From c92984e9f5a23d213765d0076ab7123378d39053 Mon Sep 17 00:00:00 2001 From: PolishPigeon Date: Sat, 6 Nov 2021 06:22:19 +0100 Subject: [PATCH] finishing touches on matlab code and transfering code and solutions to latex --- ENUME/projectA/indicatedMethod.m | 124 ++-- ENUME/projectA/projectA.aux | 71 +++ ENUME/projectA/projectA.fdb_latexmk | 152 +++++ ENUME/projectA/projectA.fls | 846 ++++++++++++++++++++++++++++ ENUME/projectA/projectA.log | 591 +++++++++++++++++++ ENUME/projectA/projectA.out | 38 ++ ENUME/projectA/projectA.pdf | Bin 232481 -> 254784 bytes ENUME/projectA/projectA.synctex.gz | Bin 0 -> 154065 bytes ENUME/projectA/projectA.tex | 259 ++++++++- ENUME/projectA/projectA.toc | 43 ++ ENUME/projectA/usefullinks.txt | 8 +- 11 files changed, 2069 insertions(+), 63 deletions(-) create mode 100644 ENUME/projectA/projectA.aux create mode 100644 ENUME/projectA/projectA.fdb_latexmk create mode 100644 ENUME/projectA/projectA.fls create mode 100644 ENUME/projectA/projectA.log create mode 100644 ENUME/projectA/projectA.out create mode 100644 ENUME/projectA/projectA.synctex.gz create mode 100644 ENUME/projectA/projectA.toc 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; + x = x - r; + new_euclideanNormOfR = norm(r); + end end % end function - -function x = improveSolution(x, new_euclideanColumnsormOfR, euclideanColumnsormOfR, Matrix, Vector) - while new_euclideanColumnsormOfR <= euclideanColumnsormOfR - euclideanColumnsormOfR = new_euclideanColumnsormOfR; - r = Matrix*x - Vector; - x = x - r; - new_euclideanColumnsormOfR = norm(r); - end % end while -end % end function \ No newline at end of file diff --git a/ENUME/projectA/projectA.aux b/ENUME/projectA/projectA.aux new file mode 100644 index 00000000..9af8525f --- /dev/null +++ b/ENUME/projectA/projectA.aux @@ -0,0 +1,71 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline 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[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 098d3b31ee5536a8ab6639af6ed98f1eeef24812..c15ed55603872887cc2c7b3f6be8ec98888f4b84 100755 GIT binary patch delta 143771 zcmZs>V{oQT)HNF0wr$(?#I~)8CzCt2ZQHhOPHatV=MK(1@2R&=o%+5%U0wUvb#-^G zwf0)OARVcH6d8?DSwf1Dm5CFMa{kZA8XPMpcLSO;C@60NDmoZ1cTznk9pFmG;Aa*O 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