diff --git a/ENUME/projectA/indicatedMethod.m b/ENUME/projectA/indicatedMethod.m index 3bce3539..76716419 100644 --- a/ENUME/projectA/indicatedMethod.m +++ b/ENUME/projectA/indicatedMethod.m @@ -1,9 +1,21 @@ -function x = indicatedMethod(Matrix, Vector) - [~,Columns] = size(Matrix); +function x = indicatedMethod(Matrix, Vector) % Name of the method as in the textbook +% x stands for obtained result checkIfMatrixIsSquareMatrix(Matrix); - [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); - [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); - x = iterativeResidualCorrection(Matrix, x, Vector); + [Matrix, Vector, x] = solveSystem(Matrix, Vector); + norm(x) + x = iterativeResidualCorrection(Matrix, x, Vector); % Improve on the solution + norm(x) +end % end function + +function [Matrix, Vector, x] = solveSystem(Matrix, Vector) + [~,Columns] = size(Matrix); % We need to know how big the matrix is in next steps + % notice the '~', since we assume we use square matrix, we do not need + % to have another variable for number of rows since it is the same as + % number of columns + [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); + % Change matrix to upper triangular matrix + [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); + % Get the solution end % end function function checkIfMatrixIsSquareMatrix(Matrix) @@ -14,79 +26,81 @@ function checkIfMatrixIsSquareMatrix(Matrix) end % end function function [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector) - for i1 = 1 : Columns - 1 - m = max(Matrix(i1:Columns,i1)); - partialPivoting(Matrix, Vector, i1, m); - [Matrix, Vector] = gaussianElimination(i1, Columns, Matrix, Vector); + for j = 1 : Columns + centralElement = max(Matrix(j:Columns,j)); + % we stay in the same row (j) but we change columns, as in the + % textbook + [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns); + % ensures that a_kk != 0 and reduces errors + [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector); + % change matrix into upper triangular matrix end % end for end % end function -function [Matrix, Vector] = partialPivoting(Matrix, Vector, i1, m) - [~,Columns] = size(Matrix); - for i3 = i1 : Columns - partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m); +function [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns) + for k = j : Columns + partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement); end % end for end % end function -function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m) - if Matrix(i3,i1) == m - swapRowMatrix(Matrix, i1, i3); - swapValueVector(Vector, i1, i3); +function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement) + if Matrix(k,j) == centralElement + swapRowMatrix(Matrix, j, k); % swap jth row with kth row + swapValueVector(Vector, j, k); % swap jth value with kth value end % end if end % end function -function Matrix = swapRowMatrix(Matrix, i1, i3) - tempRow = Matrix(i1 , :); - Matrix(i1 , :) = Matrix(i3, :); - Matrix(i3, :) = tempRow; +function Matrix = swapRowMatrix(Matrix, j, k) + temp = Matrix(j , :); % ' : ' denote "all elements in jth row" + Matrix(j , :) = Matrix(k, :); + Matrix(k, :) = temp; % temp equal to previous value of jth row end -function Vector = swapValueVector(Vector, i1, i3) - tempVal = Vector(i1); - Vector(i1) = Vector(i3); - Vector(i3) = tempVal; +function Vector = swapValueVector(Vector, j, k) + temp = Vector(j); + Vector(j) = Vector(k); + Vector(k) = temp; % temp equal to previous value of k element of vector end % end function -function [Matrix, Vector] = gaussianElimination(i1, Columns, Matrix, Vector) - for i2 = i1 + 1 : Columns - - l = Matrix(i2,i1) / Matrix(i1,i1); - [Matrix, Vector] = substractRows(Matrix, Vector, i2, l, i1, Columns); - +function [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector) + for i = j + 1 : Columns + rowMultiplier = Matrix(i,j) / Matrix(j,j); + [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns); end % end for end % end function -function [Matrix, Vector] = substractRows(Matrix, Vector, i2, l, i1, Columns) - Vector(i2,1) = Vector(i2,1) - l * Vector(i1, 1); - for t = 1 : Columns - Matrix(i2,t) = Matrix(i2,t) - l * Matrix(i1, t); +function [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns) + Vector(i) = Vector(i) - rowMultiplier * Vector(j); + for curentColumn = 1 : Columns + Matrix(i,curentColumn) = Matrix(i,curentColumn) - rowMultiplier * Matrix(j, curentColumn); end % end for end % end function function [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector) - for i3 = Columns : -1 : 1 - - E = 0; - for iter = i3+1 : Columns - E = E + Matrix(i3,iter) * x(iter,1); + for k = Columns : -1 : 1 + % Start at final column and move by -1 each iteration until we reach 1 + equation = 0; + for j = k+1 : Columns + equation = equation + Matrix(k,j) * x(j, 1); + % even though x is a vector we still need to put '1' to ensure + % that number of columns in the first matrix matches number of + % rows in second matrix end % end for - x(i3, 1) = (Vector(i3,1) - E) / Matrix(i3,i3); + x(k, 1) = (Vector(k,1) - equation) / Matrix(k,k); + % even though x is a vector we still need to put '1' to ensure + % that we do not exceed array bounds end % end for end % end function -function x = iterativeResidualCorrection(Matrix, x, Vector) - r = Matrix*x - Vector; - euclideanColumnsormOfR = norm(r); - new_euclideanColumnsormOfR = euclideanColumnsormOfR; - x = improveSolution(x, new_euclideanColumnsormOfR, euclideanColumnsormOfR, Matrix, Vector); +function x = iterativeResidualCorrection(A, x, b) + r = A*x - b; + euclideanNormOfR = norm(r); + new_euclideanNormOfR = euclideanNormOfR; + while new_euclideanNormOfR <= euclideanNormOfR + euclideanNormOfR = new_euclideanNormOfR; + r = A*x - b; 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1141 hyphenation exceptions out of 8191 + 81i,7n,88p,292b,1379s stack positions out of 5000i,500n,10000p,200000b,80000s +{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc}{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-ts1.enc}{/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc} +Output written on projectA.pdf (18 pages, 254784 bytes). +PDF statistics: + 520 PDF objects out of 1000 (max. 8388607) + 478 compressed objects within 5 object streams + 186 named destinations out of 1000 (max. 500000) + 305 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/ENUME/projectA/projectA.out b/ENUME/projectA/projectA.out new file mode 100644 index 00000000..6a02b063 --- /dev/null +++ b/ENUME/projectA/projectA.out @@ -0,0 +1,38 @@ +\BOOKMARK [0][-]{chapter.1}{Problem 1 - Finding machine epsilion}{}% 1 +\BOOKMARK [1][-]{section.1.1}{Problem}{chapter.1}% 2 +\BOOKMARK [1][-]{section.1.2}{Theoretical Introduction}{chapter.1}% 3 +\BOOKMARK [2][-]{subsection.1.2.1}{Definition of machine epsilion}{section.1.2}% 4 +\BOOKMARK [2][-]{subsection.1.2.2}{Practical applications of machine epsilion}{section.1.2}% 5 +\BOOKMARK [1][-]{section.1.3}{Solution}{chapter.1}% 6 +\BOOKMARK [2][-]{subsection.1.3.1}{Matlab code}{section.1.3}% 7 +\BOOKMARK [1][-]{section.1.4}{Discussion of the result}{chapter.1}% 8 +\BOOKMARK [0][-]{chapter.2}{Problem 2 - Solving a system of n linear equations - indicated method}{}% 9 +\BOOKMARK [1][-]{section.2.1}{Problem}{chapter.2}% 10 +\BOOKMARK [1][-]{section.2.2}{Theoretical Introduction}{chapter.2}% 11 +\BOOKMARK [2][-]{subsection.2.2.1}{Transform matrix into upper-triangular matrix}{section.2.2}% 12 +\BOOKMARK [2][-]{subsection.2.2.2}{Backward substitution}{section.2.2}% 13 +\BOOKMARK [2][-]{subsection.2.2.3}{Partial Pivoting}{section.2.2}% 14 +\BOOKMARK [1][-]{section.2.3}{Solution}{chapter.2}% 15 +\BOOKMARK [2][-]{subsection.2.3.1}{Main function}{section.2.3}% 16 +\BOOKMARK [2][-]{subsection.2.3.2}{checkIfMatrixIsSquareMatrix}{section.2.3}% 17 +\BOOKMARK [2][-]{subsection.2.3.3}{gaussianEliminationWithPartialPivoting}{section.2.3}% 18 +\BOOKMARK [2][-]{subsection.2.3.4}{partialPivoting}{section.2.3}% 19 +\BOOKMARK [2][-]{subsection.2.3.5}{partialPivotingSwapOneRow}{section.2.3}% 20 +\BOOKMARK [2][-]{subsection.2.3.6}{swapRowMatrix}{section.2.3}% 21 +\BOOKMARK [2][-]{subsection.2.3.7}{swapValueVector}{section.2.3}% 22 +\BOOKMARK [2][-]{subsection.2.3.8}{gaussianElimination}{section.2.3}% 23 +\BOOKMARK [2][-]{subsection.2.3.9}{substractRows}{section.2.3}% 24 +\BOOKMARK [2][-]{subsection.2.3.10}{backSubstitutionPhase}{section.2.3}% 25 +\BOOKMARK [2][-]{subsection.2.3.11}{iterativeResidualCorrection}{section.2.3}% 26 +\BOOKMARK [2][-]{subsection.2.3.12}{improveSolution}{section.2.3}% 27 +\BOOKMARK [1][-]{section.2.4}{Discussion of the result}{chapter.2}% 28 +\BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 29 +\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 30 +\BOOKMARK [1][-]{section.3.2}{Theoretical introduction}{chapter.3}% 31 +\BOOKMARK [1][-]{section.3.3}{Solution}{chapter.3}% 32 +\BOOKMARK [1][-]{section.3.4}{Discussion of the result}{chapter.3}% 33 +\BOOKMARK [0][-]{chapter.4}{Problem 4 - QR method of finding eigenvalues}{}% 34 +\BOOKMARK [1][-]{section.4.1}{Problem}{chapter.4}% 35 +\BOOKMARK [1][-]{section.4.2}{Theoretical introduction}{chapter.4}% 36 +\BOOKMARK [1][-]{section.4.3}{Solution}{chapter.4}% 37 +\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 38 diff --git a/ENUME/projectA/projectA.pdf b/ENUME/projectA/projectA.pdf index 098d3b31..c15ed556 100755 Binary files a/ENUME/projectA/projectA.pdf and b/ENUME/projectA/projectA.pdf differ diff --git a/ENUME/projectA/projectA.synctex.gz b/ENUME/projectA/projectA.synctex.gz new file mode 100644 index 00000000..7355006b Binary files /dev/null and b/ENUME/projectA/projectA.synctex.gz differ diff --git a/ENUME/projectA/projectA.tex b/ENUME/projectA/projectA.tex index 209c80a0..816d4f63 100755 --- a/ENUME/projectA/projectA.tex +++ b/ENUME/projectA/projectA.tex @@ -107,7 +107,6 @@ Display difference between calculated macheps and $2^{-52}$: As expected they are all equal to eachother. It means that our method of calculating macheps was correct. - \chapter{Problem 2 - Solving a system of n linear equations - indicated method} \section{Problem} @@ -115,7 +114,7 @@ Write a program solving a system of \textit{n} linear equations Ax = b using the \section{Theoretical Introduction} Gaussian elimination with partial pivoting consists of 3 main steps: -\subsection{Transform system of equation into an upper-triangular matrix} +\subsection{Transform matrix into upper-triangular matrix} \subsubsection{Starting conditions} We start with the system of linear equations looking like this: @@ -134,7 +133,9 @@ We start with the system of linear equations looking like this: \end{matrix} \] -In order for this method to work all the elements of "diagonal" line - $ a_{11}, a_{22}, \dots, a_{nn} $ must be different from zero since we will be dividing by them. +In order for this method to work all the elements of "diagonal" line: +\[ a_{11}, a_{22}, \dots, a_{nn} \] +Must be different from zero since we will be dividing by them. We will denote rows as '$w_i$' where 'i' is number of the row. @@ -222,25 +223,269 @@ We do it before each Gaussian elimination step since this will lead to smaller e Then we swap this row with k-th row. Since the matrix we use is assumed to be nonsingular then $|{a_{ik}^{k}}| \neq 0$ will be always true. After that we continue with the Gaussian elimination method. +\newpage \section{Solution} \subsection{Main function} \begin{simplechar} \begin{lstlisting} -function x = indicatedMethod(Matrix, Vector) - [~,Columns] = size(Matrix); +function x = indicatedMethod(Matrix, Vector) % Name of the method as in the textbook +% x stands for obtained result + [~,Columns] = size(Matrix); % We need to know how big the matrix is in next steps + % notice the '~', since we assume we use square matrix, we do not need + % to have another variable for number of rows since it is the same as + % number of columns checkIfMatrixIsSquareMatrix(Matrix); [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); + % Change matrix to upper triangular matrix [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); - x = iterativeResidualCorrection(Matrix, x, Vector); + % Get the solution + x = iterativeResidualCorrection(Matrix, x, Vector); % Improve on the solution +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{checkIfMatrixIsSquareMatrix} +\begin{simplechar} +\begin{lstlisting} +function checkIfMatrixIsSquareMatrix(Matrix) + [Rows,Columns] = size(Matrix); + if Rows ~= Columns + error ('Matrix is not square matrix!'); + end % end if +end % end function +\end{lstlisting} +\end{simplechar} + +\newpage +\subsection{gaussianEliminationWithPartialPivoting} +\begin{simplechar} +\begin{lstlisting} +function [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector) + for j = 1 : Columns + centralElement = max(Matrix(j:Columns,j)); + % we stay in the same row (j) but we change columns, as in the + % textbook + [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns); + % ensures that a_kk != 0 and reduces errors + [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector); + % change matrix into upper triangular matrix + end % end for +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{partialPivoting} +\begin{simplechar} +\begin{lstlisting} +function [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns) + for k = j : Columns + partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement); + end % end for +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{partialPivotingSwapOneRow} +\begin{simplechar} +\begin{lstlisting} +function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement) + if Matrix(k,j) == centralElement + swapRowMatrix(Matrix, j, k); % swap jth row with kth row + swapValueVector(Vector, j, k); % swap jth value with kth value + end % end if +end % end function +\end{lstlisting} +\end{simplechar} + +\newpage +\subsection{swapRowMatrix} +\begin{simplechar} +\begin{lstlisting} +function Matrix = swapRowMatrix(Matrix, j, k) + temp = Matrix(j , :); % ' : ' denote "all elements in jth row" + Matrix(j , :) = Matrix(k, :); + Matrix(k, :) = temp; % temp equal to previous value of jth row +end +\end{lstlisting} +\end{simplechar} + +\subsection{swapValueVector} +\begin{simplechar} +\begin{lstlisting} +function Vector = swapValueVector(Vector, j, k) + temp = Vector(j); + Vector(j) = Vector(k); + Vector(k) = temp; % temp equal to previous value of k element of vector +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{gaussianElimination} +\begin{simplechar} +\begin{lstlisting} +function [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector) + for i = j + 1 : Columns + rowMultiplier = Matrix(i,j) / Matrix(j,j); + [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns); + end % end for +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{substractRows} +\begin{simplechar} +\begin{lstlisting} +function [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns) + Vector(i) = Vector(i) - rowMultiplier * Vector(j); + for curentColumn = 1 : Columns + Matrix(i,curentColumn) = Matrix(i,curentColumn) - rowMultiplier * Matrix(j, curentColumn); + end % end for +end % end function +\end{lstlisting} +\end{simplechar} + +\newpage +\subsection{backSubstitutionPhase} +\begin{simplechar} +\begin{lstlisting} +function [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector) + for k = Columns : -1 : 1 + % Start at final column and move by -1 each iteration until we reach 1 + equation = 0; + for j = k+1 : Columns + equation = equation + Matrix(k,j) * x(j, 1); + % even though x is a vector we still need to put '1' to ensure + % that number of columns in the first matrix matches number of + % rows in second matrix + end % end for + + x(k, 1) = (Vector(k,1) - equation) / Matrix(k,k); + % even though x is a vector we still need to put '1' to ensure + % that we do not exceed array bounds + end % end for +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{iterativeResidualCorrection} +\begin{simplechar} +\begin{lstlisting} +function x = iterativeResidualCorrection(Matrix, x, Vector) + residuum = Matrix*x - Vector; % as in the book + newResiduum = residuum; + x = improveSolution(x, newResiduum, residuum, Matrix, Vector); +end % end function +\end{lstlisting} +\end{simplechar} + +\subsection{improveSolution} +\begin{simplechar} +\begin{lstlisting} +function x = improveSolution(x, newResiduum, oldResiduum, Matrix, Vector) + while newResiduum <= oldResiduum + oldResiduum = newResiduum; + residuum = Matrix*x - Vector; + x = x - residuum; + newResiduum = residuum; + end % end while end % end function \end{lstlisting} \end{simplechar} - \section{Discussion of the result} +Solutions vectors for matrix A and vector A and n = 16: +\[ x_{algorithm} = \left( \begin{array}{cc} + -0.930056653761514 \\ + -1.223503294617857 \\ + -1.273786563425385 \\ + -1.231189728991652 \\ + -1.153115956882885 \\ + -1.061491474990357 \\ + -0.964691801421511 \\ + -0.865917262607523 \\ + -0.766393319734375 \\ + -0.666596029928932 \\ + -0.566728103385765 \\ + -0.466921613561691 \\ + -0.367370070632649 \\ + -0.268521931669581 \\ + -0.171529057709426 \\ + -0.079398574792031 +\end{array} \right) +% +x_{Matlab Method} = \left( = \begin{array}{cc} + -0.930056653761507 \\ + -1.223503294617854 \\ + -1.273786563425390 \\ + -1.231189728991649 \\ + -1.153115956882891 \\ + -1.061491474990357 \\ + -0.964691801421514 \\ + -0.865917262607518 \\ + -0.766393319734373 \\ + -0.666596029928935 \\ + -0.566728103385765 \\ + -0.466921613561692 \\ + -0.367370070632646 \\ + -0.268521931669579 \\ + -0.171529057709426 \\ + -0.079398574792031 +\end{array} \right) +\] +Error for 'A' method for algorithm: \[ 3.383918772654241 \] +Error for 'A' method for matlab method: \[ 3.383918772654241 \] +\newpage +Solutions vectors for matrix B and vector B and n = 16 (Both multiplied by $ 1.0e+17 $): +\[ x_{algorithm} = \left( \begin{array}{cc} + 0.000001960155675 \\ + -0.000102773501571 \\ + 0.001959454282882 \\ + -0.018894079120425 \\ + 0.104895022735396 \\ + -0.352396798852209 \\ + 0.705545645736628 \\ + -0.728747526649489 \\ + 0.112818247768452 \\ + 0.261440075930356 \\ + 0.953713133491034 \\ + -3.080986443185790 \\ + 3.765178552913233 \\ + -2.440218622397594 \\ + 0.834768953546100 \\ + -0.118974790826378 +\end{array} \right) +% +x_{Matlab Method} = \left( = \begin{array}{cc} + 0.000001102587209 \\ + -0.000066546298462 \\ + 0.001476714765758 \\ + -0.016859999627589 \\ + 0.113920718370153 \\ + -0.488374161741872 \\ + 1.368641128884513 \\ + -2.495283439985873 \\ + 2.793405296264694 \\ + -1.547642305352008 \\ + -0.035332172403445 \\ + 0.154726025421297 \\ + 0.807694426359552 \\ + -1.133485136703852 \\ + 0.592810708065954 \\ + -0.115632364630554 +\end{array} \right) +\] + +Error for 'B' method for algorithm: \[ 5.699979882700911e+17 \] +Error for 'B' method for matlab method: \[ 4.569118543317684e+17 \] + + + + + +\addtocontents{toc}{\protect\newpage} \chapter{Problem 3 - Solving a system of n linear equations - iterative algorithm} \section{Problem} diff --git a/ENUME/projectA/projectA.toc b/ENUME/projectA/projectA.toc new file mode 100644 index 00000000..c4736621 --- /dev/null +++ b/ENUME/projectA/projectA.toc @@ -0,0 +1,43 @@ +\contentsline {chapter}{\numberline {1}Problem 1 - Finding machine epsilion}{3}{chapter.1}% +\contentsline {section}{\numberline {1.1}Problem}{3}{section.1.1}% +\contentsline {section}{\numberline {1.2}Theoretical Introduction}{3}{section.1.2}% +\contentsline {subsection}{\numberline {1.2.1}Definition of machine epsilion}{3}{subsection.1.2.1}% +\contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{3}{subsection.1.2.2}% +\contentsline {section}{\numberline {1.3}Solution}{4}{section.1.3}% +\contentsline {subsection}{\numberline {1.3.1}Matlab code}{4}{subsection.1.3.1}% +\contentsline {section}{\numberline {1.4}Discussion of the result}{4}{section.1.4}% +\contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{6}{chapter.2}% +\contentsline {section}{\numberline {2.1}Problem}{6}{section.2.1}% +\contentsline {section}{\numberline {2.2}Theoretical Introduction}{6}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Transform matrix into upper-triangular matrix}{6}{subsection.2.2.1}% +\contentsline {subsubsection}{Starting conditions}{6}{section*.2}% +\contentsline {subsubsection}{Zeroing first column}{7}{section*.3}% +\contentsline {subsubsection}{Zeroing second column}{7}{section*.4}% +\contentsline {subsubsection}{Zeroing next columns}{7}{section*.5}% +\contentsline {subsection}{\numberline {2.2.2}Backward substitution}{8}{subsection.2.2.2}% +\contentsline {subsection}{\numberline {2.2.3}Partial Pivoting}{8}{subsection.2.2.3}% +\contentsline {section}{\numberline {2.3}Solution}{9}{section.2.3}% +\contentsline {subsection}{\numberline {2.3.1}Main function}{9}{subsection.2.3.1}% +\contentsline {subsection}{\numberline {2.3.2}checkIfMatrixIsSquareMatrix}{9}{subsection.2.3.2}% +\contentsline {subsection}{\numberline {2.3.3}gaussianEliminationWithPartialPivoting}{10}{subsection.2.3.3}% +\contentsline {subsection}{\numberline {2.3.4}partialPivoting}{10}{subsection.2.3.4}% +\contentsline {subsection}{\numberline {2.3.5}partialPivotingSwapOneRow}{10}{subsection.2.3.5}% +\contentsline {subsection}{\numberline {2.3.6}swapRowMatrix}{11}{subsection.2.3.6}% +\contentsline {subsection}{\numberline {2.3.7}swapValueVector}{11}{subsection.2.3.7}% +\contentsline {subsection}{\numberline {2.3.8}gaussianElimination}{11}{subsection.2.3.8}% +\contentsline {subsection}{\numberline {2.3.9}substractRows}{11}{subsection.2.3.9}% +\contentsline {subsection}{\numberline {2.3.10}backSubstitutionPhase}{12}{subsection.2.3.10}% +\contentsline {subsection}{\numberline {2.3.11}iterativeResidualCorrection}{12}{subsection.2.3.11}% +\contentsline {subsection}{\numberline {2.3.12}improveSolution}{12}{subsection.2.3.12}% +\contentsline {section}{\numberline {2.4}Discussion of the result}{13}{section.2.4}% +\newpage +\contentsline {chapter}{\numberline {3}Problem 3 - Solving a system of n linear equations - iterative algorithm}{15}{chapter.3}% +\contentsline {section}{\numberline {3.1}Problem}{15}{section.3.1}% +\contentsline {section}{\numberline {3.2}Theoretical introduction}{15}{section.3.2}% +\contentsline {section}{\numberline {3.3}Solution}{15}{section.3.3}% +\contentsline {section}{\numberline {3.4}Discussion of the result}{15}{section.3.4}% +\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{16}{chapter.4}% +\contentsline {section}{\numberline {4.1}Problem}{16}{section.4.1}% +\contentsline {section}{\numberline {4.2}Theoretical introduction}{16}{section.4.2}% +\contentsline {section}{\numberline {4.3}Solution}{16}{section.4.3}% +\contentsline {section}{\numberline {4.4}Discussion of the result}{16}{section.4.4}% diff --git a/ENUME/projectA/usefullinks.txt b/ENUME/projectA/usefullinks.txt index 16490bc9..9b497fe8 100755 --- a/ENUME/projectA/usefullinks.txt +++ b/ENUME/projectA/usefullinks.txt @@ -32,4 +32,10 @@ https://tex.stackexchange.com/questions/526025/problem-with-vdotswithin - vdots https://www.overleaf.com/learn/latex/Brackets_and_Parentheses - Curly brackets latex -https://tex.stackexchange.com/questions/130516/how-to-write-something-vertically-below-another-math-symbol - Symbol vertically below another +https://tex.stackexchange.com/questions/130516/how-to-write-something-vertically-below-another-math-symbol - Symbol vertically below another + +https://tex.stackexchange.com/questions/103998/how-to-force-a-table-of-contents-to-start-new-page - new page at the start of chapter in table of contents + +https://www.codegrepper.com/code-examples/whatever/column+vector+latex - Vector in latex + +https://tex.stackexchange.com/questions/9901/how-to-have-matrices-side-by-side-in-latex - Matrices side by side in LaTeX