diff --git a/ENUME/projectA/backSubstitutionPhase.m b/ENUME/projectA/backSubstitutionPhase.m deleted file mode 100644 index 654e09c8..00000000 --- a/ENUME/projectA/backSubstitutionPhase.m +++ /dev/null @@ -1,11 +0,0 @@ -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); - end % end for - - x(i3, 1) = (Vector(i3,1) - E) / Matrix(i3,i3); - end % end for -end % end function \ No newline at end of file diff --git a/ENUME/projectA/checkIfMatrixIsSquareMatrix.m b/ENUME/projectA/checkIfMatrixIsSquareMatrix.m deleted file mode 100644 index 66d7a6f4..00000000 --- a/ENUME/projectA/checkIfMatrixIsSquareMatrix.m +++ /dev/null @@ -1,5 +0,0 @@ -function checkIfMatrixIsSquareMatrix(Matrix) -[Rows,Columns] = size(Matrix); -if Rows ~= Columns - error ('Matrix is not square matrix!'); -end % end if \ No newline at end of file diff --git a/ENUME/projectA/fileAttrib.m b/ENUME/projectA/fileAttrib.m deleted file mode 100644 index ce9b8a4b..00000000 --- a/ENUME/projectA/fileAttrib.m +++ /dev/null @@ -1 +0,0 @@ -fileattrib debilMode.m \ No newline at end of file diff --git a/ENUME/projectA/findmacheps.mat b/ENUME/projectA/findmacheps.mat deleted file mode 100755 index 9b268693..00000000 Binary files a/ENUME/projectA/findmacheps.mat and /dev/null differ diff --git a/ENUME/projectA/gaussianElimination.asv b/ENUME/projectA/gaussianElimination.asv deleted file mode 100755 index 435cfd01..00000000 --- a/ENUME/projectA/gaussianElimination.asv +++ /dev/null @@ -1,61 +0,0 @@ -function x = gaussianElimination(A, b) % A - matrix, b - vector - [~, Height] = size(A); - % Find column with greatest value element - for i = 1 : Height - 1 - m = max(A(i:Height, i)); - % Find row with greatest value element - for k = i : Height - if A(k, i) == m % If the element has the biggest value we now - % have the value of two rows that we want to swap - % Swap row in matrix A % - swapRow(A, i, k); - swapValue(b, i, k); - break; - end % end if - end % end for - % Swap value in vector B % - - end % end for - - for j = i + 1 : Height - l = A(j, i) / A(i, i); - b(j, 1) = b(j, 1) - l * b(i, 1); - - for t = 1 : Height - A(j, t) = A(j, t) - l * A(i, t); - end % end for - end % end for - - - -x = zeros(Height, 1); - - - -for k = Height: -1:1 - E = 0; - for iter = k + 1 : Height - E = E + A(k, iter) * x(iter, 1); - end - - x(k, 1) = (b(k, 1) - E) / A(k, k); -end - - -end % end function - - - - -function swapRow(A, j, k) - temp = A(j, :); - A(j, :) = A(k, :); - A(k, :) = temp; -end % end function - -function swapValue(b, j, k) - temp = b(j); - b(j) = b(k); - b(k) = temp; -end % end function - diff --git a/ENUME/projectA/gaussianElimination.m b/ENUME/projectA/gaussianElimination.m deleted file mode 100755 index da90cef0..00000000 --- a/ENUME/projectA/gaussianElimination.m +++ /dev/null @@ -1,8 +0,0 @@ -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); - - end % end for -end % end function \ No newline at end of file diff --git a/ENUME/projectA/gaussianEliminationWithPartialPivoting.m b/ENUME/projectA/gaussianEliminationWithPartialPivoting.m deleted file mode 100644 index 71556374..00000000 --- a/ENUME/projectA/gaussianEliminationWithPartialPivoting.m +++ /dev/null @@ -1,7 +0,0 @@ -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); - end % end for -end % end function \ No newline at end of file diff --git a/ENUME/projectA/improveSolution.m b/ENUME/projectA/improveSolution.m deleted file mode 100644 index f535adfa..00000000 --- a/ENUME/projectA/improveSolution.m +++ /dev/null @@ -1,8 +0,0 @@ -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/indicatedMethod.asv b/ENUME/projectA/indicatedMethod.asv deleted file mode 100644 index 25552cd7..00000000 --- a/ENUME/projectA/indicatedMethod.asv +++ /dev/null @@ -1,7 +0,0 @@ -function x = indicatedMethod(Matrix, Vector) - [~,Columns] = size(Matrix); - checkIfMatrixIsSquareMatrix(Matrix); - [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); - [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); - x = iterativeResidualCorrection(Matrix, x, Vector); -end % end function \ No newline at end of file diff --git a/ENUME/projectA/indicatedMethod.m b/ENUME/projectA/indicatedMethod.m index 25552cd7..3bce3539 100644 --- a/ENUME/projectA/indicatedMethod.m +++ b/ENUME/projectA/indicatedMethod.m @@ -4,4 +4,89 @@ function x = indicatedMethod(Matrix, Vector) [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); x = iterativeResidualCorrection(Matrix, x, Vector); +end % end function + +function checkIfMatrixIsSquareMatrix(Matrix) + [Rows,Columns] = size(Matrix); + if Rows ~= Columns + error ('Matrix is not square matrix!'); + end % end if +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); + 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); + 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); + end % end if +end % end function + +function Matrix = swapRowMatrix(Matrix, i1, i3) + tempRow = Matrix(i1 , :); + Matrix(i1 , :) = Matrix(i3, :); + Matrix(i3, :) = tempRow; +end + +function Vector = swapValueVector(Vector, i1, i3) + tempVal = Vector(i1); + Vector(i1) = Vector(i3); + Vector(i3) = tempVal; +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); + + 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); + 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); + end % end for + + x(i3, 1) = (Vector(i3,1) - E) / Matrix(i3,i3); + 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); +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/iterativeResidualCorrection.m b/ENUME/projectA/iterativeResidualCorrection.m deleted file mode 100644 index 77af1d37..00000000 --- a/ENUME/projectA/iterativeResidualCorrection.m +++ /dev/null @@ -1,6 +0,0 @@ -function x = iterativeResidualCorrection(Matrix, x, Vector) - r = Matrix*x - Vector; - euclideanColumnsormOfR = norm(r); - new_euclideanColumnsormOfR = euclideanColumnsormOfR; - x = improveSolution(x, new_euclideanColumnsormOfR, euclideanColumnsormOfR, Matrix, Vector); -end % end function \ No newline at end of file diff --git a/ENUME/projectA/partialPivoting.m b/ENUME/projectA/partialPivoting.m deleted file mode 100644 index 00fb8c5f..00000000 --- a/ENUME/projectA/partialPivoting.m +++ /dev/null @@ -1,6 +0,0 @@ -function [Matrix, Vector] = partialPivoting(Matrix, Vector, i1, m) - [~,Columns] = size(Matrix); - for i3 = i1 : Columns - partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m); - end % end for -end % end function \ No newline at end of file diff --git a/ENUME/projectA/partialPivotingSwap.m b/ENUME/projectA/partialPivotingSwap.m deleted file mode 100644 index 2c6b3d83..00000000 --- a/ENUME/projectA/partialPivotingSwap.m +++ /dev/null @@ -1,6 +0,0 @@ -function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m) - if Matrix(i3,i1) == m - swapRowMatrix(Matrix, i1, i3); - swapValueVector(Vector, i1, i3); - end % end if -end % end function \ No newline at end of file diff --git a/ENUME/projectA/partialPivotingSwapOneRow.m b/ENUME/projectA/partialPivotingSwapOneRow.m deleted file mode 100644 index 2c6b3d83..00000000 --- a/ENUME/projectA/partialPivotingSwapOneRow.m +++ /dev/null @@ -1,6 +0,0 @@ -function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m) - if Matrix(i3,i1) == m - swapRowMatrix(Matrix, i1, i3); - swapValueVector(Vector, i1, i3); - end % end if -end % end 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302 PDF objects out of 1000 (max. 8388607) - 269 compressed objects within 3 object streams - 63 named destinations out of 1000 (max. 500000) - 209 words of extra memory for PDF output out of 10000 (max. 10000000) - diff --git a/ENUME/projectA/projectA.out b/ENUME/projectA/projectA.out deleted file mode 100755 index b44ae505..00000000 --- a/ENUME/projectA/projectA.out +++ /dev/null @@ -1,26 +0,0 @@ -\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 system of equation into an 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 [1][-]{section.2.4}{Discussion of the result}{chapter.2}% 16 -\BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 17 -\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 18 -\BOOKMARK [1][-]{section.3.2}{Theoretical introduction}{chapter.3}% 19 -\BOOKMARK [1][-]{section.3.3}{Solution}{chapter.3}% 20 -\BOOKMARK [1][-]{section.3.4}{Discussion of the result}{chapter.3}% 21 -\BOOKMARK [0][-]{chapter.4}{Problem 4 - QR method of finding eigenvalues}{}% 22 -\BOOKMARK [1][-]{section.4.1}{Problem}{chapter.4}% 23 -\BOOKMARK [1][-]{section.4.2}{Theoretical introduction}{chapter.4}% 24 -\BOOKMARK [1][-]{section.4.3}{Solution}{chapter.4}% 25 -\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 26 diff --git a/ENUME/projectA/projectA.pdf b/ENUME/projectA/projectA.pdf index 72e20282..098d3b31 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 deleted file mode 100644 index a58c1a8f..00000000 Binary files a/ENUME/projectA/projectA.synctex.gz and /dev/null differ diff --git a/ENUME/projectA/projectA.tex b/ENUME/projectA/projectA.tex index 91b20756..209c80a0 100755 --- a/ENUME/projectA/projectA.tex +++ b/ENUME/projectA/projectA.tex @@ -224,6 +224,21 @@ Then we swap this row with k-th row. Since the matrix we use is assumed to be no \section{Solution} +\subsection{Main function} +\begin{simplechar} +\begin{lstlisting} +function x = indicatedMethod(Matrix, Vector) + [~,Columns] = size(Matrix); + checkIfMatrixIsSquareMatrix(Matrix); + [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); + [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); + x = iterativeResidualCorrection(Matrix, x, Vector); +end % end function +\end{lstlisting} +\end{simplechar} + + + \section{Discussion of the result} \chapter{Problem 3 - Solving a system of n linear equations - iterative algorithm} diff --git a/ENUME/projectA/projectA.toc b/ENUME/projectA/projectA.toc deleted file mode 100755 index 4e7626fc..00000000 --- a/ENUME/projectA/projectA.toc +++ /dev/null @@ -1,30 +0,0 @@ -\contentsline {chapter}{\numberline {1}Problem 1 - Finding machine epsilion}{2}{chapter.1}% -\contentsline {section}{\numberline {1.1}Problem}{2}{section.1.1}% -\contentsline {section}{\numberline {1.2}Theoretical Introduction}{2}{section.1.2}% -\contentsline {subsection}{\numberline {1.2.1}Definition of machine epsilion}{2}{subsection.1.2.1}% -\contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{2}{subsection.1.2.2}% -\contentsline {section}{\numberline {1.3}Solution}{3}{section.1.3}% -\contentsline {subsection}{\numberline {1.3.1}Matlab code}{3}{subsection.1.3.1}% -\contentsline {section}{\numberline {1.4}Discussion of the result}{3}{section.1.4}% -\contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{5}{chapter.2}% -\contentsline {section}{\numberline {2.1}Problem}{5}{section.2.1}% -\contentsline {section}{\numberline {2.2}Theoretical Introduction}{5}{section.2.2}% -\contentsline {subsection}{\numberline {2.2.1}Transform system of equation into an upper-triangular matrix}{5}{subsection.2.2.1}% -\contentsline {subsubsection}{Starting conditions}{5}{section*.2}% -\contentsline {subsubsection}{Zeroing first column}{6}{section*.3}% -\contentsline {subsubsection}{Zeroing second column}{6}{section*.4}% -\contentsline {subsubsection}{Zeroing next columns}{6}{section*.5}% -\contentsline {subsection}{\numberline {2.2.2}Backward substitution}{7}{subsection.2.2.2}% -\contentsline {subsection}{\numberline {2.2.3}Partial Pivoting}{7}{subsection.2.2.3}% -\contentsline {section}{\numberline {2.3}Solution}{7}{section.2.3}% -\contentsline {section}{\numberline {2.4}Discussion of the result}{7}{section.2.4}% -\contentsline {chapter}{\numberline {3}Problem 3 - Solving a system of n linear equations - iterative algorithm}{8}{chapter.3}% -\contentsline {section}{\numberline {3.1}Problem}{8}{section.3.1}% -\contentsline {section}{\numberline {3.2}Theoretical introduction}{8}{section.3.2}% -\contentsline {section}{\numberline {3.3}Solution}{8}{section.3.3}% -\contentsline {section}{\numberline {3.4}Discussion of the result}{8}{section.3.4}% -\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{9}{chapter.4}% -\contentsline {section}{\numberline {4.1}Problem}{9}{section.4.1}% -\contentsline {section}{\numberline {4.2}Theoretical introduction}{9}{section.4.2}% -\contentsline {section}{\numberline {4.3}Solution}{9}{section.4.3}% -\contentsline {section}{\numberline {4.4}Discussion of the result}{9}{section.4.4}% diff --git a/ENUME/projectA/substractRows b/ENUME/projectA/substractRows deleted file mode 100644 index 1c4c4ed9..00000000 --- a/ENUME/projectA/substractRows +++ /dev/null @@ -1,6 +0,0 @@ -function [Matrix, Vector] substractRows(Vector, i2, l, i1, t, 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); - end % end for -end % end function \ No newline at end of file diff --git a/ENUME/projectA/substractRows.m b/ENUME/projectA/substractRows.m deleted file mode 100644 index cf3e92cc..00000000 --- a/ENUME/projectA/substractRows.m +++ /dev/null @@ -1,6 +0,0 @@ -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); - end % end for -end % end function \ No newline at end of file diff --git a/ENUME/projectA/swapRowMatrix.m b/ENUME/projectA/swapRowMatrix.m deleted file mode 100644 index 916a57a3..00000000 --- a/ENUME/projectA/swapRowMatrix.m +++ /dev/null @@ -1,5 +0,0 @@ -function Matrix = swapRowMatrix(Matrix, i1, i3) - tempRow = Matrix(i1 , :); - Matrix(i1 , :) = Matrix(i3, :); - Matrix(i3, :) = tempRow; -end \ No newline at end of file diff --git a/ENUME/projectA/swapValueVector b/ENUME/projectA/swapValueVector deleted file mode 100644 index 2a91a010..00000000 --- a/ENUME/projectA/swapValueVector +++ /dev/null @@ -1,5 +0,0 @@ -function Vector = swapValueVector(Vector, i1, i3) - tempVal = Vector(i1); - Vector(i1) = Vector(i3); - Vector(i3) = tempVal; -end % end function \ No newline at end of file diff --git a/ENUME/projectA/swapValueVector.m b/ENUME/projectA/swapValueVector.m deleted file mode 100644 index 2a91a010..00000000 --- a/ENUME/projectA/swapValueVector.m +++ /dev/null @@ -1,5 +0,0 @@ -function Vector = swapValueVector(Vector, i1, i3) - tempVal = Vector(i1); - Vector(i1) = Vector(i3); - Vector(i3) = tempVal; -end % end function \ No newline at end of file diff --git a/ENUME/projectA/untitled.m b/ENUME/projectA/untitled.m deleted file mode 100755 index 6d2fff4a..00000000 --- a/ENUME/projectA/untitled.m +++ /dev/null @@ -1 +0,0 @@ -jebaćdsia \ No newline at end of file