feat: Beautifying backsubstitution code

ENUME/projectC/backSubstitution.m

@@ -1,16 +1,20 @@
-% solves system with triangular matrix
-function result = backSubstitution(eqsys)
-    for col = size(eqsys, 1):-1:1
-        % normalize diagonal coefficients to 1
-        eqsys(col, :) = eqsys(col, :) / eqsys(col, col);
-        
-        % eliminate factor from other rows
-        for row = (col - 1):-1:1
-            reductor = eqsys(row, col) / eqsys(col, col);
-            eqsys(row, :) = eqsys(row, :) - eqsys(col, :) * reductor;
-        end
-    end
-    
-    % rightmost column is now the result
-    result = eqsys(:, size(eqsys, 2));
+function solution = backSubstitution(Matrix)
+    Columns = size(Matrix, 1);
+    Matrix = backSubstitutionOuterLoop(Matrix, Columns);
+    % rightmost column of the Matrix is now our result
+    solution = Matrix(:, size(Matrix, 2));
+end
+
+function Matrix = backSubstitutionOuterLoop(Matrix, Columns)
+    for k = Columns : -1 : 1
+        % Diagonal coefficients of matrix need to be equal to 1
+        Matrix(k, :) = Matrix(k, :) / Matrix(k, k);
+        Matrix = eliminateFactors(Matrix, k);
+    end
+end
+
+function Matrix = eliminateFactors(Matrix, k)
+    for row = (k - 1) : -1 : 1
+        Matrix(row, :) = Matrix(row, :) - Matrix(k, :) * (Matrix(row, k) / Matrix(k, k));
+    end
 end

ENUME/projectC/task2ode.asv

@@ -1,104 +0,0 @@
-% functions of the ODE system and initial values
-sysfuncts = {
-    @(x) x(2) + x(1) * (0.5 - x(1)^2 - x(2)^2);
-    @(x) -x(1) + x(2) * (0.5 - x(1)^2 - x(2)^2)
-};
-initvalues = [0; 12];
-interval = [0; 15];
-
-% define available algorithms
-algorithms = {
-    'RK4', @rk4, [0.01, 0.013408];
-    'Adams PC', @adamspc, [0.002, 0.01305]
-};
-
-% solve ODE using different algorithms and step sizes
-for alg = 1:size(algorithms, 1)
-    [algname, algfunc, stepsizes] = algorithms{alg, :};
-    
-    % solve using the given algorithm for each step size
-    stepresults = cell(size(stepsizes, 2), 3);
-    stepnames = {'optimal step', 'larger step'};
-    for stepno = 1:size(stepsizes, 2)
-        result = algfunc(sysfuncts, initvalues, interval, stepsizes(stepno));
-        stepresults(stepno, :) = { ...
-            stepsizes(stepno), ...
-            stepnames{stepno}, ...
-            result};
-    end
-    
-    % plot each component against time
-    for eqnum = 1:size(sysfuncts, 1)
-        % begin plot
-        figure; grid on; hold on;
-        title([algname, ', x_', num2str(eqnum), ' against time']);
-        set(gcf, 'PaperPosition', [0 0 6 4]);
-        set(gcf, 'PaperSize', [6 4]);
-        
-        % plot component for each step size
-        for stepresult = stepresults'
-            plot(stepresult{3}(1, :), stepresult{3}(eqnum + 1, :));
-        end
-        
-        % finish plot
-        hold off;
-        legend(stepresults{:, 2});
-%        %print(['report/', func2str(algfunc), 'x', num2str(eqnum)], '-dpdf');
-    end
-    
-    % plot first two components against each other
-    figure; grid on; hold on;
-    title([algname, ' trajectory plot (x_2 against x_1)']);
-    set(gcf, 'PaperPosition', [0 0 6 4]);
-    set(gcf, 'PaperSize', [6 4]);
-    
-    % plot for each step size
-    for stepresult = stepresults'
-        plot(stepresult{3}(2, :), stepresult{3}(3, :));
-    end
-    
-    % finish plot
-    hold off;
-    legend(stepresults{:, 2});
-%    %print(['report/', func2str(algfunc), 'traj'], '-dpdf');
-end
-
-% solve ODE using RK4 with automatic step size
-[result, sizes, errors] = rk4auto(sysfuncts, initvalues, interval, ...
-    1e-5, 10e-10, 10e-10);
-
-% plot trajectory
-figure;
-plot(result(2, :), result(3, :));
-grid on;
-title('RK4 auto-step trajectory plot (x_2 against x_1)');
-set(gcf, 'PaperPosition', [0 0 6 4]);
-set(gcf, 'PaperSize', [6 4]);
-%%print(['report/', 'rk4autotraj'], '-dpdf');
-
-% plot statistics
-stats = {
-    "RK4 auto-step step size", "rk4sizes", sizes;
-    "RK4 auto-step approximation error", "rk4errors", errors
-};
-for stat = stats'
-    figure;
-    plot(result(1, 2:(end - 1)), stat{3});
-    grid on;
-    title(stat{1});
-    set(gcf, 'PaperPosition', [0 0 6 4]);
-    set(gcf, 'PaperSize', [6 4]);
-%    %print(strcat("report/", stat{2}), '-dpdf');
-end
-
-% compare results with ODE45
-odefun = @(t, x) [ sysfuncts{1}(x); sysfuncts{2}(x) ];
-odeoptions = odeset('RelTol', 10e-10, 'AbsTol', 10e-10);
-[t, x] = ode45(odefun, interval, initvalues, odeoptions);
-figure;
-plot(x(:, 1), x(:, 2));
-grid on;
-title('ODE45 trajectory plot (x_2 against x_1)');
-set(gcf, 'PaperPosition', [0 0 6 4]);
-set(gcf, 'PaperSize', [6 4]);
-%print('report/ode45', '-dpdf');