diff --git a/exo4/molecule_problem.m b/exo4/molecule_problem.m
index 1b7b38f..94953ab 100644
--- a/exo4/molecule_problem.m
+++ b/exo4/molecule_problem.m
@@ -1,302 +1,302 @@
-nTwo = true;
+nTwo = false;
 nThree = false;
 nFour = false;
 nFive = false;
-nSix = false;
+nSix = true;
 nSeven = false;
 
 nEight = false;
 k_lower = 0.5;
 k_high = 3.0;
 k_highest = 7.0;
 
 global kz
 kz = 0;
 
 %%%%%%%%%%%%%
 % test 2,3,4, random configuration
 
 if nTwo
 
 E = [];
 for N = 2:4
     X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
     %X = solve_nonl_CG_num(@(x) f(x), 3*N, []);
     E = [E f(X)]
 
     % remap solution
     X =reshape(X, 3, N)'
 
     visualize_molecule(X, 3, N-1, sprintf('output/molecule%d',N))
 end
 
 figure
 plot(2:4,E)
 title("Case 2,3,4")
 xlabel("N")
 ylabel("E")
 
 end
 
 %%%%%%%%%%
 
 if nThree
 
 N = 5 % , locate minimum
 
 aptempt = 100;
 E = zeros(aptempt,1);
 %N = 5;
 norm_range = [0.1,2.0]
 X = cell(aptempt,1);
 for k = 1:aptempt
     k
     hint = rand(3*5,1);
     hnorm = rand() * (norm_range(2) - norm_range(1)) + norm_range(1)
     hint = hint * hnorm / norm(hint);
     X{k} = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, hint);
     U = f(X{k})
     X{k} = reshape(X{k}, 3, N)';
     E(k) = U;
 end
 
 cases = input('Case to visualise? ');
 
 for i = 1:length(cases)
     visualize_molecule(X{i}, 3, 17 + i, sprintf('output/molecule5_statcase_%d',i))
 end
 
 outcols('output/molecule_stat.out', {'case', 'E'}, [(1:aptempt)', E]);
 
 figure
 plot(1:aptempt,E)
 title("Case N = 5, energy statistics")
 xlabel("case number")
 ylabel("E")
 
 end
 
 %%%%%%%%%%%%%%%%%
 N = 5;
 kz = 0.5;
 
 if nFour || nFive || nSix
 
 hint = [];
 
 if nSix
     kz = 0.5
     hint = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
     % start by tetrahedron
 end
 
 if ~nFour
     kz = 2.0
 end
 
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, hint);
 
 E = f(X)
 
 % remap solution
 X =reshape(X, 3, N)'
 
 visualize_molecule(X, 3, 6, 'output/molecule')
 
 end
 
 if nSeven
     disp("compute hint (2)")
     kz = 0.5;
     hint2 = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
     fprintf("hint2: E = %g\n", f(hint2))
 
     disp("compute hint (3)")
     kz = 2.0;
     hint3 = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
     fprintf("hint3: E = %g\n", f(hint3))
 
     disp("compute hint (4)")
     hint4 = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, hint2);
     fprintf("hint4: E = %g\n", f(hint4))
 
     kz = 0;
     disp("Set kz = 0, compute results using the previous hints")
 
     X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, hint2);
     fprintf("hint2: E = %g\n", f(X))
     X =reshape(X, 3, N)'
     visualize_molecule(X, 3, 7, 'output/molecule_hint2')
 
     X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, hint3);
     fprintf("hint3: E = %g\n", f(X))
     X =reshape(X, 3, N)'
     visualize_molecule(X, 3, 8, 'output/molecule_hint3')
 
     X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, hint4);
     fprintf("hint4: E = %g\n", f(X))
     X =reshape(X, 3, N)'
     visualize_molecule(X, 3, 9, 'output/molecule_hint4')
 end
 
 if nEight
 N = 6;
 kz = 0.0;
 fprintf("N = 6\n\n")
 
 fprintf("Random configuration\n")
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
 E = f(X)
 X =reshape(X, 3, N)'
 visualize_molecule(X, 3, 10, 'output/molecule_rand6')
 
 fprintf("Softly increase energy, kz = %g\n", k_lower)
 kz = k_lower;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
 E = f(X)
 Y =reshape(X, 3, N)'
 visualize_molecule(Y, 3, 11, 'output/molecule_inc6')
 
 fprintf("Decrease energy, kz = 0, set previous as hint\n")
 kz = 0;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, X);
 E = f(X)
 X =reshape(X, 3, N)'
 visualize_molecule(X, 3, 12, 'output/molecule_incdec6')
 
 fprintf("Strongly increase energy, kz = %g\n", k_highest)
 kz = k_highest;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, []);
 E = f(X)
 Y =reshape(X, 3, N)'
 visualize_molecule(Y, 3, 13, 'output/molecule_incstrong6')
 Z = X(:); % copy it
 
 fprintf("Decrease energy, kz = %g, set previous as hint\n", k_high)
 kz = k_high;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, X);
 E = f(X)
 Y =reshape(X, 3, N)'
 visualize_molecule(Y, 3, 14, 'output/molecule_incstrongdec6')
 
 fprintf("Decrease energy, kz = %g, set previous as hint\n", k_lower)
 kz = k_lower;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, X);
 E = f(X)
 Y =reshape(X, 3, N)'
 visualize_molecule(Y, 3, 15, 'output/molecule_incstrongdecdec6')
 
 fprintf("Decrease energy level, kz = 0, set previous as hint\n")
 kz = 0.0;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, X);
 E = f(X)
 Y =reshape(X, 3, N)'
 visualize_molecule(Y, 3, 16, 'output/molecule_softfall6')
 
 
 fprintf("Jump to zero from highest energy level, kz = %g -> 0\n", k_highest)
 kz = 0.0;
 X = solve_nonl_CG(@(x) f(x), @(i,x) grad_f(i,x), @(i,j,x) hess_f(i,j,x), 3*N, Z);
 E = f(X)
 Y =reshape(X, 3, N)'
 visualize_molecule(Y, 3, 17, 'output/molecule_strongfall6')
 
 end
 
 % problem specific function definition
 
 % scalar field, nothing to say
 function y = f(X)
     global kz
     N = fix(length(X)/ 3);
     y = X' * X + kz * X(1:3:N)' * X(1:3:N);
     for i = 1:N
         for j = (i+1):N
             y = y + 1.0 / norm(X(threeV(i)) - X(threeV(j)));
         end
     end
 end
 
 % i is the index w.r.t the gradient of f
 % i can be an array
 % y is returned as a column vector
 function y = grad_f(i, X)
     global kz
     N = length(X)/ 3;
     % compute result as a column vector
     y = zeros(length(i),1);
 
     for arg = 1:length(i)
         [i_i, l] = threeD(i(arg), 3);
 
         y_y = 2 * X(three(i_i, l));
         if l == 3
             y_y = 2 * (1.0 + kz) * X(three(i_i, l));
         end
         for k = 1:N
             if k ~= i_i
                 r = X(threeV(i_i)) - X(threeV(k));
                 y_y = y_y - (r(l) / norm(r)^3);
             end
         end
         y(arg) = y_y;
     end
 end
 
 % i,n are the row and resp. column indexes referring
 % to the hessian matrix
 % i,n can be arrays
 function y = hess_f(i, n, X)
     global kz
     N = length(X) / 3;
     y = zeros(length(i),length(n));
 
     for arg_i = 1:length(i)
         for arg_n = 1:length(n)
 
     [i_i, l] = threeD(i(arg_i), 3);
     [n_n, m] = threeD(n(arg_n), 3);
 
     if i_i == n_n
         y_y = 0;
         for k = 1:N
             if k ~= i_i
                 r = X(threeV(i_i)) - X(threeV(k));
                 y_y = y_y + 3 * r(m) * r(l) / norm(r)^5;
                 if l == m
                     y_y = y_y - 1.0 / norm(r)^3;
                 end
             end
         end
 
         if l == m
             y_y = y_y + 2 * (1.0 + (l == 3) * kz);
         end
     else
         r = X(threeV(i_i)) - X(threeV(n_n));
         y_y = -3 * r(m) * r(l) / norm(r)^5;
         if l == m
             y_y = y_y + 1.0 / norm(r)^3;
         end
     end % if
 
     y(arg_i, arg_n) = y_y;
 
         end % for
     end % for
 end
 
 function [i,j] = threeD(k, m)
     i = fix((k-1) / m) + 1;
     j = mod((k-1), m) + 1;
     % idea: k = m*(i-1) + j
 end
 
 function I = threeV(k)
     I = 3*(k-1)+1:3*k;
 end
 
 % j = 0,1,2
 function k = three(i,j)
     k = 3 * (i-1) + j;
 end
diff --git a/exo4/output/A1CG.out b/exo4/output/A1CG.out
index 662e3f7..cad7bc1 100644
--- a/exo4/output/A1CG.out
+++ b/exo4/output/A1CG.out
@@ -1,12 +1,12 @@
 N  err  fact  
 1  1  1.76797  
-2  0.317076  1.56285  
-3  0.281238  1.38154  
-4  0.198871  1.22126  
-5  0.176138  1.07957  
-6  1.05137e-12  0.954322  
-7  9.89518e-15  0.843605  
-8  1.35109e-15  0.745733  
-9  5.67865e-16  0.659215  
-10  7.8168e-16  0.582736  
-11  1.107e-15  0.515129  
+2  0.484999  1.56285  
+3  0.392753  1.38154  
+4  0.301866  1.22126  
+5  0.218447  1.07957  
+6  1.19354e-12  0.954322  
+7  1.34649e-14  0.843605  
+8  1.27773e-15  0.745733  
+9  9.93177e-16  0.659215  
+10  2.09711e-15  0.582736  
+11  2.12493e-15  0.515129  
diff --git a/exo4/output/A1SD.out b/exo4/output/A1SD.out
index e2c3fe9..01dbd5e 100644
--- a/exo4/output/A1SD.out
+++ b/exo4/output/A1SD.out
@@ -1,3001 +1,3001 @@
 N  err  fact  
 1  1  0.992444  
-2  0.332464  0.984946  
-3  0.263002  0.977504  
-4  0.243833  0.970118  
-5  0.232913  0.962788  
-6  0.225107  0.955514  
-7  0.219183  0.948294  
-8  0.214384  0.941129  
-9  0.21033  0.934019  
-10  0.206774  0.926961  
-11  0.203589  0.919958  
-12  0.200688  0.913007  
-13  0.198019  0.906108  
-14  0.19554  0.899262  
-15  0.193225  0.892468  
-16  0.191049  0.885724  
-17  0.188993  0.879032  
-18  0.187041  0.872391  
-19  0.185179  0.865799  
-20  0.183395  0.859257  
-21  0.181681  0.852765  
-22  0.180026  0.846322  
-23  0.178424  0.839927  
-24  0.176868  0.833581  
-25  0.175354  0.827283  
-26  0.173876  0.821032  
-27  0.17243  0.814829  
-28  0.171014  0.808672  
-29  0.169624  0.802562  
-30  0.168257  0.796498  
-31  0.166912  0.79048  
-32  0.165586  0.784508  
-33  0.164279  0.77858  
-34  0.162989  0.772698  
-35  0.161714  0.766859  
-36  0.160454  0.761065  
-37  0.159208  0.755315  
-38  0.157975  0.749608  
-39  0.156754  0.743944  
-40  0.155545  0.738323  
-41  0.154348  0.732745  
-42  0.153161  0.727208  
-43  0.151986  0.721714  
-44  0.15082  0.716261  
-45  0.149665  0.710849  
-46  0.148519  0.705478  
-47  0.147383  0.700148  
-48  0.146256  0.694858  
-49  0.145138  0.689608  
-50  0.14403  0.684397  
-51  0.14293  0.679226  
-52  0.141839  0.674094  
-53  0.140757  0.669001  
-54  0.139683  0.663946  
-55  0.138618  0.65893  
-56  0.13756  0.653951  
-57  0.136512  0.64901  
-58  0.135471  0.644106  
-59  0.134438  0.63924  
-60  0.133413  0.63441  
-61  0.132396  0.629616  
-62  0.131387  0.624859  
-63  0.130386  0.620138  
-64  0.129393  0.615453  
-65  0.128407  0.610802  
-66  0.127428  0.606187  
-67  0.126457  0.601607  
-68  0.125494  0.597062  
-69  0.124537  0.592551  
-70  0.123589  0.588073  
-71  0.122647  0.58363  
-72  0.121712  0.57922  
-73  0.120785  0.574844  
-74  0.119865  0.570501  
-75  0.118952  0.56619  
-76  0.118045  0.561912  
-77  0.117146  0.557667  
-78  0.116254  0.553453  
-79  0.115368  0.549271  
-80  0.114489  0.545121  
-81  0.113617  0.541003  
-82  0.112751  0.536915  
-83  0.111892  0.532858  
-84  0.11104  0.528832  
-85  0.110194  0.524836  
-86  0.109354  0.520871  
-87  0.108521  0.516935  
-88  0.107694  0.51303  
-89  0.106874  0.509153  
-90  0.10606  0.505306  
-91  0.105252  0.501489  
-92  0.10445  0.497699  
-93  0.103654  0.493939  
-94  0.102865  0.490207  
-95  0.102081  0.486503  
-96  0.101303  0.482827  
-97  0.100531  0.479179  
-98  0.0997655  0.475559  
-99  0.0990055  0.471966  
-100  0.0982512  0.4684  
-101  0.0975027  0.464861  
-102  0.0967599  0.461348  
-103  0.0960228  0.457862  
-104  0.0952912  0.454403  
-105  0.0945653  0.45097  
-106  0.0938448  0.447562  
-107  0.0931299  0.444181  
-108  0.0924204  0.440825  
-109  0.0917163  0.437494  
-110  0.0910176  0.434188  
-111  0.0903242  0.430908  
-112  0.0896361  0.427652  
-113  0.0889532  0.424421  
-114  0.0882755  0.421214  
-115  0.087603  0.418031  
-116  0.0869356  0.414873  
-117  0.0862733  0.411738  
-118  0.0856161  0.408627  
-119  0.0849638  0.40554  
-120  0.0843165  0.402476  
-121  0.0836742  0.399435  
-122  0.0830367  0.396417  
-123  0.0824041  0.393422  
-124  0.0817763  0.390449  
-125  0.0811533  0.387499  
-126  0.0805351  0.384571  
-127  0.0799215  0.381666  
-128  0.0793127  0.378782  
-129  0.0787084  0.37592  
-130  0.0781088  0.37308  
-131  0.0775138  0.370261  
-132  0.0769232  0.367463  
-133  0.0763372  0.364687  
-134  0.0757557  0.361931  
-135  0.0751785  0.359197  
-136  0.0746058  0.356483  
-137  0.0740374  0.353789  
-138  0.0734734  0.351116  
-139  0.0729136  0.348463  
-140  0.0723582  0.34583  
-141  0.0718069  0.343217  
-142  0.0712599  0.340624  
-143  0.070717  0.338051  
-144  0.0701782  0.335496  
-145  0.0696436  0.332961  
-146  0.069113  0.330446  
-147  0.0685865  0.327949  
-148  0.068064  0.325471  
-149  0.0675455  0.323012  
-150  0.0670309  0.320571  
-151  0.0665202  0.318149  
-152  0.0660134  0.315745  
-153  0.0655105  0.31336  
-154  0.0650114  0.310992  
-155  0.0645162  0.308642  
-156  0.0640247  0.30631  
-157  0.0635369  0.303996  
-158  0.0630529  0.301699  
-159  0.0625725  0.29942  
-160  0.0620958  0.297157  
-161  0.0616227  0.294912  
-162  0.0611533  0.292684  
-163  0.0606874  0.290472  
-164  0.0602251  0.288278  
-165  0.0597662  0.2861  
-166  0.0593109  0.283938  
-167  0.0588591  0.281793  
-168  0.0584107  0.279663  
-169  0.0579657  0.27755  
-170  0.0575241  0.275453  
-171  0.0570858  0.273372  
-172  0.0566509  0.271307  
-173  0.0562194  0.269257  
-174  0.0557911  0.267222  
-175  0.055366  0.265203  
-176  0.0549442  0.263199  
-177  0.0545256  0.261211  
-178  0.0541103  0.259237  
-179  0.053698  0.257279  
-180  0.0532889  0.255335  
-181  0.052883  0.253405  
-182  0.0524801  0.251491  
-183  0.0520803  0.249591  
-184  0.0516835  0.247705  
-185  0.0512898  0.245833  
-186  0.050899  0.243976  
-187  0.0505113  0.242132  
-188  0.0501264  0.240303  
-189  0.0497446  0.238487  
-190  0.0493656  0.236685  
-191  0.0489895  0.234897  
-192  0.0486163  0.233122  
-193  0.0482459  0.231361  
-194  0.0478784  0.229613  
-195  0.0475136  0.227878  
-196  0.0471516  0.226156  
-197  0.0467924  0.224447  
-198  0.046436  0.222752  
-199  0.0460822  0.221068  
-200  0.0457311  0.219398  
-201  0.0453827  0.21774  
-202  0.045037  0.216095  
-203  0.0446939  0.214463  
-204  0.0443534  0.212842  
-205  0.0440155  0.211234  
-206  0.0436802  0.209638  
-207  0.0433474  0.208054  
-208  0.0430172  0.206482  
-209  0.0426894  0.204922  
-210  0.0423642  0.203374  
-211  0.0420415  0.201837  
-212  0.0417212  0.200312  
-213  0.0414033  0.198799  
-214  0.0410879  0.197296  
-215  0.0407749  0.195806  
-216  0.0404643  0.194326  
-217  0.040156  0.192858  
-218  0.0398501  0.191401  
-219  0.0395465  0.189955  
-220  0.0392452  0.18852  
-221  0.0389462  0.187095  
-222  0.0386495  0.185681  
-223  0.0383551  0.184279  
-224  0.0380629  0.182886  
-225  0.0377729  0.181504  
-226  0.0374851  0.180133  
-227  0.0371996  0.178772  
-228  0.0369162  0.177421  
-229  0.0366349  0.176081  
-230  0.0363558  0.17475  
-231  0.0360789  0.17343  
-232  0.035804  0.17212  
-233  0.0355312  0.170819  
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 /b4_Inc_state save def         % Save state for cleanup
 /dict_count countdictstack def % Count objects on dict stack
 /op_count count 1 sub def      % Count objects on operand stack
 userdict begin                 % Push userdict on dict stack
 /showpage { } def              % Redefine showpage, { } = null proc
 0 setgray 0 setlinecap         % Prepare graphics state
 1 setlinewidth 0 setlinejoin
 10 setmiterlimit [ ] 0 setdash newpath
 /languagelevel where           % If level not equal to 1 then
 {pop languagelevel             % set strokeadjust and
 1 ne                           % overprint to their defaults.
 {false setstrokeadjust false setoverprint
 } if
 } if
 } bd
 /EndEPSF { %def
 count op_count sub {pop} repeat            % Clean up stacks
 countdictstack dict_count sub {end} repeat
 b4_Inc_state restore
 } bd
 %%EndResource
 %%EndProlog
 %%Page: 1 1
 %%PageBoundingBox: 0 0 1022 487
 %%BeginPageSetup
 [1 0 0 -1 0 487] CT
 %%EndPageSetup
 GS
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 0 0 L
 cp
 clip
 1 GC
 N
 0 0 1362 649 re
 f
 GR
 GS
 [0.48003 0 0 0.48003 0 175.46082] CT
 [1 0 0 1 0 0] CT
 N
 0 -365 M
 2128 -365 L
 2128 649 L
 0 649 L
 0 -365 L
 cp
 clip
 GS
 0 0 translate
 1362 649 scale
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 {{
 /RawData currentfile /ASCII85Decode filter def
 /Data RawData /FlateDecode filter def
 /DeviceRGB setcolorspace
 <<
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   /Decode [0 1 0 1 0 1]
   /DataSource Data
   /Height 649
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   /Width 1362
   /BitsPerComponent 8
 >> image
 } stopped {handleerror} if
   RawData flushfile
 } exec
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diff --git a/exo4/report.aux b/exo4/report.aux
index db8ecb6..7d4dd78 100644
--- a/exo4/report.aux
+++ b/exo4/report.aux
@@ -1,123 +1,123 @@
 \relax 
 \@writefile{toc}{\contentsline {section}{Problem 1}{3}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{(1) Steepest Descent \& Conjugate Gradient}{3}\protected@file@percent }
 \newlabel{solve_SD_debug}{{1}{3}}
-\@writefile{lol}{\contentsline {lstlisting}{\numberline {1}Steepest Descent method implementation}{3}\protected@file@percent }
+\@writefile{lol}{\contentsline {lstlisting}{\numberline {1}Steepest Descent (SD) method implementation}{3}\protected@file@percent }
 \newlabel{solve_CG_debug}{{2}{3}}
-\@writefile{lol}{\contentsline {lstlisting}{\numberline {2}Conjugate Greadient method implementation}{3}\protected@file@percent }
+\@writefile{lol}{\contentsline {lstlisting}{\numberline {2}Conjugate Gradient (CG) method implementation}{3}\protected@file@percent }
 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
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 \newlabel{sub@star}{{a}{3}}
 \newlabel{square}{{1b}{3}}
 \newlabel{sub@square}{{b}{3}}
 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Comparing SD and CG methods for solving linear problem of size $5\times 5$ \relax }}{3}\protected@file@percent }
 \newlabel{A1}{{1}{3}}
 \newlabel{SDCG_thres_def}{{2}{4}}
 \newlabel{SDCG_delta_def}{{3}{4}}
 \newlabel{SDCG_ineq}{{4}{4}}
 \newlabel{star}{{2a}{4}}
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diff --git a/exo4/report.pdf b/exo4/report.pdf
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diff --git a/exo4/report.tex b/exo4/report.tex
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 \begin{document}
 \begin{spacing}{1.1}
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 % Uncomment the \tableofcontents and \newpage lines to get a Contents page
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 \tableofcontents
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 % \clearpage before each homeworkProblem environment
 
 \newpage
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{homeworkProblem}
 
 \begin{homeworkSection}{(1) Steepest Descent \& Conjugate Gradient}
 \end{homeworkSection}
 The steepest descent SD and the conjugate gradient CG algorithms are iterative methods used to solve systems of linear equations.
 Their raw implementation is shown in the code listings (\ref{solve_SD_debug}) and (\ref{solve_CG_debug}). 
 A more detailed and general implementation is presented in the attached scripts \texttt{solve\_SD} and \texttt{solve\_CG}, which are the ones used for the next points.
 
 \begin{multicols}{2}
-\matlabscript{solve_SD_debug}{Steepest Descent method implementation}
+\matlabscript{solve_SD_debug}{Steepest Descent (SD) method implementation}
 
 \vfill\null
 \columnbreak
 
-\matlabscript{solve_CG_debug}{Conjugate Greadient method implementation}
+\matlabscript{solve_CG_debug}{Conjugate Gradient (CG) method implementation}
 \end{multicols}
 
 \begin{figure}[!h]
 \begin{subfigure}{0.45\textwidth}
     \hspace{-0.1cm}
     %\includegraphics[scale=0.35]{SD1.eps}
     \resizebox{\textwidth}{!}{
         \input{graphs/SDCG/SDA1.tex}
     }
     \caption{SD method}
     \label{star}
 \end{subfigure}
 \hspace{1cm}
 \begin{subfigure}{0.45\textwidth}
     \centering
     %\includegraphics[scale=0.35]{CG1.eps}
     \resizebox{\textwidth}{!}{
         \input{graphs/SDCG/CGA1.tex}
     }
     \caption{CG method}
     \label{square}
 \end{subfigure}
 \captionsetup{justification=centering}
 \centering
 \caption{Comparing SD and CG methods for solving linear problem of size $5\times5$ }
 \label{A1}
 \end{figure}
 
 The purpose of those two iterative methods is to approximate the solution $\mathbf{x}$ of a linear equation presenting in the form:
 \begin{equation}
     A\mathbf{x}=\mathbf{b}, \quad i=1,2
 \end{equation}
 where $A$ is a symmetric and positive matrix.
 
 If $\lambda_{max}$ and $\lambda_{min}$ are the maximal and respectively the minimal values of the matrix $A$ in absolute value,
 let then be $\kappa$ the condition value of $A$ given by $\kappa = \frac{\lambda_{max}}{\lambda_{min}}$.
 Let be further defined the following suites:
 
 \begin{align}
     & \epsilon(i) = \sqrt{\frac{f(x_i) - f(x)}{f(x_0) - f(x)}} & \label{SDCG_thres_def} \\
     \delta_{SD}(i) = \left(\frac{\kappa - 1}{\kappa + 1}\right)^{i} & &
     \delta_{CG}(i) = \left(\frac{\sqrt{\kappa} - 1}{\sqrt{\kappa} + 1}\right)^{i} \label{SDCG_delta_def} \\
     \epsilon(i) \leq \delta_{SD}(i) & & \epsilon(i) \leq \delta_{CG}(i) \label{SDCG_ineq}
 \end{align}
 
 The value of $\epsilon(i)$ is defined in the equation (\ref{SDCG_thres_def}) and it corresponds to the relative error with respect the initial vector $x_0$ (taken as random) and the exact solution $x$
 computed using the matlab routine \texttt{A \textbackslash b}.
 
 Furthermore, two matrices are taken: $A_1$ of size $5\times5$ and $A_2$ of size $50\times50$.
 The figure (\ref{A1}) shows the number of iterations that the two algorithms need in order to reach an acceptable result.
 
 The same procedure is performed on $A_2$ and the results are presented in figure (\ref{A2}). 
 
 The first standing out observation is that the CG algorithm is much more efficient than the SD method. 
 When SD requires more than $10^7$ iterations ($\approx 1.75\times10^7$ iterations) before convergence,
 the CG only requires about 100 iterations. 
 The second method is clearly more advantageous, but it's more difficult to get a theoretical estimation of its convergence.
 Indeed the inequality in equation (\ref{SDCG_ineq}), although it's holding well for both methods,
 it is a good indicator of convergence only for the SD method, because the value of $\delta_{SD}(i)$ is right a little bit bigger than $\epsilon(i)$.
 Contrary, the value of $\delta_{CG}(i)$ tends to strongly diverge from $\epsilon(i)$ which means that's not a good indicator for the conjugate gradient method.
 
 Checking the condition number of both matrices using Mathlab's function \texttt{cond} the values are:
 
 \begin{equation}
     \kappa (A_1) \approx 2.63\times10^2\quad <<  \quad \kappa(A_2) \approx 2.88\times10^6.
 \end{equation}
 
 The huge between the two values means that the convergence time is different as well:
 $A_2$ in fact requires many more iterations than $A_1$.
 
 \begin{figure}[!h]
 \begin{subfigure}{0.45\textwidth}
     \hspace{-0.1cm}
     %\includegraphics[scale=0.35]{SD2.eps}
     \resizebox{\textwidth}{!}{
         \input{graphs/SDCG/SDA2.tex}
     }
     \caption{SD method}
     \label{star}
 \end{subfigure}
 \hspace{1cm}
 \begin{subfigure}{0.45\textwidth}
     \centering
     %\includegraphics[scale=0.35]{CG2.eps}
     \resizebox{\textwidth}{!}{
         \input{graphs/SDCG/CGA2.tex}
     }
     \caption{CG method}
     \label{square}
 \end{subfigure}
 \captionsetup{justification=centering}
 \centering
 \caption{Comparing SD and CG methods for solving linear problem of size $50\times50$}
 \label{A2}
 \end{figure}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newpage
 \begin{homeworkSection}{(2) Heat Transport across a Square Platelet}
 \end{homeworkSection}
 
 The modelization of the heat transport at the equilibrium involves a linear equation: precisely $\nabla^2 T = 0$, where $\nabla^2$ is the laplace operator.
 Letting the system be a thermically isolated square plate of length $L = 10$ and let there be two heat sources
 attached at opposite boundaries of the plate, then the system can be parametrized over an $N\times N$ grid of discretization
 and by appling a finite differentiation on the laplace operator one obtains:
 
 \begin{equation}
     \nabla^2 T_{i,j} \approx \frac{T_{i-1,j} + T_{i,j-1} + 4T{i,j} + T_{i+1,j} +T_{i,j+1}}{\Delta x ^2}
 \end{equation}
 
 The boundary conditions are implemented in order to set the outer value equal to the diagonal one $T{i,j}$ where
 no source is present. Otherwise, that value takes the temperature of the source. 
 More clearly applying a full computation the problem formalization leads to a simple form $A \cdot x = \vec{b}$,
 where the square $N^2\times N^2$ matrix $A$ and the $N^2$ vector $\vec{b}$ are formed as follow taking $i$ and $j$ as grid indexes:
 
 \begin{align} \label{heat_form}
     &
     \begin{cases}
         b_{i,j} = 0, \quad A_{i,j;i,j} = 4, \quad A_{i,j;i,j\pm 1} = A_{i,j;i\pm 1,j} = -1, & \quad \text{(i,j) inside}  \\
         A_{i,j;i,j} = contacts(i,j) + \#bars(i,j), b_{i,j} = \sum_{bars} T_{i\pm 1, j\pm 1}, \{A_{i,j;i \pm 1,j \pm 1} = -1 \text{ if } defined(i \pm 1,j \pm 1) \}, & \quad \text{(i,j) boundary}
     \end{cases} 
     \\
     &contacts(i,j) = \text{number of connected cells to (i,j)} \\
     &\#bars(i,j) = \text{number of bars present next to (i,j)} \\
     &defined(i,j) = \text{true if (i,j) is not an outer point of the grid} 
 \end{align}
 
 The implementation of this problem can be found in the attached script \texttt{heat\_problem.m},
 where this formalization is strictly applied.
 
 Because of the linear nature of this problem, the solution can be easily found using the SD and CG method discussed in the previous section.
 Again, the SD method is much slower getting over $100000$ iterations while the CG exits about $180$ iterations.
 Compared to the exact solution provided by matlab \texttt{A \textbackslash b}, the reached difference threshold is verified to be less than $1 \cdot 10^{-13}$ which corresponds to the expected result.
 
 \begin{figure}[!h]
     \centering
     %\includegraphics[scale =0.45]{heat.eps}
     \resizebox{\textwidth}{!}{
         \input{graphs/heat/quiver.tex}
     }
     \caption{Temperature profile and heat stream over the plates}
     \label{heat}
 \end{figure}
 
 The Fig.\,\ref{heat} is presents the resulting temperature profile and the heat current. 
 The visualized result is coherent to the physics expectations:
 the temperature is bounded between $T_2$ and $T_1 < T_2$ and it's continously descending from
 the left to the right, determining a right directioned heat stream.
 The hottest region is found close to the left source, while respectively the coldest on the right one.
 The heat stream is further null in all the boundary domain except for the source regions, which means that the isolation condition is well verified.
 
 About the convergence, the Jacobi proconditioning effectively lowers the condition number $\kappa$ of the matrix, passing from 
 $1195$ to $1162$. Despite that the difference isn't high enough to get a real advantage and the condition number is still much bigger than $1$,
 implying that the Jacobi preconditioning is useless in this context.
 That's proved by the fact that the number of iterations performed by the SD are still above $100000$ and 
 surpisingly, the CG iterations passed from $< 200$ to $> 800$. This last result, together with the one provided by the figure (\ref{A2}),
 empiricly prove that the condition number is not really a determinant factor for the convergence of the CG algorithm. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{homeworkSection}{(3) The Nonlinear Conjugate Gradient Method} 
 \end{homeworkSection}
 
 Simulating the formation of a molecular structure basing on the submitting potential is foundamental in order to 
 understand the origin of a lattice and the probability to be found in nature.
 In particular, the interest is to study the ground states energies of a system composed of $N$ identical charged particles in a 3D harmonic potential. 
 The total energy of the system can be set as:
 
 \begin{equation}\label{Etot}
     E_{tot}(\mathbf{x}_1,...,\mathbf{x}_N) = E_{harmonic} + E_{Coulomb} = k\sum_i \mathbf{x}_i^2 + 
     \sum_{i\neq j}U(|\mathbf{x}_i - \mathbf{x}_j|)
 \end{equation}
 
 with $U(x)  = \epsilon\frac{\sigma}{x}$ being the Coulomb potential term.
 Simplifying by setting $k=\epsilon = \sigma = 1$, then Eq.(\ref{Etot}) becomes:
 
 \begin{equation}
     E_{tot}(\mathbf{x}_1,...,\mathbf{x}_N) = \sum_i \mathbf{x}_i^2 + \sum_{i\neq j}\frac{1}{|\mathbf{x}_i - \mathbf{x}_j|}
 \end{equation}
 
 The principle behind the equilibrium configurations is that the energy has to be minimized.
 By using the non linear conjugate gradient method, the minimal energy can be found, but
 at the condition that it must be approximated to a second order polinomial using a Taylor expansion:
 
 \begin{equation}
     E(\mathbf{x}_i + \alpha\mathbf{d}) \approx E(\mathbf{x}_i) + \alpha[E'(\mathbf{x}_i)]^T \mathbf{d} + \frac{\alpha^2}{2}\mathbf{d}^TE''(\mathbf{x}_i)\mathbf{d}
 \end{equation}
 
 Precisely, the method used to find a minimum by following a direction is called the Newton-Raphson (NR) line search method.
 Analytically the first and second derivative are:
 
 \begin{equation}
     \frac{\partial E}{\partial x_i^{(l)}} = 2x_i^{(l)} - \sum_{j\neq i} \frac{x_i^{(l)} - x_j^{(l)}}{||\mathbf{x}_i - \mathbf{x}_j||^3} \quad\text{and}\quad 
 \end{equation}
 
 \begin{equation}
 \frac{\partial^2 E}{\partial x_i^{(l)}\partial x_j^m} = - 3 \frac{x_i^{(m)} - x_j^{(m)} + x_i^{(l)} - x_j^{(l)}}{||\mathbf{x}_i - \mathbf{x}_j||^5 }
     + \frac{\delta_{ij}}{||\mathbf{x}_i - \mathbf{x}_j||^3}
 \end{equation}
 
 \newpage
 
 \subsubsection{Study of the configurations, $N = 2,3,4$}
 
 \begin{figure}[h!]
 \begin{subfigure}{0.45\textwidth}
     %\hspace{-6cm}
     \centering
     \includegraphics[scale=0.45]{output/molecule2.pdf}
     \caption{$N = 2$}
     \label{seg}
 \end{subfigure}
 \hspace{1cm}
 \begin{subfigure}{0.45\textwidth}
     \centering
     %\hspace{-4cm}
     \includegraphics[scale=0.45]{output/molecule3.pdf}
     \caption{$N = 3$}
     \label{tri}
 \end{subfigure}
 \caption{Equilibrium configuration for a system of $N$ identical particles. The blue stings correspond to the intra-molecular bond.}
 \label{E4}
 \end{figure}
 
  
 \begin{wrapfigure}[16]{r}{0.5\textwidth}
 \centering
   \includegraphics[width=\linewidth]{output/molecule4.pdf}
   \centering
   \caption{Equilibrium configuration for a system of 4 identical particles}
   \label{tetra}
 \end{wrapfigure}
 
 It is prefered to implement an analytical scheme over an finite difference approximation for a matter of precision.
 The script containing the method implementation is \texttt{solve\_nonl\_CG.m}.
 
 Because the case $N = 2$ has an analytic solution, the method could be tested looking at the correspondence with it.
-Or rather, the obtained configurations are two particles equally distant (Fig.\ref{seg}) lying over a circle of radius $0.5$ and the systeme energy is found to be
+Or rather, the obtained configurations are two particles equally distant (figure \ref{seg}) lying over a circle of radius $0.5$ and the systeme energy is found to be
 exactly $1.5$ as expected.
 
 To the previous simulation scheme it can be easily added a 
 one dimensional potential term along the $\hat{\mathbf{z}}$.
 The equation (\ref{Etot}) becomes then:
  
 \begin{equation}
     E_{tot}(\mathbf{x}_1,...,\mathbf{x}_N) = E_{harm} + E_{Coul} + k_z\sum_iz_i^2
 \end{equation}
 
 \begin{table}[!h]
 \centering
 \begin{tabular}{cc|c|c|c|}
 \cline{3-5}
 \rowcolor[HTML]{C0C0C0} 
 \cellcolor[HTML]{FFFFFF}                                               & \cellcolor[HTML]{FFFFFF}                                                           & \multicolumn{3}{c|}{\cellcolor[HTML]{C0C0C0}Ground state energy $E_0$}                                                                                                         \\ \hline
 \rowcolor[HTML]{C0C0C0} 
 \multicolumn{1}{|l|}{\cellcolor[HTML]{C0C0C0}{\color[HTML]{000000} N}} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}{\color[HTML]{000000} Configurations}} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}$k_z = 0.0$} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}$k_z = 0.5$} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}$k_z = 2.0$} \\ \hline
 \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF}2}                        & Segment                                                                            & 1.50                                                     & 1.56                                                     & 2.00                                                     \\ \hline
 \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF}3}                        & Triangle                                                                           & 3.93                                                     & 4.04                                                     & 4.45                                                     \\ \hline
 \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF}4}                        & Tetrahedron                                                                        & 7.14                                                     & 7.65                                                     & 8.57                                                     \\ \hline
 \end{tabular}
     \caption{Energies found for the first three $N$ and different values of $k_z$}
     \label{energies}
 \end{table}
 
 One notices that with this contribution and setting an addiction small term $k_z = 0.5$, figure (\ref{tetra}) aligns in the ($O_{xy}$) plane. 
 With this along-z-new-contribution one observes that one of the faces of any configuration shape also aligns to the ($O_{xy}$) plane. 
 Another important observation is that with this new contribution, 
 the ground state energy of the equilibrium configuration increases with $k_z$ as shown in table (\ref{energies}).
 
 %% TODO here
   
 % N = 5, backwards
 \subsubsection{Case $N = 5$}
 
 The case $N=5$ allows different configurations available. However, the one that minimise the energy is as expected the configuration that corresponds to the square based pyramid (see figure (\ref{pyr})). 
 The second more stable configuration is a pentagon belonging to the ($O_{xy}$) plane as shown on figure (\ref{star}). The third more stable configuration is a square with central particle (figure (\ref{square})). 
 
 \begin{figure}[h!]
 \begin{subfigure}{.48\textwidth}
 \centering
     \resizebox{\textwidth}{!}{
         \input{graphs/molecule/stat.tex}
     }
     \caption{A statistic of the energy performed on various $N = 5$ random configurations. The modulus of the hint vector $X_0$ is bounded between $0.1$ and $6$.}
     \label{stat}
 \end{subfigure}
 \hspace{1.2cm}
 \begin{subfigure}{.48\textwidth}
   \centering
   \includegraphics[width=\linewidth]{output/molecule5_statcase_1.pdf}
   \caption{A possible configuration holding the minimum of energy $E = 11.23$. The corresponding shape is a square based pyramid.}
   \label{pyr}
 \end{subfigure}
   \caption{Simulation of the molecule taking $N = 5$}
 \end{figure}
 
 \begin{wrapfigure}[17]{r}{0.45\textwidth}
     \centering
     \includegraphics[width=\linewidth]{output/molecule_K2_extra.pdf}
-    \caption{Configuration obtained setting the input hint to the configuration in figure (\ref{}).
+    \caption{Configuration obtained setting the input hint to the configuration in figure (\ref{k05}).
            In this case the pyramid summit of the previous configuration was allowed to move along the $\hat{z}$ direction,
            meaning that during the transition the other components remained unvariated. Because the most stable solution
            has to lie on the $O_{xy}$ plane, the shape ends up to be a square centered pattern.
            Its energy is $E = 13.80$.}
     \label{square_centered}
 \end{wrapfigure}
 
 One notices that with small variation of $k_z$ the system converges to the same equilibrium configurations. 
 Geometrically the more visible feature is the alignment of the figure along the ($O_{xy}$) plane as already observed in the cases $N\leq 3$. 
 The most important detail of lowering the energy is that the configuration is well maintained taking $k_z$ to zero, this is an important realization because it
 physically explaines the origin of more improbable lattice configurations, which are simply stabler for higher energies.
 The graph in figure (\ref{stat}) presents over $100$ simulations a statistic of the obtained energies. This task was necessary in order to assure that the 
 computed energy is really the ground state energy, confirming in fact that the minimum is $11.23$.
 
 \begin{figure}[h!]
 \begin{subfigure}{.48\textwidth}
   \centering
   \includegraphics[width=\linewidth]{output/molecule_K05.pdf}
   \centering
   \caption{$k_z = 0.5$. Its energy is $E = 11.54$. The pyramid shape is maintained, allowing the summit to take only two possible orientations: the top or the bottom.}
   \label{k05}
 \end{subfigure}
 \hspace{1.2cm}
 \begin{subfigure}{.48\textwidth}
   \centering
   \includegraphics[width=\linewidth]{output/molecule_K2.pdf}
   \centering
     \caption{$k_z = 2$. Its energy is $E = 13.50$. The shape has turned into a pentagonal distribution lying on the $O_{xy}$ plane.}
   \label{k2}
 \end{subfigure}
     \caption{Resulting molecule taking two different values of $k_z > 0$. The asymmetry in the harmonic potential constraints the atom at the summit to locate over the $\hat{z}$ direction. The other four atoms are still free to rotate keeping the square structure fixed.}
     \label{kzmajor0}
 \end{figure}
 
 \newpage
 
 \begin{figure}[!h]
 \centering
 \begin{subfigure}{.3\textwidth}
   \centering
   \includegraphics[width=\linewidth]{output/molecule_hint2.pdf}
   \centering
   \caption{From $k_z = 0.5$. Its energy is $E = 11.23$, the minimum.}
   \label{hint2}
 \end{subfigure}
 \hspace{0.5cm}
 \begin{subfigure}{.3\textwidth}
   \centering
   \includegraphics[width=\linewidth]{output/molecule_hint3.pdf}
   \centering
   \caption{From $k_z = 2$. Its energy is $E = 11.69$.}
   \label{hint3}
 \end{subfigure}
 \hspace{0.5cm}
 \begin{subfigure}{.3\textwidth}
   \centering
   \includegraphics[width=\linewidth]{output/molecule_hint4.pdf}
   \centering
-  \caption{From the configuration given in figure (\ref{}). Its energy is $E = 11.82$.}
+  \caption{From the configuration given in figure (\ref{square_centered}). Its energy is $E = 11.82$.}
   \label{hint4}
 \end{subfigure}
   \caption{The three figures above show the configurations which can be taken with $k_z = 0$. 
            Starting from higher energy configurations as hint, same more energetic molecular structures can be easily triggered and
            the resulting one will have an higher energy too, compared to the ground state.}
   \label{take_five}
 \end{figure}
 
 \newpage
 
 % N = 6
 \subsubsection{Case $N = 6$}
 
 In the case $N = 6 $, similarly to case $N = 5$ there are several equilibrium configurations available for the system. 
 The table (\ref{N6table}) presents an overview of various experiments performed for this number of particles.
 
 \begin{figure}[h!]
 \begin{subfigure}{.45\textwidth}
     \centering
     \includegraphics[width=\linewidth]{output/case6/random.pdf}
     \caption{Random configuration for the harmonic potential}
     \label{octahedron}
 \end{subfigure}
 \hspace{1.2cm}
 \begin{subfigure}{.45\textwidth}
     \centering
     \includegraphics[width=\linewidth]{output/case6/strongly_fall.pdf}
     \caption{The same result as (\ref{consecutive_decrease}) is obtained equally by a one shot fall of $k_z$ from $7$ to $0$.}
     \label{strongly_fall}
 \end{subfigure}
     \caption{For $N = 6$, the ground state contrasted to the third stable state}
 \end{figure}
 
 The first one is just detecting the ground state configuration which is found to be an octahedron (see figure (\ref{octahedron})),
-while the second softly increases $k_z$ and then relowers it finding a new configuration for $k_z = 0$: a pentagonal base holding an addiction summit point (figure (\ref{pentagonal_dec})).
+while the second softly increases $k_z$ and then relowers it finding a new configuration for $k_z = 0$: a pentagonal base holding an addictional summit point (figure (\ref{pentagon_dec})).
 The same is done for $k_z = 7$, but this time the energy was lowered progressively in order to preserve the most as possible the found hexagonal configuration (figure (\ref{softly_fall})).
 Finally exactly the same result is proved to be obtained by striking down $k_z$ from $7$ directly to zero, that's what the last entry of the table and the figure (\ref{strongly_fall}) show.
 
 \begin{table}[h]
 \centering
 \begin{tabular}{cc|c|c|c|c|}
 \Xhline{2\arrayrulewidth}
 \rowcolor[HTML]{C0C0C0} 
 \multicolumn{1}{|l|}{\cellcolor[HTML]{C0C0C0}{\color[HTML]{000000} Configuration}} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}{\color[HTML]{000000} Geometry}} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}{\color[HTML]{000000} Figure}} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0} $k_z$ as hint} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0}$k_z$} & \multicolumn{1}{l|}{\cellcolor[HTML]{C0C0C0} Energy} \\ \Xhline{2\arrayrulewidth}
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Random}                 &  Octahedron     & (\ref{octahedron})                                                                           & 0                                                     & 0                                                     & 15.92                                                     \\ \Xhline{2\arrayrulewidth}
 %
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Softly increase}        &  Pentagon base + summit & (\ref{pentagon})                                                                           & 0                                                     & 0.5                                                     & 16.82                                                     \\ \hline
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Softly decrease}        &  Pentagon base + summit & (\ref{pentagon_dec})                                                                        & 0.5                                                     & 0                                                     & 16.20                                                     \\ \Xhline{2\arrayrulewidth}
 %
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Extreme increase}       &  Planar hexagon  & (\ref{extreme_inc})                                                                        & 0                                                     & 7                                                     & 21.00                                                     \\ \hline
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Half decrease}          &  Planar hexagon  & (\ref{half_dec})                                                                        & 7                                                     & 3                                                     & 18.68                                                     \\ \hline
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Quarter decrease}       &  Planar hexagon  & (\ref{quarter_dec})                                                                        & 3                                                     & 0.5                                                     & 17.24                                                     \\ \hline
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Soft fall}              &  Planar hexagon  & (\ref{softly_fall})                                                                        & 0.5                                                     & 0                                                     & 16.95                                                     \\ \Xhline{2\arrayrulewidth}
 %
     \multicolumn{1}{|c|}{\cellcolor[HTML]{EFEFEF} Strong fall}            &  Planar hexagon  & (\ref{strongly_fall})                                                                        & 7                                                     & 0                                                     & 16.95                                                     \\ \Xhline{2\arrayrulewidth}
 \end{tabular}
 \caption{For $N = 6$ all configuration grouped by consequentiality.}
     \label{N6table}
 \end{table}
 
 \begin{figure}[p]
 \begin{subfigure}{0.5\textwidth}
     \centering
     \includegraphics[scale=0.4]{output/case6/softly_increase.pdf}
     \caption{Configuration given by a random starting point, $k_z = 0.5$. It's characterized by a pentagonal base and a summit.}
     \label{pentagon}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.5\textwidth}
     \centering
     \includegraphics[scale=0.4]{output/case6/softly_decrease.pdf}
     \caption{Configuration given by taking the one in figure (\ref{pentagon}) as hint and $k_z = 0$. This result corresponds to a soft decrease of energy maintaining the previous form.}
     \label{pentagon_dec}
 \end{subfigure}
 \caption{Increase and consecutive decrease of the value $k_z$ from $0$ to $0.5$.}
 \label{softly_incdec}
 \end{figure}
 
 \begin{figure}[p]
 \begin{subfigure}{0.5\textwidth}
     \centering
     \includegraphics[scale=0.4]{output/case6/extreme_increase.pdf}
     \caption{$k_z = 7$, random starting configuration}
     \label{extreme_inc}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.5\textwidth}
     \centering
     \includegraphics[scale=0.4]{output/case6/half_decrease.pdf}
     \caption{$k_z = 3$, starting from (\ref{extreme_inc})}
     \label{half_dec}
 \end{subfigure}
 
 \begin{subfigure}{0.5\textwidth}
     \centering
     \includegraphics[scale=0.4]{output/case6/quarter_decrease.pdf}
     \caption{$k_z = 0.5$, starting from (\ref{half_dec})}
     \label{quarter_dec}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.5\textwidth}
     \centering
     \includegraphics[scale=0.4]{output/case6/softly_fall.pdf}
     \caption{$k_z = 0$, starting from (\ref{quarter_dec})}
     \label{softly_fall}
 \end{subfigure}
 \caption{Rasing up the value of $k_z$ to $7$ and then progressively lowering it to $0$ leads to an hexagonal configuration for the harmonic oscillator potential.}
 \label{consecutive_decrease}
 \end{figure}
 
 
 \end{homeworkProblem}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \newpage
 
 \begin{homeworkProblem}
 \begin{homeworkSection}{(1) Over defined system of linear equations} 
 \end{homeworkSection}
 
 A system of linear equations which has more equations then incognita
 is called an over-determined system.
 For example the following system is a case:
 
 \begin{equation} \label{over}
         \begin{pmatrix}
         1 & 1 \\
         1 & 2 \\
         1 & 3\\
         \end{pmatrix} \times
          \left(\begin{array}{c}
   x_1 \\ 
   x_2 \\ 
   \end{array}\right) = \left(\begin{array}{c}
   1\\ 
   4\\ 
   1\\ 
   \end{array}\right) 
   \end{equation}
 
 In the equation (\ref{over}), there are three equations for two incognitas. 
 Logically, an exact solution doesn't forcely exist.
 A possible approach is to minimize the quantity $\| A \vec{x} - \vec{b} \|$, or rather finding the best approximation. 
 This mimimum is obtained using the matrix Moore-Penrose inverse $A^+\equiv (A^\star A)^{-1} A $.
 
 \begin{equation*}
     A\mathbf{x} = \mathbf{b} \quad\Leftrightarrow\quad A^\star A\mathbf{x} = A^\star \mathbf{b} \quad\implies\quad (A^\star A)^{-1} A \mathbf{b} = A^+ \mathbf{b}.
 \end{equation*}
 
 \begin{figure}[!h]
 \centering
 \captionsetup{justification=centering}
 \begin{minipage}{.45\textwidth}
 
 This matrix in matlab can be easily obtained using the \texttt{pinv} function.
 Taking the one defined in the equation (\ref{over}), one obtains:
 
 
 \[    % <-- start math environment
 A^+ = 
     \begin{pmatrix}
     0.7509 & -0.2941 & 0.0588\\
     -0.586 & 0.4314 & -0.0196
     \end{pmatrix}
 \] ,
 
 the solution is then:
 
 \begin{equation}
     \mathbf{x} = \left(\begin{array}{c}
   -0.4118 \\ 
   1.1373 \\ 
   \end{array}\right)
 \end{equation}
 
 The above proposed solution minimises well the norm of the residual $r(\vec{x}) = \|A\vec{x} - \vec{b}\|$. 
 The plot ot the magnitude of the residual is presented on figure (\ref{res}). One can see that the numerical solution atches the minimun of the residual function.
 \end{minipage}
 \hspace{1.2cm}
 \begin{minipage}{.45\textwidth}
   \centering
   \resizebox{1.1\textwidth}{!}{
       \input{graphs/penrose/quiver.tex}
   }
     \caption{Residual magnitude $r(\vec{x})$ and location of the numerical solution. The bluest central zone corresponds to the found mininum.}
   \label{res}
 \end{minipage}%
 \end{figure}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{homeworkSection}{(2) (SVD) decomposition } 
 \end{homeworkSection}
 
 The singular value decomposition (SVD) is used for the compression of images. By SVD is meant that any $m\times n\, (m\geq n)$ matrix $A$ can be decomposed as:
 
 \begin{equation}
     A = U\Sigma V^\star, \quad U \in M_{m\times m}, \quad V \in M_{n\times n}
 \end{equation}
 
 where $U$ and $V$ are unitary and $\Sigma$ is the $m\times n$ diagonal matrix which of the diagonal elements are called the singular values. 
 By convention the singular values are ordered in a descending order:
 
 \begin{equation}
     \sigma_1\geq\sigma_2\geq...\sigma_p>\sigma_{p+1}= ... = \sigma_n = 0
 \end{equation}
 
 with $p$ the rank matrix of $A$. If $p = n$ then the matrix is \textit{full-rank}.
 
 \subsubsection{Compressing images}
 A color picture of Julia Yeomans (figure \ref{original})  
 is loaded in a $1080\times 810\times 3$ tensor (each RGB color channel requires a matrix $1080\times 810$).  
 To do the singular value decomposition, matlab function \texttt{svd} is used. 
 The singular values obtained for each color are plotted on a logarithmic scale on figure (\ref{yeomans_plot}).
 The rank of the image matrix for each channel is $p = 810$ because the latest 
 singular value $\sigma_p$ is non-zero and there are exactly $810$, meaning that the matrix is determined in its n-size (number of columns).
 
 \begin{wrapfigure}[18]{r}{0.5\textwidth}
   \resizebox{0.5\textwidth}{!}{
       \input{graphs/ising/yeomans.tex}
   }
   \caption{Singular values $\sigma_i$ plotted for the three color channels (red, green, blue)}
   \label{yeomans_plot}
 \end{wrapfigure}
 
 In order to be furthermore compressed the image was loaded using the class \texttt{imgcomp}: this class loads the image in all its channels, automatically performs the (SVD) decomposition and 
 allows to further lower the rank.
 In principle, the compression (with loss) process is characterized by the rank-lowering of the matrix by setting to zero some singular values starting by the smallest ones.
 The result after the compression is that the matrix $\Sigma$ holds a controlled number of diagonal non-zero values and after the 
 reconstruction, the matrix will result structurally similar to the original one with a difference in the values distribution.
 
 In the example of a compressed image, the final result will appear more unclear than the original, although image is still visible in its shape.
 On Fig.\,\ref{Ycomp} are presented the original and compressed pictures of J.Yeomans for matrix of rank $N = 100,\, 20,\, 4$. 
 The lower is $N$, the less the compressed image will be clear: there is in fact an evident difference between the figures (\ref{original}) and (\ref{N4}). 
 
 \begin{figure}[!h]
 \centering
 \captionsetup{justification=centering}
 \begin{subfigure}{0.2\textwidth}
     %\hspace{-0.2cm}
     \includegraphics[scale=0.07]{output/ising/yeomans.jpg}
     \centering
     \caption{Original picture of Julia Yeomans}
     \label{original}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.2\textwidth}
     \centering
     \includegraphics[scale=0.17]{output/ising/yeomans100.eps}
     \caption{Compressed picture: $N = 100$}
     \label{N100}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.2\textwidth}
     \centering
     %\includegraphics[scale=0.17]{Y20.eps}
     \includegraphics[scale=0.17]{output/ising/yeomans20.eps}
     \caption{Compressed picture: $N = 20$}
     \label{N20}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.2\textwidth}
     \centering
     %\includegraphics[scale=0.17]{Y100.eps}
     \includegraphics[scale=0.17]{output/ising/yeomans4.eps}
     \caption{Compressed picture: $N = 4$}
     \label{N4}
 \end{subfigure}
 \caption{J.Yeomans picture: original and compressed versions}
 \label{Ycomp}
 \end{figure}
 
 However, the increase in rank leads to reconstructed image being more and more accurate. 
 The difference is very little when the original picture is compared to the compressed one with $N=100$ (figure (\ref{N100})). 
 This means that the transformations allows to store an image, very difficult to distinguish from the original one, using less memory.
 
 Precisely the responsible operation to the storage has to be capable to discard the useless informations: if $\chi$ is the rank of the compressed image, then
 $U$ can be reduced to a $m\times \chi$ matrix, as well as $V$ can become a $\chi \times n$ matrix and, finally,
 $\Sigma$ holds only $\chi$ diagonal terms making a total storage size of $\chi \cdot (m + n + 1)$.
 
 \newpage
 
 \subsubsection{Ising model interpretation}
 
 An interesting application of the image compression is the study of a square lattice Ising model obtained by using Monte carlo simulations (figure (\ref{ising})). 
 Physically, at temperature higher than the critical temperature $T_c$, the entropic 
 contribution dominates the exchanged energy between nearest neighbours. 
 
 \begin{wrapfigure}[18]{r}{0.5\textwidth}
     \resizebox{0.5\textwidth}{!}{
       \input{graphs/ising/ising.tex}
     }
   \caption{SVD for three pictures of MonteCarlo simulations of Ising model}
   \label{svdising}
 \end{wrapfigure}
 
 This means that there are no big clusters appearing. Whereas, on temperature 
 lower then $T_c$, interaction between neighbours dominates, spins that are close 
 together tend to align in the same direction forming that way big clusters of same alignment spins. 
 Therefore, one would say that figures (\ref{is1}) and (\ref{is3}), where clusters 
 are present, appear to be at temperature below $T_c$. 
 Consequently, figure (\ref{is2}) seems to be at temperature above $T_c$. 
 
 The detail to be noticed is that figure (\ref{is3}) seems to be a zoom on (\ref{is1}). 
 If this is the case, this would confirm the hypothesis of figure (\ref{is1}) 
 beeing at a temperature \textit{strictly} below $T_c$. Since no scale invariance 
 is visible, the latter beeing a major characterisitic of the system at $T_c$.
 On figure (\ref{svdising}) are presented the singular values decomposition of the 
 original three pictures (compressed version on figure (\ref{ising})). 
 
 The second picture one has globally higher singular values with respect 
 to the others but lower ones near $N = 0$. It seems that more the picture is of a 
 disordered system, the higher globally the singular values are. Since figure (\ref{is2}) 
 classifies as the most disordered this confirms qualitatively and by comparison with 
 the other singular values decomposition, the prediciton on the latter being at 
 temperature above $T_c$.
 
 \begin{figure}[!h]
 \centering
 \captionsetup{justification=centering}
 \begin{subfigure}{0.3\textwidth}
     %\hspace{-0.2cm}
     \includegraphics[scale=0.3]{output/ising/ising1.pdf}
     \centering
     \caption{$T < T_c$}
     \label{is1}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.3\textwidth}
     \centering
     \includegraphics[scale=0.3]{output/ising/ising2.pdf}
     \caption{$T > T_c$}
     \label{is2}
 \end{subfigure}
 \hspace{0.1cm}
 \begin{subfigure}{0.3\textwidth}
     \centering
     \includegraphics[scale=0.3]{output/ising/ising3.pdf}
     \caption{$T < T_c$}
     \label{is3}
 \end{subfigure}
 \caption{Ising model pictures approximated ($N = 50$)}
 \label{ising}
 \end{figure}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newpage
-\begin{homeworkSection}{(3) SVD and quantum ground state compression} 
+\begin{homeworkSection}{(3) SVD and quantum ground state} 
 \end{homeworkSection}
 
 Let's consider a quantum ground state, defined as the lowest energy state.
 The objective of this section is to find the least quantity of information for 
 a wave function to be stored and
 then exactly reconstructed. Notice that in the previous point, 
 the image compression implied losses; it won't be the case in this section:
 for some ground state solutions to particular hemiltonians the compression 
 becomes evident analizing the system symmetries.
 
 Let's consider a spin chain with $N$ sites and local Hilbert space of dimension $d$. 
 Then, the Hilbert space dimension of the full chain is $d^N$. 
 Given a certain two-site wavefunction:
 
 \begin{equation}
     \psi = \left( \begin{matrix} 0 & 0 & -\frac{1}{2} \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{matrix} \right)
 \end{equation}
 
 One can apply to it the singular value decomposition (SVD), ie: writing the matrix has the product of three matrices: $\psi = UsV^\star$ (for more details on SVD see previous exercise) with:
  
 \begin{equation}
     U = \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right) \equiv\mathbb{I}_3; \quad s = \left( \begin{matrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 \end{matrix} \right); \quad \text{and} \quad V = \left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{matrix} \right).
 \end{equation}
 
 \begin{multicols}{2}
 
 Notice that the rank of the singular value matrix $s$, also called bond dimension is 2. 
 Using the matrices $U, s$ and $V$ and using the provided \texttt{ncon} function 
 one recovers the original wave function $\psi$.
 For a simple case like \texttt{aklt2sites}, the script \texttt{aklt2sites.m} applies a raw implementation of the
 reconstruction of $\psi$ after a (SVD) decomposition. 
 
 Let's consider more then 2 sites wave functions with open boundary conditions provided in the storage files \texttt{alkt2sites\_open}
 and \texttt{alkt\_open}, defined over 2 sites and 8 respectively. 
 This states are decomposed into a series of tensors using the iterative schmidt algorithm.
 The network contractor function (\texttt{ncon}) is used to recover the wave function after decomposition. 
 Precisely, specifying the tensors' connections by indexes, where by convention the positive symbolize the contracted ones and
 the negative the free ones, the (\texttt{ncon}) function just applies the specified contractions returning the resulting tensor or scalar.
 
 In the case of a wave function reconstruction, the given tensors are $A_1, ..., A_N$ (\texttt{Alist}) followed by another transition tensor $V$ (\texttt{Vlist\{end\}})
 and the function \texttt{recpsi} is programmed to perform this task first setting up the indexes, then using ncon the recover the original wave function.
 
 The precise process of indexing is the following: to the first tensor $A_1$ are given the indices $[-1,-2,1]$  
 (ie: $A_1\longrightarrow[-1,-2,1]$), 
 
 
 \vfill\null
 \columnbreak
 
 \matlabscript{recpsi}{Method recovering a wave function from a previously decomposed one with the \texttt{iterative-schmidt} algorithm}
 \end{multicols}
 
 \begin{equation*}
     \text{then for}\quad 2 \leq k \leq N-1, \quad A_k \longrightarrow [k-1, -k-1, k] \quad\text{and}\quad V \longrightarrow [N-1, -N-1, -N-2].
 \end{equation*}
  
 Originally \texttt{psialkt} is described with $2^2\times3^8 = 26244$ numbers and \texttt{psialkt2} with $2^2\times3^2 = 36$ numbers. 
 In there compressed form, \texttt{Alist} and \texttt{V}  need respectively 825 and $5\times3\times2 = 30$ numbers. 
 Hence the interest for this procedure: it enables to reduce considerably memory cost of the descrpition of the system whitout loosing information. \\
 
 \subsubsection{Further analysis and the translation symmetry}
 
 Finally if one considers that the wave fucntion can be described as in equation (4) of the question sheet, 
 using the \texttt{ncon} one recovers once again the ground states, as the script (\texttt{ground\_state.m}) does. 
 It is observed that this procedure is even more efficient then the iterative schmidt method, in the sense that
 the most compressed information is already provided and any solution for the $N-th$ case cells can be computed
 by concatenating $N$ contractions of the same tensor $A$. 
 Indeed this way the states are even more compressed. The description of the ground state solution 
 for the AKLT hamiltonian needs $2^2\times3 = 12$ numbers (ie: the size of $A$, it is the maximal compression possible of $\psi$).  
 This optimization is clearly possible because of the translation invariance of the system: it's not a case
 that the tensor $A$ is the same for every cell. The fact that a spin-1 cell can be decomposed into two spin-$\frac{1}{2}$ nodes
 and anti-symetrized with their neighbouring nodes shows that all cells present in the chain are identical: which translates
 to the fact that the needed information the describe the chain limits itself to how a cell it's constructed, how it
 binds with its neightbourhood and how many there are of them, indeed nothing more than $A$ and $N$ (as the function \texttt{ground\_recover} takes as arguments).
 
 \end{homeworkProblem}
 
 \end{spacing}
 
 \end{document}
 
diff --git a/exo4/report.toc b/exo4/report.toc
index e7dbaf5..2244a44 100644
--- a/exo4/report.toc
+++ b/exo4/report.toc
@@ -1,14 +1,14 @@
 \contentsline {section}{Problem 1}{3}% 
 \contentsline {subsection}{(1) Steepest Descent \& Conjugate Gradient}{3}% 
 \contentsline {subsection}{(2) Heat Transport across a Square Platelet}{5}% 
 \contentsline {subsection}{(3) The Nonlinear Conjugate Gradient Method}{6}% 
 \contentsline {subsubsection}{Study of the configurations, $N = 2,3,4$}{7}% 
 \contentsline {subsubsection}{Case $N = 5$}{8}% 
 \contentsline {subsubsection}{Case $N = 6$}{10}% 
 \contentsline {section}{Problem 2}{12}% 
 \contentsline {subsection}{(1) Over defined system of linear equations}{12}% 
 \contentsline {subsection}{(2) (SVD) decomposition }{12}% 
 \contentsline {subsubsection}{Compressing images}{12}% 
 \contentsline {subsubsection}{Ising model interpretation}{14}% 
-\contentsline {subsection}{(3) SVD and quantum ground state compression}{15}% 
+\contentsline {subsection}{(3) SVD and quantum ground state}{15}% 
 \contentsline {subsubsection}{Further analysis and the translation symmetry}{16}% 
diff --git a/exo4/solve_SD.m b/exo4/solve_SD.m
index e0673ab..2effb20 100644
--- a/exo4/solve_SD.m
+++ b/exo4/solve_SD.m
@@ -1,42 +1,42 @@
 function [X, varargout] = solve_SD(A, b, varargin)
     [Ni, Nj] = size(A);
     assert(Ni == Nj || Ni == 0, 'Matrix must be square  or non-zero')
     X = rand(Ni,1);
     X = X / norm(X);
     r = b - A * X;
 
     if length(varargin) > 0
         limit = varargin{1};
     else
-        limit = floor(5000 * log(1 + Ni));
+        limit = 1000000;
     end
     
     X0 = X(:); % force copy
     solution = A \ b;
     errors = zeros(limit,1);
     k = 1;
     xnorm = norm(solution);
 
     while norm(r) > eps && k < limit
         errors(k) = err_A(A, b, X, X0, solution);
         alpha = (r' * r) / (r' * A * r);
         X = X + alpha * r;
         r = b - A * X;
         k = k + 1;
     end
 
     errors(k) = err_A(A, b, X, X0, solution);
     errors = errors(1:k);
 
     if length(varargin) > 0
         varargout{1} = errors;
     end
 end
 
 function err = err_A(A, b, Xi, X0, X)
     err = sqrt(abs(diff_A(A, b, Xi, X) / diff_A(A, b, X0, X)));
 end
 
 function err = diff_A(A, b, Xi, X)
     err = 0.5 * ( Xi' * A * Xi - X' * A * X ) - b' * (Xi - X);
 end