_a),it_z_n);
mt_pol_a_cur = mt_pol_a - 1;

Initialize Convergence Conditions

bl_vfi_continue = true;
it_iter = 0;
ar_val_diff_norm = zeros([it_maxiter_val, 1]);
ar_pol_diff_norm = zeros([it_maxiter_val, 1]);
mt_pol_perc_change = zeros([it_maxiter_val, it_z_n]);

Iterate Value Function

Loop solution with 4 nested loops

  1. loop 1: over exogenous states
  2. loop 2: over endogenous states
  3. loop 3: over choices
  4. loop 4: add future utility, integration--loop over future shocks
% Start Profile
if (bl_profile)
    close all;
    profile off;
    profile on;
end

% Start Timer
if (bl_time)
    tic;
end

% Utility at-Default/at-limiting-case-when-nodefault
if (fl_crra == 1)
    fl_u_cmin = f_util_log(fl_c_min);
else
    fl_u_cmin = f_util_crra(fl_c_min);
end


% Value Function Iteration
while bl_vfi_continue
    it_iter = it_iter + 1;

Iterate over a and z states

Solve Problem if Default: coh < borr_bound, then: u(c_min) + EV(a'=0,z') these are the values for defaulter if do not borrow enough to pay debt: a'=0 if save more than what you have: a'=0 if borrowing, possible that all ar_val_cur values are u(cmin), when that is the case, utility equal to c_min, this means given current choice set, can only default, get utility at c_min level, and a' go to 0. See maximization over loop 3 choices for loop 1+2 states. see README_cminymin_borrsave for additional discussions.

when default is not allowed, there should be exactly one element of the state space where we enter this condition. That is at a_min and z_min. At this point, the natural borrowing constraint when fully used up leads to c = 0. When default is not allowed, this is the limiting case. We solve the limiting case as if default is possible. see the figure here: ffs_abz_get_funcgrid_nodefault. For the figure under section Generate Borrowing A Grid with Default, there are no points to the lower right of the red horizontal and verticle lines, because no default is allowed. But the intersection of the two red lines is this limiting case where c = 0.

when default is allowed. See again ffs_abz_get_funcgrid_nodefault. Again under section Generate Borrowing A Grid with Default, see that there is an area to the right bottom of the two blue horizontal and vertical lines. For all points in that area, the household has to default, they can not borrow even at max to get positive consumption. At other points higher than the blue horizontal line, it is also possible for defaulting to be the optimal choice.

    % loop 1: over exogenous states
    for it_z_i = 1:it_z_n

        % Current Shock
        fl_z_r_borr = ar_z_r_borr_mesh_wage(it_z_i);
        fl_z_wage = ar_z_wage_mesh_r_borr(it_z_i);

        % loop 2: over endogenous states
        for it_a_j = 1:length(ar_a)
            fl_a = ar_a(it_a_j);
            ar_val_cur = zeros(size(ar_a));

            % calculate cash on hand
            fl_coh = f_coh(fl_z_wage, fl_a);

            % loop 3: over choices
            for it_ap_k = 1:length(ar_a)

                % get next period asset choice
                fl_ap = ar_a(it_ap_k);

                % calculate consumption
                fl_c = f_cons_coh(fl_coh, fl_z_r_borr, fl_ap);

                % assign u(c)
                if (fl_c <= fl_c_min)
                    if (bl_default)
                        % defaults
                        % current utility: only today u(cmin)
                        ar_val_cur(it_ap_k) = fl_u_cmin;
                        % transition out next period, debt wiped out
                        for it_az_q = 1:it_z_n
                            ar_val_cur(it_ap_k) = ar_val_cur(it_ap_k) + ...
                                fl_beta*mt_z_trans(it_z_i, it_az_q)*mt_val_cur((ar_a == fl_default_aprime), it_az_q);
                        end
                    else
                        % if default is not allowed: v = fl_nan_replace
                        ar_val_cur(it_ap_k) = fl_nan_replace;
                    end
                else
                    % Solve Optimization Problem: max_{a'} u(c(a,a',z)) + beta*EV(a',z')
                    % borrowed enough to pay debt (and borrowing limit not exceeded)
                    % saved only the coh available.
                    % current utility
                    if (fl_crra == 1)
                        ar_val_cur(it_ap_k) = f_util_log(fl_c);
                    else
                        ar_val_cur(it_ap_k) = f_util_crra(fl_c);
                    end
                    % loop 4: add future utility, integration--loop over future shocks
                    for it_az_q = 1:it_z_n
                        ar_val_cur(it_ap_k) = ar_val_cur(it_ap_k) + ...
                            fl_beta*mt_z_trans(it_z_i, it_az_q)*mt_val_cur(it_ap_k, it_az_q);
                    end
                end
            end

            % optimal choice value
            [fl_opti_val_z, fl_opti_idx_z] = max(ar_val_cur);
            fl_opti_aprime_z = ar_a(fl_opti_idx_z);
            fl_opti_c_z = f_cons_coh(fl_coh, fl_z_r_borr, fl_opti_aprime_z);

            % Handle Default is optimal or not
            if (fl_opti_c_z <= fl_c_min)
                if (bl_default)
                    % if defaulting is optimal choice, at these states, not required
                    % to default, non-default possible, but default could be optimal
                    fl_opti_aprime_z = fl_default_aprime;
                else
                    % if default is not allowed, then next period same state as now
                    % this is absorbing state, this is the limiting case, single
                    % state space point, lowest a and lowest shock has this.
                    fl_opti_aprime_z = fl_a;
                end
            end

Store Optimal Choices and Value Given(a,z)

            mt_val(it_a_j,it_z_i) = fl_opti_val_z;
            mt_pol_a(it_a_j,it_z_i) = fl_opti_aprime_z;
        end
    end

Check Tolerance and Continuation

    % Difference across iterations
    ar_val_diff_norm(it_iter) = norm(mt_val - mt_val_cur);
    ar_pol_diff_norm(it_iter) = norm(mt_pol_a - mt_pol_a_cur);
    mt_pol_perc_change(it_iter, :) = sum((mt_pol_a ~= mt_pol_a_cur))/(it_a_n);

    % Update
    mt_val_cur = mt_val;
    mt_pol_a_cur = mt_pol_a;

    % Print Iteration Results
    if (bl_display && (rem(it_iter, it_display_every)==0))
        fprintf('VAL it_iter:%d, fl_diff:%d, fl_diff_pol:%d\n', ...
            it_iter, ar_val_diff_norm(it_iter), ar_pol_diff_norm(it_iter));
        tb_valpol_iter = array2table([mean(mt_val_cur,1); mean(mt_pol_a_cur,1); ...
            mt_val_cur(it_a_n,:); mt_pol_a_cur(it_a_n,:)]);
        tb_valpol_iter.Properties.VariableNames = strcat('z', string((1:size(mt_val_cur,2))));
        tb_valpol_iter.Properties.RowNames = {'mval', 'map', 'Hval', 'Hap'};
        disp('mval = mean(mt_val_cur,1), average value over a')
        disp('map  = mean(mt_pol_a_cur,1), average choice over a')
        disp('Hval = mt_val_cur(it_a_n,:), highest a state val')
        disp('Hap = mt_pol_a_cur(it_a_n,:), highest a state choice')
        disp(tb_valpol_iter);
    end

    % Continuation Conditions:
    % 1. if value function convergence criteria reached
    % 2. if policy function variation over iterations is less than
    % threshold
    if (it_iter == (it_maxiter_val + 1))
        bl_vfi_continue = false;
    elseif ((it_iter == it_maxiter_val) || ...
            (ar_val_diff_norm(it_iter) < fl_tol_val) || ...
            (sum(ar_pol_diff_norm(max(1, it_iter-it_tol_pol_nochange):it_iter)) < fl_tol_pol))
        % Fix to max, run again to save results if needed
        it_iter_last = it_iter;
        it_iter = it_maxiter_val;
    end
end

% End Timer
if (bl_time)
    toc;
end

% End Profile
if (bl_profile)
    profile off
    profile viewer
    st_file_name = [st_profile_prefix st_profile_name_main st_profile_suffix];
    profsave(profile('info'), strcat(st_profile_path, st_file_name));
end
Elapsed time is 612.536238 seconds.

Process Optimal Choices

result_map = containers.Map('KeyType','char', 'ValueType','any');
result_map('mt_val') = mt_val;

result_map('cl_mt_val') = {mt_val, zeros(1)};
result_map('cl_mt_coh') = {f_coh(ar_z_wage_mesh_r_borr, ar_a'), zeros(1)};
result_map('cl_mt_pol_a') = {f_awithr_to_anor(ar_z_r_borr_mesh_wage, mt_pol_a), zeros(1)};
result_map('cl_mt_pol_c') = {f_cons_checkcmin(ar_z_r_borr_mesh_wage, ar_z_wage_mesh_r_borr, ar_a', mt_pol_a), zeros(1)};
result_map('ar_st_pol_names') = ["cl_mt_val", "cl_mt_pol_a", "cl_mt_coh", "cl_mt_pol_c"];

if (bl_post)
    bl_input_override = true;
    result_map('ar_val_diff_norm') = ar_val_diff_norm(1:it_iter_last);
    result_map('ar_pol_diff_norm') = ar_pol_diff_norm(1:it_iter_last);
    result_map('mt_pol_perc_change') = mt_pol_perc_change(1:it_iter_last, :);
    result_map = ff_az_vf_post(param_map, support_map, armt_map, func_map, result_map, bl_input_override);
end
valgap = norm(mt_val - mt_val_cur): value function difference across iterations
polgap = norm(mt_pol_a - mt_pol_a_cur): policy function difference across iterations
z1 = z1 perc change: sum((mt_pol_a ~= mt_pol_a_cur))/(it_a_n): percentage of state space points conditional on shock where the policy function is changing across iterations
                valgap     polgap    zi1_zr_0_025_zw_0_33942    zi2_zr_0_025_zw_0_5596    zi3_zr_0_025_zw_0_92263    zi4_zr_0_025_zw_1_5212    zi5_zr_0_025_zw_2_508    zi6_zr_0_06_zw_0_33942    zi7_zr_0_06_zw_0_5596    zi8_zr_0_06_zw_0_92263    zi9_zr_0_06_zw_1_5212    zi10_zr_0_06_zw_2_508    zi11_zr_0_095_zw_0_33942    zi12_zr_0_095_zw_0_5596    zi13_zr_0_095_zw_0_92263    zi14_zr_0_095_zw_1_5212    zi15_zr_0_095_zw_2_508
               ________    ______    _______________________    ______________________    _______________________    ______________________    _____________________    ______________________    _____________________    ______________________    _____________________    _____________________    ________________________    _______________________    ________________________    _______________________    ______________________

    iter=1       65.172    633.35                  1                          1                         1                          1                        1                         1                        1                         1                        1                        1                          1                          1                           1                          1                          1       
    iter=2       55.481    697.53            0.98667                          1                   0.98667                          1                        1                   0.98667                  0.98667                         1                  0.98667                        1                    0.97333                       0.96                     0.97333                    0.98667                          1       
    iter=3       42.718    222.16               0.96                    0.97333                      0.96                    0.97333                        1                   0.97333                     0.96                   0.98667                  0.97333                  0.97333                    0.94667                    0.97333                        0.96                       0.96                    0.98667       
    iter=4       34.252     114.2               0.92                    0.94667                   0.93333                    0.94667                     0.96                      0.92                     0.92                   0.93333                  0.93333                     0.96                    0.90667                    0.94667                        0.92                    0.94667                       0.96       
    iter=5       28.491    65.961            0.86667                       0.88                      0.84                       0.88                  0.89333                   0.85333                  0.85333                      0.88                     0.88                  0.90667                    0.86667                       0.88                     0.85333                       0.88                       0.92       
    iter=6       24.283     45.62            0.82667                    0.77333                   0.77333                    0.81333                  0.85333                   0.82667                  0.77333                   0.78667                  0.81333                  0.86667                        0.8                    0.77333                        0.76                    0.82667                    0.82667       
    iter=7       20.984    33.283               0.72                    0.70667                   0.70667                       0.72                  0.78667                      0.72                  0.74667                   0.70667                  0.69333                  0.77333                    0.74667                       0.72                        0.72                       0.72                       0.76       
    iter=8       18.295    24.642            0.61333                    0.62667                       0.6                       0.64                  0.65333                       0.6                      0.6                   0.62667                  0.61333                  0.69333                        0.6                    0.61333                         0.6                    0.61333                    0.66667       
    iter=9       16.057     22.22               0.44                    0.49333                      0.48                    0.50667                     0.56                      0.44                     0.48                   0.49333                  0.49333                     0.56                    0.45333                       0.48                        0.48                    0.50667                    0.54667       
    iter=10       14.16    16.068               0.36                    0.42667                   0.42667                    0.41333                  0.45333                   0.38667                  0.41333                       0.4                  0.38667                     0.48                    0.38667                    0.41333                         0.4                    0.41333                    0.45333       
    iter=11      12.527    12.742               0.28                       0.28                   0.29333                       0.32                  0.37333                   0.30667                  0.26667                   0.30667                  0.33333                  0.38667                    0.29333                       0.28                     0.30667                    0.34667                    0.38667       
    iter=12      11.082     11.61            0.25333                    0.26667                   0.26667                    0.26667                  0.29333                      0.24                     0.24                      0.28                  0.22667                  0.29333                       0.24                       0.24                        0.28                    0.22667                    0.26667       
    iter=13      9.7712     10.41            0.22667                    0.21333                   0.22667                       0.24                     0.24                   0.22667                      0.2                   0.21333                  0.22667                  0.25333                    0.22667                    0.21333                     0.21333                    0.22667                    0.22667       
    iter=14      8.6441    9.1353            0.18667                    0.21333                       0.2                    0.17333                  0.22667                   0.13333                  0.21333                   0.21333                  0.17333                  0.22667                    0.14667                    0.18667                     0.22667                       0.16                    0.21333       
    iter=15      7.6535    9.3668            0.18667                    0.13333                      0.16                    0.18667                  0.18667                   0.17333                  0.14667                   0.18667                  0.18667                      0.2                    0.17333                    0.13333                     0.18667                    0.18667                    0.17333       
    iter=16      6.7818    7.5957               0.12                    0.13333                      0.16                    0.13333                  0.17333                  0.093333                  0.13333                   0.14667                  0.14667                  0.18667                   0.093333                    0.10667                     0.14667                       0.12                       0.16       
    iter=17      6.0432    7.3872               0.12                    0.10667                      0.12                    0.13333                     0.12                   0.10667                  0.13333                      0.12                     0.12                     0.12                    0.10667                    0.10667                     0.13333                    0.10667                       0.12       
    iter=18      5.4139    7.8713           0.093333                    0.14667                  0.093333                    0.10667                  0.14667                  0.093333                  0.14667                      0.08                     0.12                     0.16                    0.10667                    0.13333                        0.08                       0.12                    0.14667       
    iter=19      4.8819    6.9847            0.10667                       0.08                      0.08                    0.10667                     0.08                   0.10667                 0.093333                      0.08                     0.12                 0.093333                       0.12                   0.066667                        0.08                   0.093333                       0.12       
    iter=20      4.4358    6.0895           0.053333                       0.08                  0.066667                    0.10667                 0.093333                  0.053333                 0.093333                  0.066667                 0.093333                 0.093333                   0.053333                   0.093333                        0.08                   0.093333                    0.10667       
    iter=21       4.038    4.3313           0.026667                       0.08                  0.093333                   0.053333                 0.053333                      0.04                     0.08                  0.093333                 0.066667                 0.053333                       0.04                   0.066667                    0.093333                       0.04                   0.066667       
    iter=22       3.676    5.8123           0.093333                   0.053333                  0.066667                   0.066667                 0.093333                  0.066667                 0.066667                      0.08                 0.066667                  0.10667                       0.08                   0.053333                    0.066667                    0.10667                   0.093333       
    iter=23        3.35    4.5998               0.04                   0.053333                  0.093333                   0.053333                 0.066667                  0.053333                     0.04                  0.093333                 0.066667                 0.066667                       0.04                       0.04                    0.093333                   0.066667                       0.08       
    iter=24      3.0579    4.0683               0.04                   0.053333                      0.04                       0.04                 0.053333                      0.04                 0.053333                      0.04                     0.04                 0.053333                       0.04                   0.053333                        0.04                       0.04                   0.053333       
    iter=25      2.8022    4.4267               0.04                   0.053333                  0.026667                   0.053333                 0.053333                  0.053333                 0.053333                  0.026667                 0.053333                 0.066667                       0.04                   0.066667                    0.026667                   0.053333                   0.053333       
    iter=26      2.5729    3.5821               0.04                       0.04                  0.026667                       0.04                 0.053333                      0.04                 0.053333                  0.026667                     0.04                 0.053333                       0.04                       0.04                        0.04                   0.053333                   0.053333       
    iter=27      2.3659    3.3645               0.04                       0.04                  0.026667                       0.04                 0.053333                      0.04                     0.04                  0.026667                     0.04                     0.04                       0.04                       0.04                    0.026667                       0.04                   0.053333       
    iter=28      2.1864    3.4895           0.026667                          0                      0.04                   0.053333                 0.013333                  0.026667                        0                      0.04                 0.053333                 0.013333                   0.026667                   0.013333                        0.04                   0.053333                   0.013333       
    iter=29      2.0277    2.9335               0.04                   0.026667                  0.026667                   0.013333                 0.026667                      0.04                 0.026667                  0.026667                 0.013333                 0.026667                   0.053333                   0.026667                    0.026667                   0.013333                   0.026667       
    iter=30      1.8867    2.3488           0.013333                   0.013333                  0.026667                   0.026667                 0.026667                  0.013333                 0.013333                  0.026667                 0.026667                 0.026667                   0.013333                   0.013333                    0.026667                   0.026667                   0.026667       
    iter=31      1.7625    2.3488           0.026667                   0.026667                  0.026667                   0.013333                 0.013333                  0.026667                 0.026667                  0.026667                 0.013333                 0.013333                   0.026667                   0.026667                    0.026667                   0.013333                   0.013333       
    iter=32      1.6519    2.3488                  0                   0.013333                  0.013333                   0.013333                 0.026667                         0                 0.013333                  0.013333                 0.013333                 0.026667                          0                   0.013333                    0.013333                   0.013333                   0.026667       
    iter=33      1.5532    2.8767           0.013333                   0.013333                      0.04                   0.013333                 0.026667                  0.013333                 0.013333                      0.04                 0.013333                 0.026667                   0.013333                   0.013333                        0.04                   0.013333                   0.026667       
    iter=34      1.4635    2.3488           0.026667                   0.013333                  0.026667                   0.026667                 0.013333                  0.026667                 0.013333                  0.026667                 0.026667                 0.013333                   0.026667                   0.013333                    0.026667                   0.026667                   0.013333       
    iter=35      1.3818    2.6873                  0                   0.026667                  0.013333                   0.013333                 0.013333                         0                 0.026667                  0.013333                 0.013333                 0.013333                          0                   0.026667                    0.013333                   0.013333                   0.013333       
    iter=36      1.3067    1.6609           0.013333                   0.013333                  0.013333                          0                 0.013333                  0.013333                 0.013333                  0.013333                        0                 0.013333                   0.013333                   0.013333                    0.013333                          0                   0.013333       
    iter=37      1.2375    2.3488           0.013333                          0                  0.013333                   0.026667                 0.026667                  0.013333                        0                  0.013333                 0.026667                 0.026667                   0.013333                          0                    0.013333                   0.026667                   0.026667       
    iter=38      1.1731    2.3488                  0                   0.026667                  0.026667                   0.013333                        0                         0                 0.026667                  0.026667                 0.013333                        0                          0                   0.026667                    0.026667                   0.013333                          0       
    iter=39      1.1133    2.3488           0.013333                          0                         0                          0                 0.026667                  0.013333                        0                         0                        0                 0.026667                   0.013333                          0                           0                          0                   0.026667       
    iter=40      1.0572    2.3488                  0                   0.026667                  0.026667                   0.013333                 0.013333                         0                 0.013333                  0.026667                 0.013333                 0.013333                          0                   0.013333                    0.026667                   0.013333                   0.013333       
    iter=41      1.0048    2.3488           0.013333                          0                  0.013333                   0.026667                        0                  0.013333                        0                  0.013333                 0.026667                        0                   0.013333                          0                    0.013333                   0.026667                          0       
    iter=42     0.95538    2.3488           0.013333                   0.013333                  0.013333                          0                 0.026667                  0.013333                 0.013333                  0.013333                        0                 0.026667                   0.013333                   0.013333                    0.013333                          0                   0.026667       
    iter=43     0.90874    2.3488           0.013333                   0.026667                         0                   0.013333                        0                  0.013333                 0.026667                         0                 0.013333                        0                   0.013333                   0.026667                           0                   0.013333                          0       
    iter=44     0.86469    1.6609                  0                          0                  0.013333                          0                 0.013333                         0                        0                  0.013333                        0                 0.013333                          0                          0                    0.013333                          0                   0.013333       
    iter=45     0.82291    1.6609           0.013333                   0.013333                         0                   0.013333                 0.013333                  0.013333                 0.013333                         0                 0.013333                 0.013333                   0.013333                   0.013333                           0                   0.013333                   0.013333       
    iter=46     0.78326    1.6609                  0                          0                         0                          0                 0.013333                         0                        0                         0                        0                 0.013333                          0                          0                           0                          0                   0.013333       
    iter=47     0.74564    1.6609                  0                   0.013333                         0                   0.013333                        0                         0                 0.013333                         0                 0.013333                        0                          0                   0.013333                           0                   0.013333                          0       
    iter=48     0.70974    1.6609           0.013333                          0                  0.013333                          0                 0.013333                  0.013333                        0                  0.013333                        0                 0.013333                   0.013333                          0                    0.013333                          0                   0.013333       
    iter=49     0.67549    1.6609                  0                          0                         0                   0.013333                        0                         0                        0                         0                 0.013333                        0                          0                          0                           0                   0.013333                          0       
    iter=50     0.64274    1.6609                  0                   0.013333                         0                          0                 0.013333                         0                 0.013333                         0                        0                 0.013333                          0                   0.013333                           0                          0                   0.013333       
    iter=51     0.61134    1.6609           0.013333                          0                         0                   0.013333                 0.013333                  0.013333                        0                         0                 0.013333                 0.013333                   0.013333                          0                           0                   0.013333                   0.013333       
    iter=52     0.58127    1.6609                  0                          0                         0                          0                 0.013333                         0                        0                         0                        0                 0.013333                          0                          0                           0                          0                   0.013333       
    iter=53     0.55243    1.6609                  0                   0.013333                         0                          0                        0                         0                 0.013333                         0                        0                        0                          0                   0.013333                           0                          0                          0       
    iter=54     0.52475    1.6609                  0                          0                         0                   0.013333                        0                         0                        0                         0                 0.013333                        0                          0                          0                           0                   0.013333                          0       
    iter=55     0.49815    1.6609           0.013333                          0                         0                          0                 0.013333                  0.013333                        0                         0                        0                 0.013333                   0.013333                          0                           0                          0                   0.013333       
    iter=56     0.47262    1.6609                  0                   0.013333                         0                          0                        0                         0                 0.013333                         0                        0                        0                          0                   0.013333                           0                          0                          0       
    iter=57     0.44812         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=58     0.42461    1.6609                  0                          0                  0.013333                   0.013333                        0                         0                        0                  0.013333                 0.013333                        0                          0                          0                    0.013333                   0.013333                          0       
    iter=59     0.40209    1.6609                  0                          0                         0                          0                 0.013333                         0                        0                         0                        0                 0.013333                          0                          0                           0                          0                   0.013333       
    iter=60     0.38054         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=61     0.35996         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=62      0.3403    1.6609                  0                   0.013333                         0                          0                        0                         0                 0.013333                         0                        0                        0                          0                   0.013333                           0                          0                          0       
    iter=63     0.32156    2.3488                  0                          0                         0                   0.026667                        0                         0                        0                         0                 0.026667                        0                          0                          0                           0                   0.026667                          0       
    iter=64     0.30371         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=65     0.28674         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=66     0.27061         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=67     0.25531         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=68      0.2408         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=69     0.22704         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=70     0.21402         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=71      0.2017         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=72     0.19006         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=73     0.17904         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=74     0.16864         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=75     0.15882         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=76     0.14955         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=77      0.1408         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=78     0.13254         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=79     0.12476         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=80     0.11742         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=81      0.1105         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=82     0.10398         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=83    0.097838         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=84    0.092051         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=85    0.086601         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=86    0.081469         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=87    0.076637         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=88    0.072088         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       
    iter=89    0.067805         0                  0                          0                         0                          0                        0                         0                        0                         0                        0                        0                          0                          0                           0                          0                          0       

tb_val: V(a,z) value at each state space point
                     zi1_zr_0_025_zw_0_33942    zi2_zr_0_025_zw_0_5596    zi3_zr_0_025_zw_0_92263    zi4_zr_0_025_zw_1_5212    zi5_zr_0_025_zw_2_508    zi6_zr_0_06_zw_0_33942    zi7_zr_0_06_zw_0_5596    zi8_zr_0_06_zw_0_92263    zi9_zr_0_06_zw_1_5212    zi10_zr_0_06_zw_2_508    zi11_zr_0_095_zw_0_33942    zi12_zr_0_095_zw_0_5596    zi13_zr_0_095_zw_0_92263    zi14_zr_0_095_zw_1_5212    zi15_zr_0_095_zw_2_508
                     _______________________    ______________________    _______________________    ______________________    _____________________    ______________________    _____________________    ______________________    _____________________    _____________________    ________________________    _______________________    ________________________    _______________________    ______________________

    a1=-20                   -18.809                    -17.407                   -15.294                    -13.317                  -10.304                   -18.809                  -17.407                   -15.849                  -13.876                  -10.785                    -18.809                    -17.407                    -15.849                     -14.415                     -11.392      
    a2=-19.0411              -17.247                    -15.914                   -14.447                    -12.632                  -9.4616                   -18.809                  -16.793                   -14.884                  -13.134                  -9.9416                    -18.809                    -17.407                    -15.849                     -13.509                     -10.541      
    a3=-18.0822               -16.61                    -15.444                   -13.965                    -11.825                  -8.6469                   -16.956                  -15.711                   -14.318                  -12.276                  -9.0896                    -17.691                    -16.188                    -14.604                     -12.775                     -9.6088      
    a4=-17.1233              -15.994                    -14.654                   -13.175                    -11.103                  -7.8244                     -16.5                  -15.331                   -13.533                  -11.508                  -8.2666                    -16.738                    -15.546                    -14.204                     -11.925                     -8.7072      
    a5=-16.1644              -15.256                     -14.09                   -12.604                    -10.343                  -6.9898                    -15.68                  -14.429                   -12.998                  -10.706                  -7.4369                    -16.204                    -14.791                    -13.257                     -11.164                     -7.8125      
    a6=-15.2055               -14.61                     -13.33                   -11.863                    -9.6209                   -6.103                   -14.932                  -13.856                   -12.156                  -9.9467                  -6.5962                    -15.379                    -14.172                    -12.678                     -10.369                     -6.9173      
    a7=-14.2466                -13.9                    -12.728                   -11.215                    -8.8908                  -5.1663                   -14.277                  -13.089                   -11.515                  -9.1821                   -5.704                    -14.661                    -13.365                    -11.885                     -9.6156                     -5.9788      
    a8=-13.2877              -13.179                    -11.973                   -10.477                     -8.081                  -4.2372                    -13.56                  -12.383                   -10.756                  -8.3719                  -4.7351                     -13.89                    -12.721                    -11.128                     -8.7864                     -4.9977      
    a9=-12.3288              -12.481                    -11.315                   -9.6894                    -7.1681                    -3.31                   -12.832                  -11.676                   -10.014                  -7.4583                  -3.7326                    -13.116                    -11.929                    -10.329                      -7.804                     -4.0304      
    a10=-11.3699             -11.737                    -10.569                   -8.8809                    -6.2673                  -2.3639                   -12.128                  -10.886                   -9.2232                  -6.5477                  -2.7212                    -12.373                    -11.238                     -9.491                     -6.8365                     -3.0699      
    a11=-10.411              -11.034                    -9.8619                   -8.0584                    -5.3751                  -1.4136                   -11.378                  -10.139                    -8.412                  -5.6496                  -1.7138                    -11.589                    -10.462                    -8.6392                     -5.8908                     -2.0948      
    a12=-9.45205             -10.287                    -9.1078                    -7.233                    -4.4969                 -0.46125                   -10.644                  -9.3495                   -7.5873                  -4.7604                 -0.71103                    -10.853                    -9.7065                    -7.7797                     -4.9613                     -1.0642      
    a13=-8.49315             -9.5665                    -8.3601                    -6.404                    -3.6155                  0.47175                   -9.8705                  -8.5691                   -6.7602                  -3.8855                  0.26661                    -10.075                    -8.8589                    -6.9224                     -4.0518                  -0.0023497      
    a14=-7.53425             -8.7964                    -7.5605                   -5.5635                    -2.6411                   1.4272                   -9.0536                  -7.7398                   -5.8632                  -2.9911                   1.2618                    -9.3275                    -7.9745                    -6.0658                     -3.1441                      1.0589      
    a15=-6.57534             -8.0216                    -6.7326                   -4.7003                     -1.479                   2.2706                    -8.237                  -6.8847                   -4.9444                  -1.7541                   2.1408                    -8.5285                    -7.0737                    -5.2012                     -2.1371                      1.9907      
    a16=-5.61644             -7.2167                    -5.8886                   -3.6317                   -0.38738                   3.0307                   -7.3947                  -6.0157                   -3.8272                 -0.59839                    2.933                    -7.6203                    -6.1665                    -4.0756                    -0.86567                      2.8257      
    a17=-4.65753             -6.3557                    -4.9828                   -2.4737                    0.60261                   3.6916                      -6.5                  -5.0869                    -2.626                  0.44692                   3.6228                    -6.6725                    -5.2052                    -2.8069                     0.26485                      3.5507      
    a18=-3.69863             -5.4505                    -3.9496                   -1.3854                     1.5011                   4.2901                   -5.5643                  -4.0925                   -1.4994                   1.3935                   4.2476                    -5.6934                    -4.1831                    -1.6269                       1.276                      4.2048      
    a19=-2.73973             -4.5039                    -2.9184                  -0.36632                     2.3172                   4.7864                   -4.5899                  -3.0739                  -0.44614                   2.2517                   4.7676                    -4.6831                    -3.1405                    -0.5308                      2.1844                      4.7495      
    a20=-1.78082             -3.5319                    -1.9048                   0.58667                     3.0549                   5.2448                   -3.5927                  -2.0159                   0.53752                   3.0266                   5.2448                    -3.6558                    -2.1048                    0.48783                      2.9989                      5.2448      
    a21=-0.821918              -2.55                   -0.72707                    1.5625                     3.6999                   5.6293                   -2.5878                 -0.77398                    1.5409                   3.6999                   5.6293                    -2.6255                   -0.82049                     1.5201                      3.6999                      5.6293      
    a22=0                    -1.5699                    0.22988                    2.3102                     4.1611                    5.955                   -1.5894                  0.22988                    2.3102                   4.1611                    5.955                    -1.6083                    0.22988                     2.3102                      4.1611                       5.955      
    a23=0.136986             -1.4544                    0.42786                    2.4167                     4.2142                   5.9967                   -1.4544                  0.42786                    2.4167                   4.2142                   5.9967                    -1.4544                    0.42786                     2.4167                      4.2142                      5.9967      
    a24=1.09589             -0.34206                     1.2141                    2.9567                     4.6541                   6.3419                  -0.34206                   1.2141                    2.9567                   4.6541                   6.3419                   -0.34206                     1.2141                     2.9567                      4.6541                      6.3419      
    a25=2.05479              0.59383                     1.9235                    3.4527                     5.0613                   6.6701                   0.59383                   1.9235                    3.4527                   5.0613                   6.6701                    0.59383                     1.9235                     3.4527                      5.0613                      6.6701      
    a26=3.0137                1.4012                     2.5638                    3.9091                     5.4394                   6.9875                    1.4012                   2.5638                    3.9091                   5.4394                   6.9875                     1.4012                     2.5638                     3.9091                      5.4394                      6.9875      
    a27=3.9726                2.1099                      3.143                    4.3505                     5.8055                   7.2941                    2.1099                    3.143                    4.3505                   5.8055                   7.2941                     2.1099                      3.143                     4.3505                      5.8055                      7.2941      
    a28=4.93151               2.7399                      3.673                    4.7786                     6.1605                   7.5891                    2.7399                    3.673                    4.7786                   6.1605                   7.5891                     2.7399                      3.673                     4.7786                      6.1605                      7.5891      
    a29=5.89041               3.3067                     4.1629                    5.1908                     6.5027                   7.8724                    3.3067                   4.1629                    5.1908                   6.5027                   7.8724                     3.3067                     4.1629                     5.1908                      6.5027                      7.8724      
    a30=6.84932               3.8228                     4.6194                    5.5859                     6.8313                   8.1439                    3.8228                   4.6194                    5.5859                   6.8313                   8.1439                     3.8228                     4.6194                     5.5859                      6.8313                      8.1439      
    a31=7.80822               4.2974                     5.0471                    5.9635                     7.1462                   8.4041                    4.2974                   5.0471                    5.9635                   7.1462                   8.4041                     4.2974                     5.0471                     5.9635                      7.1462                      8.4041      
    a32=8.76712               4.7375                     5.4495                    6.3238                     7.4474                   8.6534                    4.7375                   5.4495                    6.3238                   7.4474                   8.6534                     4.7375                     5.4495                     6.3238                      7.4474                      8.6534      
    a33=9.72603               5.1485                     5.8292                    6.6674                     7.7355                   8.8924                    5.1485                   5.8292                    6.6674                   7.7355                   8.8924                     5.1485                     5.8292                     6.6674                      7.7355                      8.8924      
    a34=10.6849               5.5343                     6.1885                    6.9948                      8.011                   9.1216                    5.5343                   6.1885                    6.9948                    8.011                   9.1216                     5.5343                     6.1885                     6.9948                       8.011                      9.1216      
    a35=11.6438               5.8979                     6.5292                    7.3069                     8.2745                   9.3417                    5.8979                   6.5292                    7.3069                   8.2745                   9.3417                     5.8979                     6.5292                     7.3069                      8.2745                      9.3417      
    a36=12.6027               6.2419                     6.8528                    7.6046                     8.5267                   9.5533                    6.2419                   6.8528                    7.6046                   8.5267                   9.5533                     6.2419                     6.8528                     7.6046                      8.5267                      9.5533      
    a37=13.5616               6.5681                     7.1607                    7.8886                     8.7682                    9.757                    6.5681                   7.1607                    7.8886                   8.7682                    9.757                     6.5681                     7.1607                     7.8886                      8.7682                       9.757      
    a38=14.5205               6.8782                      7.454                    8.1597                     8.9997                   9.9537                    6.8782                    7.454                    8.1597                   8.9997                   9.9537                     6.8782                      7.454                     8.1597                      8.9997                      9.9537      
    a39=15.4795               7.1804                     7.7337                    8.4189                      9.222                   10.144                    7.1804                   7.7337                    8.4189                    9.222                   10.144                     7.1804                     7.7337                     8.4189                       9.222                      10.144      
    a40=16.4384               7.4708                     8.0011                    8.6668                     9.4355                    10.33                    7.4708                   8.0011                    8.6668                   9.4355                    10.33                     7.4708                     8.0011                     8.6668                      9.4355                       10.33      
    a41=17.3973               7.7515                      8.257                    8.9041                     9.6412                   10.512                    7.7515                    8.257                    8.9041                   9.6412                   10.512                     7.7515                      8.257                     8.9041                      9.6412                      10.512      
    a42=18.3562               8.0209                     8.5023                    9.1316                      9.844                   10.691                    8.0209                   8.5023                    9.1316                    9.844                   10.691                     8.0209                     8.5023                     9.1316                       9.844                      10.691      
    a43=19.3151               8.2805                      8.739                    9.3505                     10.043                   10.865                    8.2805                    8.739                    9.3505                   10.043                   10.865                     8.2805                      8.739                     9.3505                      10.043                      10.865      
    a44=20.274                8.5296                      8.971                    9.5616                     10.237                   11.035                    8.5296                    8.971                    9.5616                   10.237                   11.035                     8.5296                      8.971                     9.5616                      10.237                      11.035      
    a45=21.2329               8.7698                     9.1949                    9.7659                     10.426                   11.202                    8.7698                   9.1949                    9.7659                   10.426                   11.202                     8.7698                     9.1949                     9.7659                      10.426                      11.202      
    a46=22.1918               9.0024                     9.4134                    9.9638                     10.609                   11.366                    9.0024                   9.4134                    9.9638                   10.609                   11.366                     9.0024                     9.4134                     9.9638                      10.609                      11.366      
    a47=23.1507               9.2268                     9.6244                    10.156                     10.788                   11.528                    9.2268                   9.6244                    10.156                   10.788                   11.528                     9.2268                     9.6244                     10.156                      10.788                      11.528      
    a48=24.1096               9.4449                     9.8301                    10.342                     10.962                   11.685                    9.4449                   9.8301                    10.342                   10.962                   11.685                     9.4449                     9.8301                     10.342                      10.962                      11.685      
    a49=25.0685               9.6555                     10.029                    10.523                     11.131                   11.838                    9.6555                   10.029                    10.523                   11.131                   11.838                     9.6555                     10.029                     10.523                      11.131                      11.838      
    a50=26.0274               9.8604                     10.223                    10.699                     11.295                   11.987                    9.8604                   10.223                    10.699                   11.295                   11.987                     9.8604                     10.223                     10.699                      11.295                      11.987      
    a51=26.9863               10.058                      10.41                     10.87                     11.455                   12.132                    10.058                    10.41                     10.87                   11.455                   12.132                     10.058                      10.41                      10.87                      11.455                      12.132      
    a52=27.9452               10.251                     10.593                    11.037                     11.611                   12.274                    10.251                   10.593                    11.037                   11.611                   12.274                     10.251                     10.593                     11.037                      11.611                      12.274      
    a53=28.9041               10.438                      10.77                      11.2                     11.763                   12.411                    10.438                    10.77                      11.2                   11.763                   12.411                     10.438                      10.77                       11.2                      11.763                      12.411      
    a54=29.863                 10.62                     10.943                    11.361                     11.911                   12.546                     10.62                   10.943                    11.361                   11.911                   12.546                      10.62                     10.943                     11.361                      11.911                      12.546      
    a55=30.8219               10.796                     11.111                    11.518                     12.055                   12.678                    10.796                   11.111                    11.518                   12.055                   12.678                     10.796                     11.111                     11.518                      12.055                      12.678      
    a56=31.7808               10.968                     11.275                    11.672                     12.197                   12.807                    10.968                   11.275                    11.672                   12.197                   12.807                     10.968                     11.275                     11.672                      12.197                      12.807      
    a57=32.7397               11.135                     11.434                    11.822                     12.335                   12.933                    11.135                   11.434                    11.822                   12.335                   12.933                     11.135                     11.434                     11.822                      12.335                      12.933      
    a58=33.6986               11.298                      11.59                     11.97                     12.471                   13.056                    11.298                    11.59                     11.97                   12.471                   13.056                     11.298                      11.59                      11.97                      12.471                      13.056      
    a59=34.6575               11.456                     11.742                    12.114                     12.603                   13.178                    11.456                   11.742                    12.114                   12.603                   13.178                     11.456                     11.742                     12.114                      12.603                      13.178      
    a60=35.6164                11.61                     11.891                    12.255                     12.734                   13.297                     11.61                   11.891                    12.255                   12.734                   13.297                      11.61                     11.891                     12.255                      12.734                      13.297      
    a61=36.5753               11.761                     12.036                    12.394                     12.861                   13.413                    11.761                   12.036                    12.394                   12.861                   13.413                     11.761                     12.036                     12.394                      12.861                      13.413      
    a62=37.5342               11.908                     12.178                    12.529                     12.986                   13.527                    11.908                   12.178                    12.529                   12.986                   13.527                     11.908                     12.178                     12.529                      12.986                      13.527      
    a63=38.4932               12.051                     12.317                    12.662                     13.108                   13.639                    12.051                   12.317                    12.662                   13.108                   13.639                     12.051                     12.317                     12.662                      13.108                      13.639      
    a64=39.4521               12.192                     12.452                    12.791                     13.228                   13.749                    12.192                   12.452                    12.791                   13.228                   13.749                     12.192                     12.452                     12.791                      13.228                      13.749      
    a65=40.411                12.329                     12.585                    12.918                     13.346                   13.857                    12.329                   12.585                    12.918                   13.346                   13.857                     12.329                     12.585                     12.918                      13.346                      13.857      
    a66=41.3699               12.463                     12.715                    13.043                     13.461                   13.962                    12.463                   12.715                    13.043                   13.461                   13.962                     12.463                     12.715                     13.043                      13.461                      13.962      
    a67=42.3288               12.594                     12.842                    13.165                     13.574                   14.066                    12.594                   12.842                    13.165                   13.574                   14.066                     12.594                     12.842                     13.165                      13.574                      14.066      
    a68=43.2877               12.723                     12.967                    13.284                     13.685                   14.168                    12.723                   12.967                    13.284                   13.685                   14.168                     12.723                     12.967                     13.284                      13.685                      14.168      
    a69=44.2466                12.85                     13.088                    13.401                     13.793                   14.268                     12.85                   13.088                    13.401                   13.793                   14.268                      12.85                     13.088                     13.401                      13.793                      14.268      
    a70=45.2055               12.975                     13.208                    13.516                       13.9                   14.365                    12.975                   13.208                    13.516                     13.9                   14.365                     12.975                     13.208                     13.516                        13.9                      14.365      
    a71=46.1644               13.097                     13.325                    13.628                     14.004                   14.462                    13.097                   13.325                    13.628                   14.004                   14.462                     13.097                     13.325                     13.628                      14.004                      14.462      
    a72=47.1233               13.217                      13.44                    13.739                     14.107                   14.556                    13.217                    13.44                    13.739                   14.107                   14.556                     13.217                      13.44                     13.739                      14.107                      14.556      
    a73=48.0822               13.335                     13.552                    13.847                     14.207                   14.649                    13.335                   13.552                    13.847                   14.207                   14.649                     13.335                     13.552                     13.847                      14.207                      14.649      
    a74=49.0411                13.45                     13.663                    13.953                     14.307                   14.741                     13.45                   13.663                    13.953                   14.307                   14.741                      13.45                     13.663                     13.953                      14.307                      14.741      
    a75=50                    13.564                     13.772                    14.056                     14.405                   14.831                    13.564                   13.772                    14.056                   14.405                   14.831                     13.564                     13.772                     14.056                      14.405                      14.831      

tb_pol_a: optimal asset choice for each state space point
                     zi1_zr_0_025_zw_0_33942    zi2_zr_0_025_zw_0_5596    zi3_zr_0_025_zw_0_92263    zi4_zr_0_025_zw_1_5212    zi5_zr_0_025_zw_2_508    zi6_zr_0_06_zw_0_33942    zi7_zr_0_06_zw_0_5596    zi8_zr_0_06_zw_0_92263    zi9_zr_0_06_zw_1_5212    zi10_zr_0_06_zw_2_508    zi11_zr_0_095_zw_0_33942    zi12_zr_0_095_zw_0_5596    zi13_zr_0_095_zw_0_92263    zi14_zr_0_095_zw_1_5212    zi15_zr_0_095_zw_2_508
                     _______________________    ______________________    _______________________    ______________________    _____________________    ______________________    _____________________    ______________________    _____________________    _____________________    ________________________    _______________________    ________________________    _______________________    ______________________

    a1=-20                         0                          0                   -19.512                    -19.512                  -17.641                         0                        0                         0                  -18.868                  -17.963                          0                          0                           0                          0                    -18.265       
    a2=-19.0411              -19.512                    -19.512                   -19.512                    -18.577                  -16.706                         0                  -18.868                   -18.868                  -17.963                  -17.059                          0                          0                           0                    -18.265                    -17.389       
    a3=-18.0822              -19.512                    -19.512                   -17.641                    -17.641                   -15.77                   -18.868                  -18.868                   -18.868                  -17.059                  -16.154                    -18.265                    -18.265                     -18.265                    -17.389                    -15.638       
    a4=-17.1233              -17.641                    -17.641                   -17.641                    -16.706                  -14.835                   -18.868                  -17.059                   -17.059                  -16.154                  -15.249                    -18.265                    -18.265                     -18.265                    -16.513                    -14.762       
    a5=-16.1644              -16.706                    -16.706                    -15.77                     -15.77                  -13.899                   -17.059                  -17.059                   -16.154                  -15.249                  -14.345                    -16.513                    -16.513                     -16.513                    -15.638                    -13.886       
    a6=-15.2055               -15.77                     -15.77                    -15.77                    -14.835                  -12.964                   -16.154                  -16.154                   -15.249                  -14.345                   -13.44                    -15.638                    -15.638                     -14.762                    -14.762                    -13.011       
    a7=-14.2466              -14.835                    -14.835                   -13.899                    -13.899                  -12.028                   -15.249                  -15.249                   -14.345                   -13.44                  -12.536                    -14.762                    -14.762                     -14.762                    -13.886                    -12.135       
    a8=-13.2877              -13.899                    -13.899                   -12.964                    -12.028                  -11.093                   -14.345                   -13.44                    -13.44                  -12.536                  -10.726                    -13.886                    -13.886                     -13.011                    -12.135                    -11.259       
    a9=-12.3288              -12.964                    -12.964                   -12.028                    -11.093                  -10.157                    -13.44                  -12.536                   -12.536                  -11.631                  -9.8217                    -13.011                    -13.011                     -12.135                    -11.259                    -10.383       
    a10=-11.3699             -12.028                    -12.028                   -11.093                    -10.157                  -9.2215                   -12.536                  -11.631                   -11.631                  -10.726                   -8.917                    -12.135                    -12.135                     -11.259                    -10.383                    -9.5077       
    a11=-10.411              -11.093                    -11.093                   -10.157                    -9.2215                   -8.286                   -11.631                  -10.726                   -10.726                  -9.8217                  -8.0124                    -11.259                    -11.259                     -10.383                    -9.5077                     -8.632       
    a12=-9.45205             -10.157                    -10.157                   -9.2215                     -8.286                  -7.3505                   -9.8217                  -9.8217                   -9.8217                   -8.917                  -7.1078                    -10.383                    -9.5077                     -9.5077                     -8.632                    -6.8806       
    a13=-8.49315             -9.2215                    -9.2215                    -8.286                    -7.3505                   -6.415                    -8.917                   -8.917                    -8.917                  -8.0124                  -6.2032                    -9.5077                     -8.632                      -8.632                    -7.7563                    -6.0049       
    a14=-7.53425              -8.286                     -8.286                   -7.3505                     -6.415                  -5.4795                   -8.0124                  -8.0124                   -7.1078                  -6.2032                  -5.2985                     -8.632                    -7.7563                     -7.7563                    -6.8806                    -5.1292       
    a15=-6.57534             -7.3505                    -7.3505                    -6.415                    -5.4795                  -4.5439                   -7.1078                  -7.1078                   -6.2032                  -5.2985                  -4.3939                    -6.8806                    -6.8806                     -6.8806                    -5.1292                    -4.2535       
    a16=-5.61644              -6.415                     -6.415                   -5.4795                    -4.5439                  -3.6084                   -6.2032                  -6.2032                   -5.2985                  -4.3939                  -3.4893                    -6.0049                    -6.0049                     -5.1292                    -4.2535                    -3.3777       
    a17=-4.65753             -5.4795                    -5.4795                   -4.5439                    -3.6084                  -2.6729                   -5.2985                  -5.2985                   -4.3939                  -3.4893                  -2.5846                    -5.1292                    -5.1292                     -4.2535                    -3.3777                     -2.502       
    a18=-3.69863             -4.5439                    -3.6084                   -3.6084                    -2.6729                  -1.7374                   -4.3939                  -4.3939                   -3.4893                  -2.5846                    -1.68                    -4.2535                    -4.2535                     -3.3777                     -2.502                    -1.6263       
    a19=-2.73973             -3.6084                    -2.6729                   -2.6729                    -1.7374                 -0.80187                   -3.4893                  -3.4893                   -2.5846                    -1.68                 -0.77539                    -3.3777                    -3.3777                      -2.502                    -1.6263                   -0.75061       
    a20=-1.78082             -2.6729                    -1.7374                   -1.7374                   -0.80187                        0                   -2.5846                    -1.68                     -1.68                 -0.77539                        0                     -2.502                     -2.502                     -1.6263                   -0.75061                          0       
    a21=-0.821918            -1.7374                   -0.80187                  -0.80187                          0                  0.13365                     -1.68                 -0.77539                  -0.77539                        0                  0.13365                    -1.6263                   -0.75061                    -0.75061                          0                    0.13365       
    a22=0                   -0.80187                          0                         0                    0.13365                   1.0692                  -0.77539                        0                         0                  0.13365                   1.0692                   -0.75061                          0                           0                    0.13365                     1.0692       
    a23=0.136986                   0                          0                   0.13365                    0.13365                   1.0692                         0                        0                   0.13365                  0.13365                   1.0692                          0                          0                     0.13365                    0.13365                     1.0692       
    a24=1.09589              0.13365                    0.13365                    1.0692                     1.0692                   2.0047                   0.13365                  0.13365                    1.0692                   1.0692                   2.0047                    0.13365                    0.13365                      1.0692                     1.0692                     2.0047       
    a25=2.05479               1.0692                     1.0692                    2.0047                     2.0047                   2.9402                    1.0692                   1.0692                    2.0047                   2.0047                   2.9402                     1.0692                     1.0692                      2.0047                     2.0047                     2.9402       
    a26=3.0137                2.0047                     2.0047                    2.9402                     2.9402                   3.8757                    2.0047                   2.0047                    2.9402                   2.9402                   3.8757                     2.0047                     2.0047                      2.9402                     2.9402                     3.8757       
    a27=3.9726                2.9402                     2.9402                    2.9402                     3.8757                   4.8112                    2.9402                   2.9402                    2.9402                   3.8757                   4.8112                     2.9402                     2.9402                      2.9402                     3.8757                     4.8112       
    a28=4.93151               3.8757                     3.8757                    3.8757                     4.8112                   5.7467                    3.8757                   3.8757                    3.8757                   4.8112                   5.7467                     3.8757                     3.8757                      3.8757                     4.8112                     5.7467       
    a29=5.89041               4.8112                     4.8112                    4.8112                     5.7467                   6.6823                    4.8112                   4.8112                    4.8112                   5.7467                   6.6823                     4.8112                     4.8112                      4.8112                     5.7467                     6.6823       
    a30=6.84932               5.7467                     5.7467                    5.7467                     6.6823                   7.6178                    5.7467                   5.7467                    5.7467                   6.6823                   7.6178                     5.7467                     5.7467                      5.7467                     6.6823                     7.6178       
    a31=7.80822               6.6823                     6.6823                    6.6823                     7.6178                   8.5533                    6.6823                   6.6823                    6.6823                   7.6178                   8.5533                     6.6823                     6.6823                      6.6823                     7.6178                     8.5533       
    a32=8.76712               7.6178                     7.6178                    7.6178                     8.5533                   9.4888                    7.6178                   7.6178                    7.6178                   8.5533                   9.4888                     7.6178                     7.6178                      7.6178                     8.5533                     9.4888       
    a33=9.72603               8.5533                     8.5533                    8.5533                     9.4888                   10.424                    8.5533                   8.5533                    8.5533                   9.4888                   10.424                     8.5533                     8.5533                      8.5533                     9.4888                     10.424       
    a34=10.6849               9.4888                     9.4888                    9.4888                     10.424                    11.36                    9.4888                   9.4888                    9.4888                   10.424                    11.36                     9.4888                     9.4888                      9.4888                     10.424                      11.36       
    a35=11.6438               10.424                     10.424                    10.424                      11.36                   12.295                    10.424                   10.424                    10.424                    11.36                   12.295                     10.424                     10.424                      10.424                      11.36                     12.295       
    a36=12.6027                11.36                      11.36                     11.36                     12.295                   13.231                     11.36                    11.36                     11.36                   12.295                   13.231                      11.36                      11.36                       11.36                     12.295                     13.231       
    a37=13.5616               12.295                     12.295                    12.295                     13.231                   14.166                    12.295                   12.295                    12.295                   13.231                   14.166                     12.295                     12.295                      12.295                     13.231                     14.166       
    a38=14.5205               13.231                     13.231                    13.231                     14.166                   15.102                    13.231                   13.231                    13.231                   14.166                   15.102                     13.231                     13.231                      13.231                     14.166                     15.102       
    a39=15.4795               13.231                     14.166                    14.166                     15.102                   16.037                    13.231                   14.166                    14.166                   15.102                   16.037                     13.231                     14.166                      14.166                     15.102                     16.037       
    a40=16.4384               14.166                     15.102                    15.102                     16.037                   16.973                    14.166                   15.102                    15.102                   16.037                   16.973                     14.166                     15.102                      15.102                     16.037                     16.973       
    a41=17.3973               15.102                     16.037                    16.037                     16.973                   17.908                    15.102                   16.037                    16.037                   16.973                   17.908                     15.102                     16.037                      16.037                     16.973                     17.908       
    a42=18.3562               16.037                     16.973                    16.973                     16.973                   18.844                    16.037                   16.973                    16.973                   16.973                   18.844                     16.037                     16.973                      16.973                     16.973                     18.844       
    a43=19.3151               16.973                     16.973                    17.908                     17.908                   19.779                    16.973                   16.973                    17.908                   17.908                   19.779                     16.973                     16.973                      17.908                     17.908                     19.779       
    a44=20.274                17.908                     17.908                    18.844                     18.844                   20.715                    17.908                   17.908                    18.844                   18.844                   20.715                     17.908                     17.908                      18.844                     18.844                     20.715       
    a45=21.2329               18.844                     18.844                    19.779                     19.779                   21.651                    18.844                   18.844                    19.779                   19.779                   21.651                     18.844                     18.844                      19.779                     19.779                     21.651       
    a46=22.1918               19.779                     19.779                    20.715                     20.715                   21.651                    19.779                   19.779                    20.715                   20.715                   21.651                     19.779                     19.779                      20.715                     20.715                     21.651       
    a47=23.1507               20.715                     20.715                    21.651                     21.651                   22.586                    20.715                   20.715                    21.651                   21.651                   22.586                     20.715                     20.715                      21.651                     21.651                     22.586       
    a48=24.1096               21.651                     21.651                    22.586                     22.586                   23.522                    21.651                   21.651                    22.586                   22.586                   23.522                     21.651                     21.651                      22.586                     22.586                     23.522       
    a49=25.0685               22.586                     22.586                    23.522                     23.522                   24.457                    22.586                   22.586                    23.522                   23.522                   24.457                     22.586                     22.586                      23.522                     23.522                     24.457       
    a50=26.0274               23.522                     23.522                    24.457                     24.457                   25.393                    23.522                   23.522                    24.457                   24.457                   25.393                     23.522                     23.522                      24.457                     24.457                     25.393       
    a51=26.9863               24.457                     24.457                    25.393                     25.393                   26.328                    24.457                   24.457                    25.393                   25.393                   26.328                     24.457                     24.457                      25.393                     25.393                     26.328       
    a52=27.9452               25.393                     25.393                    26.328                     26.328                   27.264                    25.393                   25.393                    26.328                   26.328                   27.264                     25.393                     25.393                      26.328                     26.328                     27.264       
    a53=28.9041               26.328                     26.328                    26.328                     27.264                   28.199                    26.328                   26.328                    26.328                   27.264                   28.199                     26.328                     26.328                      26.328                     27.264                     28.199       
    a54=29.863                27.264                     27.264                    27.264                     28.199                   29.135                    27.264                   27.264                    27.264                   28.199                   29.135                     27.264                     27.264                      27.264                     28.199                     29.135       
    a55=30.8219               28.199                     28.199                    28.199                     29.135                    30.07                    28.199                   28.199                    28.199                   29.135                    30.07                     28.199                     28.199                      28.199                     29.135                      30.07       
    a56=31.7808               29.135                     29.135                    29.135                      30.07                   31.006                    29.135                   29.135                    29.135                    30.07                   31.006                     29.135                     29.135                      29.135                      30.07                     31.006       
    a57=32.7397                30.07                      30.07                     30.07                     31.006                   31.941                     30.07                    30.07                     30.07                   31.006                   31.941                      30.07                      30.07                       30.07                     31.006                     31.941       
    a58=33.6986               31.006                     31.006                    31.006                     31.941                   32.877                    31.006                   31.006                    31.006                   31.941                   32.877                     31.006                     31.006                      31.006                     31.941                     32.877       
    a59=34.6575               31.941                     31.941                    31.941                     32.877                   33.812                    31.941                   31.941                    31.941                   32.877                   33.812                     31.941                     31.941                      31.941                     32.877                     33.812       
    a60=35.6164               32.877                     32.877                    32.877                     33.812                   34.748                    32.877                   32.877                    32.877                   33.812                   34.748                     32.877                     32.877                      32.877                     33.812                     34.748       
    a61=36.5753               33.812                     33.812                    33.812                     34.748                   35.683                    33.812                   33.812                    33.812                   34.748                   35.683                     33.812                     33.812                      33.812                     34.748                     35.683       
    a62=37.5342               34.748                     34.748                    34.748                     35.683                   36.619                    34.748                   34.748                    34.748                   35.683                   36.619                     34.748                     34.748                      34.748                     35.683                     36.619       
    a63=38.4932               35.683                     35.683                    35.683                     36.619                   37.554                    35.683                   35.683                    35.683                   36.619                   37.554                     35.683                     35.683                      35.683                     36.619                     37.554       
    a64=39.4521               36.619                     36.619                    36.619                     37.554                    38.49                    36.619                   36.619                    36.619                   37.554                    38.49                     36.619                     36.619                      36.619                     37.554                      38.49       
    a65=40.411                37.554                     37.554                    37.554                      38.49                   39.425                    37.554                   37.554                    37.554                    38.49                   39.425                     37.554                     37.554                      37.554                      38.49                     39.425       
    a66=41.3699                38.49                      38.49                     38.49                     39.425                   40.361                     38.49                    38.49                     38.49                   39.425                   40.361                      38.49                      38.49                       38.49                     39.425                     40.361       
    a67=42.3288                38.49                     39.425                    39.425                     40.361                   41.296                     38.49                   39.425                    39.425                   40.361                   41.296                      38.49                     39.425                      39.425                     40.361                     41.296       
    a68=43.2877               39.425                     40.361                    40.361                     41.296                   42.232                    39.425                   40.361                    40.361                   41.296                   42.232                     39.425                     40.361                      40.361                     41.296                     42.232       
    a69=44.2466               40.361                     41.296                    41.296                     42.232                   43.167                    40.361                   41.296                    41.296                   42.232                   43.167                     40.361                     41.296                      41.296                     42.232                     43.167       
    a70=45.2055               41.296                     42.232                    42.232                     43.167                   44.103                    41.296                   42.232                    42.232                   43.167                   44.103                     41.296                     42.232                      42.232                     43.167                     44.103       
    a71=46.1644               42.232                     43.167                    43.167                     44.103                   45.038                    42.232                   43.167                    43.167                   44.103                   45.038                     42.232                     43.167                      43.167                     44.103                     45.038       
    a72=47.1233               43.167                     44.103                    44.103                     45.038                   45.974                    43.167                   44.103                    44.103                   45.038                   45.974                     43.167                     44.103                      44.103                     45.038                     45.974       
    a73=48.0822               44.103                     45.038                    45.038                     45.038                   46.909                    44.103                   45.038                    45.038                   45.038                   46.909                     44.103                     45.038                      45.038                     45.038                     46.909       
    a74=49.0411               45.038                     45.038                    45.974                     45.974                   47.845                    45.038                   45.038                    45.974                   45.974                   47.845                     45.038                     45.038                      45.974                     45.974                     47.845       
    a75=50                    45.974                     45.974                    46.909                     46.909                    48.78                    45.974                   45.974                    46.909                   46.909                    48.78                     45.974                     45.974                      46.909                     46.909                      48.78       

end
ans = 

  Map with properties:

        Count: 12
      KeyType: char
    ValueType: any