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zejkyI;-NNt=Tey{jR)lcly^;Snk5)Sv@?0SC8-5u#5acXd+ zE$e!o`x)i1!}D{Tdqina<|Ef$^x`Xv;v4bD{N`tq?^~pXo&M+DJMqWGl<&)R%;)jz z=WEI5mHy}3=i71h=S|G#bJyE>&c}=XM@Cn;@5}Ax=kDhFv+c*N(A#^?=bfwXYs%+K z)8}>9`=jqu*XLW#$4$xW=I4&m=bO^!&F06e?=92kf&ORP=aujKz3-FI=i}$O?Zts_&aF16c~_qi=nmPh*=O>%MQ*z8^24zSqc~PmjLO l@Zy<&&hp=wK6i!QAOD^HCdT*i9{H1)Ht@2}(G3#p{{XWVoD%>5 diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex index 5693ea98..1353f33f 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex @@ -151,6 +151,88 @@ Then we get a very well conditioned set of normal equations: \[ \vdots \] \[ a_n\langle \psi_n, \psi_n \rangle = \langle f, \psi_n \rangle \] +\section{Results} + +I decided to simulate polynomials up to $\mathbf{10^{th}}$ degree. + +\begin{center} + \includegraphics[scale=0.25]{10.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{11.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{12.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{13.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{14.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{15.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{16.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{17.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{18.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{19.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{110.eps} +\end{center} + +\begin{center} + \begin{tabular}{| c | c | c |} +\hline + +Degree & Error & Condition number of Gram's Matrix \\ +\hline +0& 13.2040& 1\\ +\hline +1& 9.1281& 10\\ +\hline +2& 8.8257& 408.7796\\ +\hline +3& 4.2365& 8.5584e+03\\ +\hline +4& 1.5418& 3.1798e+05\\ +\hline +5& 1.5413& 7.4675e+06\\ +\hline +6& 1.1689& 2.8316e+08\\ +\hline +7& 0.9345& 7.6462e+09\\ +\hline +8& 0.8883& 3.3055e+11\\ +\hline +9& 0.8334& 1.5167e+13\\ +\hline +10& 6.3645e-13& 9.2930e+14\\ +\hline + \end{tabular} +\end{center} + + + + @@ -267,6 +349,48 @@ For RK methods we have following parameters: \[ \epsilon_i = |(y_i)_n^{(2)}| \epsilon_r + \epsilon_a \] \[ \alpha = \min_{1 \leq i \leq m} (\frac{\epsilon_i}{|\delta_n(h)_i}|)^{\frac{1}{p+1}} \] +\section{Results} + +\begin{center} + \includegraphics[scale=0.25]{20.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{21.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{22.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{23.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{24.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{25.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{26.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{27.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{28.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{29.eps} +\end{center} + \begin{thebibliography}{9} \bibitem{texbook} diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc index 57edbde9..357d5207 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc @@ -4,14 +4,16 @@ \contentsline {subsection}{\numberline {1.2.1}Linear Least Squares Problem}{3}{subsection.1.2.1}% \contentsline {subsection}{\numberline {1.2.2}Gram matrix and QR decomposition}{3}{subsection.1.2.2}% \contentsline {subsection}{\numberline {1.2.3}Gram-Schmidt algorithm}{4}{subsection.1.2.3}% -\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{6}{chapter.2}% -\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{6}{section.2.1}% -\contentsline {subsection}{\numberline {2.1.1}Problem}{6}{subsection.2.1.1}% -\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{6}{subsection.2.1.2}% -\contentsline {subsubsection}{Runge-Kutta method of order 4}{7}{section*.2}% -\contentsline {subsection}{\numberline {2.1.3}Adams PC method}{7}{subsection.2.1.3}% -\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{8}{section.2.2}% -\contentsline {subsection}{\numberline {2.2.1}Problem}{8}{subsection.2.2.1}% -\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{8}{subsection.2.2.2}% -\contentsline {subsubsection}{Error estimation using step doubling-approach}{8}{section*.3}% -\contentsline {subsubsection}{Actual correction of the step size}{10}{section*.4}% +\contentsline {section}{\numberline {1.3}Results}{5}{section.1.3}% +\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{12}{chapter.2}% +\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{12}{section.2.1}% +\contentsline {subsection}{\numberline {2.1.1}Problem}{12}{subsection.2.1.1}% +\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{12}{subsection.2.1.2}% +\contentsline {subsubsection}{Runge-Kutta method of order 4}{13}{section*.2}% +\contentsline {subsection}{\numberline {2.1.3}Adams PC method}{13}{subsection.2.1.3}% +\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{14}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Problem}{14}{subsection.2.2.1}% +\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{14}{subsection.2.2.2}% +\contentsline {subsubsection}{Error estimation using step doubling-approach}{14}{section*.3}% +\contentsline {subsubsection}{Actual correction of the step size}{16}{section*.4}% +\contentsline {section}{\numberline {2.3}Results}{17}{section.2.3}% diff --git a/ENUME/projectC/task1.m b/ENUME/projectC/task1.m index 8f2aca3a..122ddb9a 100644 --- a/ENUME/projectC/task1.m +++ b/ENUME/projectC/task1.m @@ -5,8 +5,8 @@ function task1(polynomialDegree) [LSPSolutions, error, conditionNumber] = approximate(dataPoints, currentPolynomialDegree); displayInfo(currentPolynomialDegree, error, conditionNumber); plotGraph(currentPolynomialDegree, dataPoints, LSPSolutions) - end + end function displayInfo(currentPolynomialDegree, error, conditionNumber) diff --git a/ENUME/projectC/task2.asv b/ENUME/projectC/task2.asv deleted file mode 100644 index b102f533..00000000 --- a/ENUME/projectC/task2.asv +++ /dev/null @@ -1,116 +0,0 @@ - -function task2() - [ordinaryDifferentialEquations, initialValues, interval, algorithms] = initialize(); - solveAndPrintODEs(algorithms, ordinaryDifferentialEquations, initialValues, interval); - [result, sizes, errors] = solveAndPlotODEAutomatic(ordinaryDifferentialEquations, initialValues, interval); - plotStatistics(result, sizes, errors); - plotComparisonToMatlabFunction(ordinaryDifferentialEquations, interval, initialValues); -end - -function [ordinaryDifferentialEquations, initialValues, interval, algorithms] = initialize() - ordinaryDifferentialEquations = { - @(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) - }; - initialValues = [8; 9]; - interval = [0; 15]; - - algorithms = { - 'RK4 algorithm', @RK4, [0.01, 0.011]; - 'Adams PC algorithm', @AdamsPCMethod, [0.002, 0.01305] - }; -end - -function solveAndPrintODEs(algorithms, ordinaryDifferentialEquations, initialValues, interval) - for algorithm = 1 : 2 - [algorithmName, algorithmFunction, stepSizes] = algorithms{algorithm, :}; - stepResults = solveForEachStep(stepSizes, ordinaryDifferentialEquations, initialValues, interval, algorithmFunction); - plotAgainstTime(ordinaryDifferentialEquations, algorithmName, stepResults); - plotAgainst(algorithmName, stepResults); - - end -end - -function stepResults = solveForEachStep(stepSizes, ordinaryDifferentialEquations, initialValues, interval, algorithmFunction) - stepResults = cell(size(stepSizes, 2), 3); - stepNames = {'Optimal Step Size', 'Slightly Larger Step Size'}; - for stepNumber = 1:size(stepSizes, 2) - result = algorithmFunction(ordinaryDifferentialEquations, initialValues, interval, stepSizes(stepNumber)); - stepResults(stepNumber, :) = {stepSizes(stepNumber), stepNames{stepNumber}, result}; - end -end - -function beginPlot() - figure; - grid on; - hold on; -end - -function plotAgainstTime(ordinaryDifferentialEquations, algorithmName, stepResults) - for equationNumber = 1:size(ordinaryDifferentialEquations, 1) - beginPlot(); - - title([algorithmName, ', x_', num2str(equationNumber), ' ODE']); - - for stepresult = stepResults' - plot(stepresult{3}(1, :), stepresult{3}(equationNumber + 1, :)); - end - - hold off; - legend(stepResults{:, 2}); - end -end - -function plotAgainst(algorithmName, stepResults) - beginPlot(); - title([algorithmName, ' trajectory plot (x_2 compared to x_1)']); - for stepresult = stepResults' - plot(stepresult{3}(2, :), stepresult{3}(3, :)); - end - hold off; - legend(stepResults{:, 2}); -% %print(['report/', func2str(algfunc), 'traj'], '-dpdf'); -end - -function [result, sizes, errors] = solveAndPlotODEAutomatic(ordinaryDifferentialEquations, initialValues, interval) - [result, sizes, errors] = solveRKAutomatic(ordinaryDifferentialEquations, initialValues, interval); - plotTrajectory(result); -end - -function [result, sizes, errors] = solveRKAutomatic(ordinaryDifferentialEquations, initialValues, interval) - initialStepSize = 1e-5; - relativeEpsilon = 1e-9; - absoluteEpsilon = 1e-9; - [result, sizes, errors] = RK4Automatic(ordinaryDifferentialEquations, initialValues, interval, initialStepSize, relativeEpsilon, absoluteEpsilon); -end - -function plotTrajectory(result) - figure; - plot(result(2, :), result(3, :)); - grid on; - title('RK4 auto-step trajectory plot (x_2 against x_1)'); -end - -function plotStatistics(result, sizes, errors) - 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}); - end -end - -function plotComparisonToMatlabFunction(ordinaryDifferentialEquations, interval, initialValues) - % compare results with ODE45 - odefun = @(t, x) [ ordinaryDifferentialEquations{1}(x); ordinaryDifferentialEquations{2}(x) ]; - odeoptions = odeset('RelTol', 10e-10, 'AbsTol', 10e-10); - [~, x] = ode45(odefun, interval, initialValues, odeoptions); - figure; - plot(x(:, 1), x(:, 2)); - grid on; - title('ODE45 trajectory plot (x_2 against x_1)'); -end \ No newline at end of file diff --git a/ENUME/projectC/task2.m b/ENUME/projectC/task2.m index ccc58b6c..50df2615 100644 --- a/ENUME/projectC/task2.m +++ b/ENUME/projectC/task2.m @@ -17,7 +17,7 @@ function [ordinaryDifferentialEquations, initialValues, interval, algorithms] = algorithms = { 'RK4 algorithm', @RK4, [0.01, 0.011]; - 'Adams PC algorithm', @AdamsPCMethod, [0.002, 0.01305] + 'Adams PC algorithm', @AdamsPCMethod, [0.002, 0.013] }; end

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