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Cleaning up
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@ -4,7 +4,7 @@ function [eigenValues, whatIterationAreWeOn, Matrix] = QRShifts(Matrix)
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initialMatrix = Matrix;
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[eigenValues, whatIterationAreWeOn, matrixSize, minThreshold] = initiateValues(Matrix);
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[Matrix, whatIterationAreWeOn, eigenValues] = QRShiftLoop(matrixSize, Matrix, eigenValues, minThreshold, whatIterationAreWeOn);
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%dispResults(eigenValues, Matrix, whatIterationAreWeOn, eigenFromMatlab, initialMatrix);
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dispResults(eigenValues, Matrix, whatIterationAreWeOn, eigenFromMatlab, initialMatrix);
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end
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function [eigenValues, whatIterationAreWeOn, matrixSize, minThreshold] = initiateValues(Matrix)
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@ -1,117 +0,0 @@
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function x = indicatedMethod(Matrix, Vector) % Name of the method as in the textbook
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% x stands for obtained result
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checkIfMatrixIsSquareMatrix(Matrix);
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[Matrix, Vector, x] = solveSystem(Matrix, Vector);
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originalSolution = x;
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errorBeforeResidualCorrection = norm(Matrix*x - Vector);
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x = iterativeResidualCorrection(Matrix, x, Vector); % Improve on the solution
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disp("errorBeforeResidualCorrection")
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disp(errorBeforeResidualCorrection);
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disp("errorAfterResidualCorrection")
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disp(norm(Matrix*x - Vector));
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disp("Solution before residual correction:")
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disp(originalSolution);
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disp("Solution after residual correction:")
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disp(x);
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disp("A\b solution:")
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disp(Matrix\Vector);
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disp("A\b error:")
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disp(Matrix\Vector - );
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end % end function
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function [Matrix, Vector, x] = solveSystem(Matrix, Vector)
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[~,Columns] = size(Matrix); % We need to know how big the matrix is in next steps
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% notice the '~', since we assume we use square matrix, we do not need
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% to have another variable for number of rows since it is the same as
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% number of columns
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[Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector);
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% Change matrix to upper triangular matrix
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[Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector);
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% Get the solution
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end % end function
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function checkIfMatrixIsSquareMatrix(Matrix)
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[Rows,Columns] = size(Matrix);
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if Rows ~= Columns
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error ('Matrix is not square matrix!');
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end % end if
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end % end function
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function [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector)
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for j = 1 : Columns
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centralElement = max(Matrix(j:Columns,j));
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% we stay in the same row (j) but we change columns, as in the
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% textbook
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[Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns);
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% ensures that a_kk != 0 and reduces errors
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[Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector);
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% change matrix into upper triangular matrix
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end % end for
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end % end function
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function [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns)
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for k = j : Columns
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partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement);
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end % end for
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end % end function
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function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement)
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if Matrix(k,j) == centralElement
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swapRowMatrix(Matrix, j, k); % swap jth row with kth row
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swapValueVector(Vector, j, k); % swap jth value with kth value
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end % end if
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end % end function
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function Matrix = swapRowMatrix(Matrix, j, k)
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temp = Matrix(j , :); % ' : ' denote "all elements in jth row"
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Matrix(j , :) = Matrix(k, :);
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Matrix(k, :) = temp; % temp equal to previous value of jth row
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end
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function Vector = swapValueVector(Vector, j, k)
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temp = Vector(j);
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Vector(j) = Vector(k);
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Vector(k) = temp; % temp equal to previous value of k element of vector
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end % end function
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function [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector)
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for i = j + 1 : Columns
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rowMultiplier = Matrix(i,j) / Matrix(j,j);
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[Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns);
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end % end for
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end % end function
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function [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns)
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Vector(i) = Vector(i) - rowMultiplier * Vector(j);
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for curentColumn = 1 : Columns
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Matrix(i,curentColumn) = Matrix(i,curentColumn) - rowMultiplier * Matrix(j, curentColumn);
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end % end for
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end % end function
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function [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector)
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for k = Columns : -1 : 1
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% Start at final column and move by -1 each iteration until we reach 1
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equation = 0;
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for j = k+1 : Columns
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equation = equation + Matrix(k,j) * x(j, 1);
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% even though x is a vector we still need to put '1' to ensure
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% that number of columns in the first matrix matches number of
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% rows in second matrix
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end % end for
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x(k, 1) = (Vector(k,1) - equation) / Matrix(k,k);
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% even though x is a vector we still need to put '1' to ensure
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% that we do not exceed array bounds
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end % end for
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end % end function
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function x = iterativeResidualCorrection(A, x, b)
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r = A*x - b;
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for i = 1 : 100
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[A, b, deltaX] = solveSystem(A, r);
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newX = x - deltaX;
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r = A*newX - b;
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x = newX;
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end
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end % end function
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@ -14,7 +14,7 @@ function x = indicatedMethod(Matrix, Vector) % Name of the method as in the text
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disp(errorAfterResidualCorrection);
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disp("A\b error:")
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disp(norm(Matrix * (Matrix \ Vector) - Vector));
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%disp(Matrix\Vector);
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disp(Matrix\Vector);
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end % end function
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@ -1,4 +0,0 @@
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PWD /home/kuchy/Zlew/Studia/SEM_5/enume_done/Project/matlabproject/ENUME_Project_31/projectA
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INPUT /etc/texmf/web2c/texmf.cnf
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INPUT /usr/share/texmf/web2c/texmf.cnf
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INPUT /usr/share/texlive/texmf-dist/web2c/texmf.cnf
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@ -1,34 +0,0 @@
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Z tablicy: https://studia.elka.pw.edu.pl
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If you want to discuss, come within 15 minutes
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classes to make some meetings to consult projects to discuss them
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3 projects to be completed, time table available on studio elka
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first project November 12 deadline, report should be submitted to the report module on the studia elka, tasks have already been distributed, check report on studia elka, you will get the task on PDF file. today quarter past four. then you will be given task to project assignment B and C, Statham is responsible for first two projects, Marusak last project
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first two assignment 15 points each
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last one 12 points
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assessed by the content, by report and possibly by the interview, up to the guys the handle projects, entire group has been split into 3 dates. If you want to come to the different group just let the guy know
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check two files from studia elka how to obtain Matlab, second one contains installation code. No need for full installation only basic module will do.
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Tutorial - Matlab Primer
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1. Start Matlab
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2. Study material from Tatjewski book
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3. Get acquainted with Matlab
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4. Learn how to debug (Step by fashion)
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Project A
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4 Tasks to be completed
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1. Calculate the machine epsilon (Deliver a lot of background,
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Theoretical findings necessary to deliver solution to the given task
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The Definition of machine epsilon
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Practical applications
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The value of the epsilon should be verified (Best verification is to calculate manually, use some formula, take the value of machine epsilon from mathematical formula, maybe from some documentation maybe there is some standard, maybe Matlab has formula for it))
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2.System of linear equations: First method is Gaussian elimination with partial pivoting, second with full. Discuss Gaussian elimination, do we need pivoting, can we apply it without pivoting, if so under what conditions, trade off between partial and full pivoting, maybe there is a gain, maybe there is a cost in accuracy, what kind of trade off we deal with, solution error of system of linear equation, discuss solution error. What is interesting is analysis of solution. This course is about analysis (Not implementing!). Take errors from graph and discuss them, for some it will be acceptable for other it will not be, (10^-8 vs 10^-2) what causes the big error, MOSTLY ASSESSED ANALYSIS NOT THE NUMBERS. Is it guessing or based on theory.
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3. Solving system of linear equations using iterative method. You don't have the guarantee how many iterations will reach the solutions. Issue of convergence, method may converge or not , based on what conditions give the condition, there is a sufficient condition there is necessary and sufficient conditions (which is slower). What happens if the sufficient condition is not fulfilled? Answer that. If the necessary condition is fulfilled what's going to happen. If the sufficient condition is fulfilled we know everything. If just necessary we don't know everything. Stop tests, when you are going to stop the process, well design, clarify in report
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4. Exercise is about calculating Eigen value (wartości własne). With shifts and without shifts. 2 algorithms to be implemented from book. You need to compare those algorithms performance ana accuracy generation and there is command eg and you can use this to produce them and then compare your solution with Matlab
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MOSTLY ABOUT ANALYSIS NOT CODING, scientifically grounded, if something goes wrong explain it.
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Bring preliminary reports before the deadline. (not obligatory but strong encouragement)
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Penalty for delivering after deadline: Lose 1 point per each late day.
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@ -1,147 +0,0 @@
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\relax
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\providecommand\hyper@newdestlabel[2]{}
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\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
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\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
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\global\let\oldcontentsline\contentsline
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\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
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\global\let\oldnewlabel\newlabel
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\gdef\newlabel#1#2{\newlabelxx{#1}#2}
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\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
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\AtEndDocument{\ifx\hyper@anchor\@undefined
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\let\contentsline\oldcontentsline
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\let\newlabel\oldnewlabel
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\fi}
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\fi}
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\global\let\hyper@last\relax
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\gdef\HyperFirstAtBeginDocument#1{#1}
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\providecommand\HyField@AuxAddToFields[1]{}
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\providecommand\HyField@AuxAddToCoFields[2]{}
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[16 <./errorsA-eps-converted-to.pdf>] [17 <./errorsB-eps-converted-to.pdf>] [18]
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[35] [36 <./iterations-eps-converted-to.pdf>] [37]
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Chapter 5.
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@ -1,78 +0,0 @@
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||||
\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 [1][-]{section.1.4}{Results}{chapter.1}% 7
|
||||
\BOOKMARK [0][-]{chapter.2}{Problem 2 - Solving a system of n linear equations - indicated method}{}% 8
|
||||
\BOOKMARK [1][-]{section.2.1}{Problem}{chapter.2}% 9
|
||||
\BOOKMARK [1][-]{section.2.2}{Theoretical Introduction}{chapter.2}% 10
|
||||
\BOOKMARK [2][-]{subsection.2.2.1}{Transform matrix into upper-triangular matrix}{section.2.2}% 11
|
||||
\BOOKMARK [2][-]{subsection.2.2.2}{Backward substitution}{section.2.2}% 12
|
||||
\BOOKMARK [2][-]{subsection.2.2.3}{Partial Pivoting}{section.2.2}% 13
|
||||
\BOOKMARK [1][-]{section.2.3}{Results}{chapter.2}% 14
|
||||
\BOOKMARK [2][-]{subsection.2.3.1}{2a\)}{section.2.3}% 15
|
||||
\BOOKMARK [2][-]{subsection.2.3.2}{2b\)}{section.2.3}% 16
|
||||
\BOOKMARK [1][-]{section.2.4}{Discussion of results}{chapter.2}% 17
|
||||
\BOOKMARK [2][-]{subsection.2.4.1}{Errors in b\)}{section.2.4}% 18
|
||||
\BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 19
|
||||
\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 20
|
||||
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|
||||
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|
||||
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|
||||
\BOOKMARK [2][-]{subsection.3.3.1}{Jacobi method result}{section.3.3}% 24
|
||||
\BOOKMARK [2][-]{subsection.3.3.2}{Gauss-Seidel method result}{section.3.3}% 25
|
||||
\BOOKMARK [1][-]{section.3.4}{Discussion of results}{chapter.3}% 26
|
||||
\BOOKMARK [2][-]{subsection.3.4.1}{Comparison based on table}{section.3.4}% 27
|
||||
\BOOKMARK [2][-]{subsection.3.4.2}{Convergence}{section.3.4}% 28
|
||||
\BOOKMARK [0][-]{chapter.4}{Problem 4 - QR method of finding eigenvalues}{}% 29
|
||||
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|
||||
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|
||||
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|
||||
\BOOKMARK [2][-]{subsection.4.2.2}{QR method for finding eigenvalues}{section.4.2}% 33
|
||||
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|
||||
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|
||||
\BOOKMARK [2][-]{subsection.4.3.2}{QR method with no shifts}{section.4.3}% 36
|
||||
\BOOKMARK [2][-]{subsection.4.3.3}{QR method with shifts}{section.4.3}% 37
|
||||
\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 38
|
||||
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|
||||
\BOOKMARK [2][-]{subsection.4.4.2}{Shift method superiority}{section.4.4}% 40
|
||||
\BOOKMARK [0][-]{chapter.5}{Code appendix}{}% 41
|
||||
\BOOKMARK [1][-]{section.5.1}{Task 1 Code}{chapter.5}% 42
|
||||
\BOOKMARK [2][-]{subsection.5.1.1}{Find macheps}{section.5.1}% 43
|
||||
\BOOKMARK [2][-]{subsection.5.1.2}{Display results}{section.5.1}% 44
|
||||
\BOOKMARK [1][-]{section.5.2}{Task 2 Code}{chapter.5}% 45
|
||||
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|
||||
\BOOKMARK [2][-]{subsection.5.2.2}{checkIfMatrixIsSquareMatrix}{section.5.2}% 47
|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
\BOOKMARK [2][-]{subsection.5.2.13}{plotErrorsGaussian}{section.5.2}% 58
|
||||
\BOOKMARK [1][-]{section.5.3}{Task 3 Code}{chapter.5}% 59
|
||||
\BOOKMARK [2][-]{subsection.5.3.1}{initializeValues}{section.5.3}% 60
|
||||
\BOOKMARK [2][-]{subsection.5.3.2}{decomposeMatrix}{section.5.3}% 61
|
||||
\BOOKMARK [2][-]{subsection.5.3.3}{jacobiLoop}{section.5.3}% 62
|
||||
\BOOKMARK [2][-]{subsection.5.3.4}{jacobiInsideLoop}{section.5.3}% 63
|
||||
\BOOKMARK [2][-]{subsection.5.3.5}{jacobiEquation}{section.5.3}% 64
|
||||
\BOOKMARK [2][-]{subsection.5.3.6}{gaussSeidelLoop}{section.5.3}% 65
|
||||
\BOOKMARK [2][-]{subsection.5.3.7}{gaussiInsideLoop}{section.5.3}% 66
|
||||
\BOOKMARK [2][-]{subsection.5.3.8}{gaussSeidelEquation}{section.5.3}% 67
|
||||
\BOOKMARK [2][-]{subsection.5.3.9}{checkError}{section.5.3}% 68
|
||||
\BOOKMARK [2][-]{subsection.5.3.10}{endOfLoop}{section.5.3}% 69
|
||||
\BOOKMARK [2][-]{subsection.5.3.11}{dispFinalResults}{section.5.3}% 70
|
||||
\BOOKMARK [2][-]{subsection.5.3.12}{plotIterations}{section.5.3}% 71
|
||||
\BOOKMARK [1][-]{section.5.4}{Task 4 Code}{chapter.5}% 72
|
||||
\BOOKMARK [2][-]{subsection.5.4.1}{Gram-Schmid algorithm}{section.5.4}% 73
|
||||
\BOOKMARK [2][-]{subsection.5.4.2}{task4}{section.5.4}% 74
|
||||
\BOOKMARK [2][-]{subsection.5.4.3}{QRNoShifts}{section.5.4}% 75
|
||||
\BOOKMARK [2][-]{subsection.5.4.4}{QRShifts}{section.5.4}% 76
|
||||
\BOOKMARK [2][-]{subsection.5.4.5}{task4Plot}{section.5.4}% 77
|
||||
\BOOKMARK [2][-]{subsection.5.4.6}{Matrix generation}{section.5.4}% 78
|
||||
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@ -63,10 +63,9 @@ Macheps is also essential when we calculate cumulation of errors of given mathem
|
||||
\section{Solution}
|
||||
|
||||
|
||||
Code above shifts macheps one bit to the right each iteration (by dividing by 2), it ends when we run out of mantissa bits which renders us unable to save smaller number. Due to underflow the value of macheps becomes 0 and therefore 1.0 > (macheps / 2) > 1.0 will become false.
|
||||
\hyperlink{function1_macheps}{Code for finding macheps} shifts macheps one bit to the right each iteration (by dividing by 2), it ends when we run out of mantissa bits which renders us unable to save smaller number. Due to underflow the value of macheps becomes 0 and therefore 1.0 > (macheps / 2) > 1.0 will become false.
|
||||
\newpage
|
||||
\section{Results}
|
||||
\hyperlink{function1_macheps}{Code for finding macheps}\\
|
||||
\hyperlink{function1_display}{Code for displaying results}\\
|
||||
Display calculated macheps:
|
||||
\[2.220446049250313\mathrm{e}{-16}\]
|
||||
@ -1059,7 +1058,7 @@ while 1.0 + macheps / 2 > 1.0
|
||||
macheps = macheps/2;
|
||||
end
|
||||
\end{lstlisting}
|
||||
|
||||
\newpage
|
||||
\hypertarget{function1_display}{\subsection{Display results}}
|
||||
\begin{simplechar}
|
||||
\begin{lstlisting}
|
||||
@ -1078,8 +1077,9 @@ disp("Display difference between calculated macheps and 2^-52:")
|
||||
disp(macheps - 2^-52)
|
||||
\end{lstlisting}
|
||||
\end{simplechar}
|
||||
\section{Task 2 Code}
|
||||
|
||||
\newpage
|
||||
\section{Task 2 Code}
|
||||
\subsection{Main function}
|
||||
\begin{simplechar}
|
||||
\begin{lstlisting}
|
||||
@ -1495,6 +1495,7 @@ end
|
||||
\end{lstlisting}
|
||||
\end{simplechar}
|
||||
|
||||
\newpage
|
||||
\hypertarget{function_3_dominant}{Code for checking if matrix is diagonally dominant}
|
||||
\begin{lstlisting}
|
||||
function d = checkIfDiagonallyDominant(Matrix)
|
||||
|
||||
@ -1,117 +0,0 @@
|
||||
\contentsline {chapter}{\numberline {1}Problem 1 - Finding machine epsilion}{4}{chapter.1}%
|
||||
\contentsline {section}{\numberline {1.1}Problem}{4}{section.1.1}%
|
||||
\contentsline {section}{\numberline {1.2}Theoretical Introduction}{4}{section.1.2}%
|
||||
\contentsline {subsection}{\numberline {1.2.1}Definition of machine epsilion}{4}{subsection.1.2.1}%
|
||||
\contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{5}{subsection.1.2.2}%
|
||||
\contentsline {section}{\numberline {1.3}Solution}{6}{section.1.3}%
|
||||
\contentsline {section}{\numberline {1.4}Results}{7}{section.1.4}%
|
||||
\contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{8}{chapter.2}%
|
||||
\contentsline {section}{\numberline {2.1}Problem}{8}{section.2.1}%
|
||||
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{8}{section.2.2}%
|
||||
\contentsline {subsection}{\numberline {2.2.1}Transform matrix into upper-triangular matrix}{8}{subsection.2.2.1}%
|
||||
\contentsline {subsubsection}{Starting conditions}{8}{section*.2}%
|
||||
\contentsline {subsubsection}{Zeroing first column}{9}{section*.3}%
|
||||
\contentsline {subsubsection}{Zeroing second column}{9}{section*.4}%
|
||||
\contentsline {subsubsection}{Zeroing next columns}{10}{section*.5}%
|
||||
\contentsline {subsection}{\numberline {2.2.2}Backward substitution}{10}{subsection.2.2.2}%
|
||||
\contentsline {subsection}{\numberline {2.2.3}Partial Pivoting}{11}{subsection.2.2.3}%
|
||||
\contentsline {section}{\numberline {2.3}Results}{12}{section.2.3}%
|
||||
\contentsline {subsection}{\numberline {2.3.1}2a)}{12}{subsection.2.3.1}%
|
||||
\contentsline {subsection}{\numberline {2.3.2}2b)}{14}{subsection.2.3.2}%
|
||||
\contentsline {section}{\numberline {2.4}Discussion of results}{16}{section.2.4}%
|
||||
\contentsline {subsection}{\numberline {2.4.1}Errors in b)}{17}{subsection.2.4.1}%
|
||||
\newpage
|
||||
\contentsline {chapter}{\numberline {3}Problem 3 - Solving a system of n linear equations - iterative algorithm}{19}{chapter.3}%
|
||||
\contentsline {section}{\numberline {3.1}Problem}{19}{section.3.1}%
|
||||
\contentsline {section}{\numberline {3.2}Theoretical introduction}{20}{section.3.2}%
|
||||
\contentsline {subsection}{\numberline {3.2.1}Procedure}{20}{subsection.3.2.1}%
|
||||
\contentsline {subsubsection}{Decomposing matrix}{20}{section*.6}%
|
||||
\contentsline {subsubsection}{Jacobi's method}{21}{section*.7}%
|
||||
\contentsline {paragraph}{Converging}{21}{section*.8}%
|
||||
\contentsline {subsubsection}{Gauss-Seidel method}{22}{section*.9}%
|
||||
\contentsline {paragraph}{Converging}{23}{section*.10}%
|
||||
\contentsline {subsubsection}{Stop tests}{24}{section*.11}%
|
||||
\contentsline {subsubsection}{\textbf {A} and \textbf {b}}{25}{section*.12}%
|
||||
\contentsline {section}{\numberline {3.3}Results}{26}{section.3.3}%
|
||||
\contentsline {subsection}{\numberline {3.3.1}Jacobi method result}{26}{subsection.3.3.1}%
|
||||
\contentsline {subsubsection}{Minimizing the demanded error}{28}{section*.13}%
|
||||
\contentsline {paragraph}{For original system of equations:}{29}{section*.14}%
|
||||
\contentsline {paragraph}{For task 2a) system of equations:}{30}{section*.15}%
|
||||
\contentsline {subsection}{\numberline {3.3.2}Gauss-Seidel method result}{31}{subsection.3.3.2}%
|
||||
\contentsline {subsubsection}{Minimizing the demanded error}{32}{section*.16}%
|
||||
\contentsline {paragraph}{For original system of equations:}{32}{section*.17}%
|
||||
\contentsline {paragraph}{For task 2a) system of equations:}{33}{section*.18}%
|
||||
\contentsline {section}{\numberline {3.4}Discussion of results}{34}{section.3.4}%
|
||||
\contentsline {paragraph}{Table}{34}{section*.19}%
|
||||
\contentsline {subsection}{\numberline {3.4.1}Comparison based on table}{34}{subsection.3.4.1}%
|
||||
\contentsline {subsection}{\numberline {3.4.2}Convergence}{35}{subsection.3.4.2}%
|
||||
\contentsline {subsubsection}{2b) task convergence }{35}{section*.20}%
|
||||
\contentsline {subsubsection}{Iterations as function of size of Matrix}{35}{section*.21}%
|
||||
\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{38}{chapter.4}%
|
||||
\contentsline {section}{\numberline {4.1}Problem}{38}{section.4.1}%
|
||||
\contentsline {section}{\numberline {4.2}Theoretical introduction}{38}{section.4.2}%
|
||||
\contentsline {subsection}{\numberline {4.2.1}Eigenvalues}{38}{subsection.4.2.1}%
|
||||
\contentsline {subsection}{\numberline {4.2.2}QR method for finding eigenvalues}{39}{subsection.4.2.2}%
|
||||
\contentsline {section}{\numberline {4.3}Results}{40}{section.4.3}%
|
||||
\contentsline {subsection}{\numberline {4.3.1}Starting matrix}{40}{subsection.4.3.1}%
|
||||
\contentsline {subsection}{\numberline {4.3.2}QR method with no shifts}{41}{subsection.4.3.2}%
|
||||
\contentsline {subsection}{\numberline {4.3.3}QR method with shifts}{41}{subsection.4.3.3}%
|
||||
\contentsline {section}{\numberline {4.4}Discussion of the result}{42}{section.4.4}%
|
||||
\contentsline {subsection}{\numberline {4.4.1}Plot}{42}{subsection.4.4.1}%
|
||||
\contentsline {subsection}{\numberline {4.4.2}Shift method superiority}{43}{subsection.4.4.2}%
|
||||
\newpage
|
||||
\contentsline {chapter}{\numberline {5}Code appendix}{44}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}Task 1 Code}{44}{section.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.1.1}Find macheps}{44}{subsection.5.1.1}%
|
||||
\contentsline {subsection}{\numberline {5.1.2}Display results}{44}{subsection.5.1.2}%
|
||||
\contentsline {section}{\numberline {5.2}Task 2 Code}{45}{section.5.2}%
|
||||
\contentsline {subsection}{\numberline {5.2.1}Main function}{45}{subsection.5.2.1}%
|
||||
\contentsline {subsection}{\numberline {5.2.2}checkIfMatrixIsSquareMatrix}{45}{subsection.5.2.2}%
|
||||
\contentsline {subsection}{\numberline {5.2.3}gaussianEliminationWithPartialPivoting}{46}{subsection.5.2.3}%
|
||||
\contentsline {subsection}{\numberline {5.2.4}partialPivoting}{46}{subsection.5.2.4}%
|
||||
\contentsline {subsection}{\numberline {5.2.5}partialPivotingSwapOneRow}{47}{subsection.5.2.5}%
|
||||
\contentsline {subsection}{\numberline {5.2.6}swapRowMatrix}{47}{subsection.5.2.6}%
|
||||
\contentsline {subsection}{\numberline {5.2.7}swapValueVector}{47}{subsection.5.2.7}%
|
||||
\contentsline {subsection}{\numberline {5.2.8}gaussianElimination}{48}{subsection.5.2.8}%
|
||||
\contentsline {subsection}{\numberline {5.2.9}substractRows}{48}{subsection.5.2.9}%
|
||||
\contentsline {subsection}{\numberline {5.2.10}backSubstitutionPhase}{49}{subsection.5.2.10}%
|
||||
\contentsline {subsection}{\numberline {5.2.11}iterativeResidualCorrection}{49}{subsection.5.2.11}%
|
||||
\contentsline {subsection}{\numberline {5.2.12}improveSolution}{50}{subsection.5.2.12}%
|
||||
\contentsline {subsection}{\numberline {5.2.13}plotErrorsGaussian}{51}{subsection.5.2.13}%
|
||||
\contentsline {section}{\numberline {5.3}Task 3 Code}{52}{section.5.3}%
|
||||
\contentsline {subsection}{\numberline {5.3.1}initializeValues}{53}{subsection.5.3.1}%
|
||||
\contentsline {subsection}{\numberline {5.3.2}decomposeMatrix}{53}{subsection.5.3.2}%
|
||||
\contentsline {subsection}{\numberline {5.3.3}jacobiLoop}{54}{subsection.5.3.3}%
|
||||
\contentsline {subsection}{\numberline {5.3.4}jacobiInsideLoop}{54}{subsection.5.3.4}%
|
||||
\contentsline {subsection}{\numberline {5.3.5}jacobiEquation}{54}{subsection.5.3.5}%
|
||||
\contentsline {subsection}{\numberline {5.3.6}gaussSeidelLoop}{55}{subsection.5.3.6}%
|
||||
\contentsline {subsection}{\numberline {5.3.7}gaussiInsideLoop}{55}{subsection.5.3.7}%
|
||||
\contentsline {subsection}{\numberline {5.3.8}gaussSeidelEquation}{56}{subsection.5.3.8}%
|
||||
\contentsline {subsection}{\numberline {5.3.9}checkError}{56}{subsection.5.3.9}%
|
||||
\contentsline {subsection}{\numberline {5.3.10}endOfLoop}{57}{subsection.5.3.10}%
|
||||
\contentsline {subsection}{\numberline {5.3.11}dispFinalResults}{57}{subsection.5.3.11}%
|
||||
\contentsline {subsection}{\numberline {5.3.12}plotIterations}{58}{subsection.5.3.12}%
|
||||
\contentsline {section}{\numberline {5.4}Task 4 Code}{60}{section.5.4}%
|
||||
\contentsline {subsection}{\numberline {5.4.1}Gram-Schmid algorithm}{60}{subsection.5.4.1}%
|
||||
\contentsline {subsubsection}{initializeGramSchmid}{60}{section*.22}%
|
||||
\contentsline {subsubsection}{factorizeColumnsOfQ}{60}{section*.23}%
|
||||
\contentsline {subsubsection}{normalizeColumns}{61}{section*.24}%
|
||||
\contentsline {subsection}{\numberline {5.4.2}task4}{61}{subsection.5.4.2}%
|
||||
\contentsline {subsection}{\numberline {5.4.3}QRNoShifts}{61}{subsection.5.4.3}%
|
||||
\contentsline {subsubsection}{QRNoShiftsLoop}{62}{section*.25}%
|
||||
\contentsline {subsubsection}{QRNoShiftsInsideLoop}{62}{section*.26}%
|
||||
\contentsline {subsubsection}{displayResults}{63}{section*.27}%
|
||||
\contentsline {subsubsection}{initializeValues}{63}{section*.28}%
|
||||
\contentsline {subsection}{\numberline {5.4.4}QRShifts}{64}{subsection.5.4.4}%
|
||||
\contentsline {subsubsection}{initiateValues}{64}{section*.29}%
|
||||
\contentsline {subsubsection}{QRShiftLoop}{65}{section*.30}%
|
||||
\contentsline {subsubsection}{findEigenValue}{65}{section*.31}%
|
||||
\contentsline {subsubsection}{getEigenValueFromCorner}{66}{section*.32}%
|
||||
\contentsline {subsubsection}{shiftAndIterate}{66}{section*.33}%
|
||||
\contentsline {subsubsection}{deflateMatrix}{66}{section*.34}%
|
||||
\contentsline {subsubsection}{thresholdBreached}{67}{section*.35}%
|
||||
\contentsline {subsubsection}{solveCharactersticEquation}{67}{section*.36}%
|
||||
\contentsline {subsubsection}{calculateZeros}{68}{section*.37}%
|
||||
\contentsline {subsubsection}{dispResults}{68}{section*.38}%
|
||||
\contentsline {subsection}{\numberline {5.4.5}task4Plot}{69}{subsection.5.4.5}%
|
||||
\contentsline {subsection}{\numberline {5.4.6}Matrix generation}{69}{subsection.5.4.6}%
|
||||
Binary file not shown.
@ -1,30 +0,0 @@
|
||||
# ENUME_Project1
|
||||
A Numerical Method project cointaining 4 tasks.
|
||||
## 1.
|
||||
Write a program finding macheps in the MATLAB environment on a lab computer or your
|
||||
computer.
|
||||
## 2.
|
||||
Write a general program solving a system of n linear equations Ax = b using the indicated
|
||||
method Apply the program to solve the system of linear equations for given matrix A and vector b, for increasing numbers
|
||||
of equations n = 10,20,40,80,160,… until the solution time becomes prohibitive (or the
|
||||
method fails), for:
|
||||
### a)
|
||||
Aij { for i = j -> 13, for i = j-1 or i = j+1 -> 4, other -> 0 \
|
||||
Bi = 2.4 + 0.6i, \
|
||||
i,j = 1..n;
|
||||
### b)
|
||||
Aij = 4/[5(i + j – 1)], Bi = 1/(2 i), i – odd; Bi = 0, i – even, i, j = 1,…,n;
|
||||
|
||||
## 3
|
||||
Write a general program for solving the system of n linear equations Ax = b using the
|
||||
Gauss-Seidel and Jacobi iterative algorithms. Apply it for the system: \
|
||||
13x1 + 2x2 – 8x3 + x4 =16 \
|
||||
x1 + 10x2 + 5x3 – 2x4 = 24 \
|
||||
6x1 + 2x2 – 23x3 + 15x4 = 184 \
|
||||
x1 + 2x2 – x3 + 13x4 = 82
|
||||
|
||||
## 4
|
||||
Write a program of the QR method for finding eigenvalues of 5×5 matrices: \
|
||||
a) without shifts; \
|
||||
b) with shifts calculated on the basis of an eigenvalue of the 2×2 right-lower-corner
|
||||
submatrix.
|
||||
Binary file not shown.
@ -1,30 +0,0 @@
|
||||
function [TableOfErrors] = ResidualCorrection(n, subpoint)
|
||||
TableOfErrors = zeros(n, 1);
|
||||
TableOfErrorsAC = zeros(n, 1);
|
||||
TableOfErrorsAC2 = zeros(n, 1);
|
||||
TableOfNValues = zeros(n, 1);
|
||||
for i = 1:n
|
||||
TableOfNValues(i) = 10*2^(i-1);
|
||||
disp(i);
|
||||
x = TASK2(10*2^(i-1));
|
||||
if subpoint == 'A'
|
||||
x = TaskA(x);
|
||||
elseif subpoint == 'B'
|
||||
x = TaskB(x);
|
||||
end
|
||||
TableOfErrors(i) = norm(x.errors);
|
||||
x = ResidualCorrection(x);
|
||||
x= GetErrors(x);
|
||||
TableOfErrorsAC(i) = norm(x.errors);
|
||||
x = ResidualCorrection(x);
|
||||
x= GetErrors(x);
|
||||
TableOfErrorsAC2(i) = norm(x.errors);
|
||||
end
|
||||
plot(TableOfNValues, TableOfErrors, '-o');
|
||||
hold on;
|
||||
ylabel('Maximal absolute value of an error')
|
||||
xlabel('N')
|
||||
plot(TableOfNValues, TableOfErrorsAC, '-or');
|
||||
plot(TableOfNValues, TableOfErrorsAC2, '-og');
|
||||
hold off;
|
||||
end
|
||||
@ -1,8 +0,0 @@
|
||||
function macheps = Task1()
|
||||
macheps = 1;
|
||||
while 1 + (macheps/2)>1
|
||||
macheps = macheps/2;
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
@ -1,161 +0,0 @@
|
||||
classdef TASK2
|
||||
properties
|
||||
%main array size n by n
|
||||
A;
|
||||
%second array size n
|
||||
B;
|
||||
%answer
|
||||
x;
|
||||
%errors
|
||||
errors;
|
||||
%size of a matricx
|
||||
n;
|
||||
end
|
||||
methods (Access = 'protected')
|
||||
function [obj, locx] = gaussWithPartialPiv(obj)
|
||||
locx = zeros(obj.n, 1);
|
||||
for j = 1:obj.n
|
||||
%finding the maximal value of a column
|
||||
index = j;
|
||||
locMax = abs(obj.A(j, j));
|
||||
for i = j+1:obj.n
|
||||
if abs(obj.A(i, j)) > locMax
|
||||
locMax = abs(obj.A(i, j));
|
||||
index = i;
|
||||
end
|
||||
end
|
||||
%checking neccessary condition for the further execution
|
||||
if locMax == 0
|
||||
disp(locMax);
|
||||
disp('Wrong Matrix');
|
||||
return
|
||||
end
|
||||
if index ~= j
|
||||
%swaping the rows in neccessary (highest value found
|
||||
%not in j row)
|
||||
obj.A([index, j],:) = obj.A([j, index],:);
|
||||
obj.B([index, j]) = obj.B([j, index]);
|
||||
end
|
||||
for i = j+1:obj.n
|
||||
r = obj.A(i, j)/obj.A(j,j);
|
||||
%if r == 0 we dont need to continue
|
||||
if r ~= 0
|
||||
%changing the main array A
|
||||
for locj = j+1:obj.n
|
||||
obj.A(i, locj) = obj.A(i, locj) - r*obj.A(j, locj);
|
||||
end
|
||||
%and the B array aswell
|
||||
obj.B(i) = obj.B(i) - r*obj.B(j);
|
||||
end
|
||||
end
|
||||
end
|
||||
%now finally we can obtain the results
|
||||
%Xn value is rather obvious
|
||||
locx(obj.n) = obj.B(obj.n)/obj.A(obj.n, obj.n);
|
||||
for i = obj.n-1:-1:1
|
||||
buffor = 0;
|
||||
for j = i+1:obj.n
|
||||
buffor = buffor + obj.A(i, j)*locx(j);
|
||||
end
|
||||
locx(i) = (obj.B(i) - buffor)/obj.A(i,i);
|
||||
end
|
||||
end
|
||||
function obj = TaskAArray(obj)
|
||||
[row, columns] = size(obj.A);
|
||||
if row ~= size(obj.B)
|
||||
disp('A and B array size is different!');
|
||||
return
|
||||
end
|
||||
for i = 1:row
|
||||
%setting the A array
|
||||
for j = 1:columns
|
||||
if i == j
|
||||
obj.A(i, j) = 13;
|
||||
elseif i == j-1 || i == j+1
|
||||
obj.A(i, j) = 4;
|
||||
else
|
||||
obj.A(i, j) = 0;
|
||||
end
|
||||
end
|
||||
%setting he B array
|
||||
obj.B(i) = 2.4 + 0.6*i;
|
||||
end
|
||||
end
|
||||
function obj = TaskBArray(obj)
|
||||
[row, columns] = size(obj.A);
|
||||
if row ~= size(obj.B)
|
||||
disp('A and B array size is differn!');
|
||||
return
|
||||
end
|
||||
for i = 1:row
|
||||
%setting the A array
|
||||
for j = 1:columns
|
||||
obj.A(i, j) = 4/(5*(i + j - 1));
|
||||
end
|
||||
%setting he B array
|
||||
if mod(i, 2) == 0
|
||||
obj.B(i) = 1/(2*i);
|
||||
else
|
||||
obj.B(i) = 0;
|
||||
end
|
||||
end
|
||||
end
|
||||
function obj = SetSize(obj, n)
|
||||
obj.A = zeros(n);
|
||||
obj.B = zeros(n, 1);
|
||||
obj.x = zeros(n,1);
|
||||
obj.errors = zeros(n,1);
|
||||
obj.n = n;
|
||||
end
|
||||
end
|
||||
%public methods
|
||||
methods
|
||||
function obj = TASK2(n)
|
||||
obj.n = n;
|
||||
end
|
||||
function obj = GetErrors(obj)
|
||||
%colculating the error following the formula r = Ax - b
|
||||
for i = 1:obj.n
|
||||
result = 0;
|
||||
for j = i:obj.n
|
||||
result = result + obj.A(i,j) * obj.x(j);
|
||||
end
|
||||
obj.errors(i) = result - obj.B(i);
|
||||
end
|
||||
end
|
||||
function obj = ResidualCorrection(obj)
|
||||
newObj = TASK2(obj.n);
|
||||
newObj = obj;
|
||||
newObj.B = newObj.errors;
|
||||
[newObj, newObj.x] = gaussWithPartialPiv(newObj);
|
||||
for i = 1:obj.n
|
||||
obj.x(i) = obj.x(i) - newObj.x(i);
|
||||
end
|
||||
end
|
||||
function obj = TaskA(obj)
|
||||
obj = SetSize(obj, obj.n);
|
||||
obj = TaskAArray(obj);
|
||||
[obj, obj.x] = gaussWithPartialPiv(obj);
|
||||
obj = GetErrors(obj);
|
||||
end
|
||||
function obj = TaskB(obj)
|
||||
obj = SetSize(obj, obj.n);
|
||||
obj = TaskBArray(obj);
|
||||
[obj, obj.x] = gaussWithPartialPiv(obj);
|
||||
obj = GetErrors(obj);
|
||||
end
|
||||
function DispSolutionAndError(obj)
|
||||
format long
|
||||
disp('x for n ='); disp(obj.n);
|
||||
obj.x
|
||||
disp('errors for n ='); disp(obj.n);
|
||||
obj.errors
|
||||
end
|
||||
function DispObjects(obj)
|
||||
obj.A
|
||||
obj.B
|
||||
obj.x
|
||||
obj.errors
|
||||
end
|
||||
end
|
||||
end
|
||||
@ -1,155 +0,0 @@
|
||||
classdef TASK3 < TASK2
|
||||
properties
|
||||
acc = 1e-10;
|
||||
end
|
||||
methods (Access = 'private')
|
||||
function [obj, array]= gaussSeidel(obj, type)
|
||||
array = [];
|
||||
err = inf;
|
||||
k = 1;
|
||||
m = 10000;
|
||||
while k <= m && err > obj.acc
|
||||
localMax = 0;
|
||||
for i = 1 : obj.n
|
||||
s = 0;
|
||||
for j = 1 : obj.n
|
||||
s = s+(-obj.A(i,j)*obj.x(j));
|
||||
end
|
||||
s = (s+obj.B(i))/obj.A(i,i);
|
||||
if abs(s) > localMax
|
||||
localMax = abs(s);
|
||||
end
|
||||
obj.x(i) = obj.x(i) + s;
|
||||
end
|
||||
if err > localMax
|
||||
err = localMax ;
|
||||
end
|
||||
k = k+1;
|
||||
obj = GetErrors(obj);
|
||||
if(type == 'N')%norm
|
||||
array(end+1) = norm(obj.errors);
|
||||
elseif(type == 'E')%maximal value of s
|
||||
array(end+1) = localMax;
|
||||
end
|
||||
end
|
||||
disp(k - 1);
|
||||
end
|
||||
function [obj, array]= jacobi(obj, type)
|
||||
array = [];
|
||||
err = inf;
|
||||
k = 1;
|
||||
m = 10000;
|
||||
x1 = zeros(obj.n, 1);
|
||||
while k <= m && err > obj.acc
|
||||
localMax = 0;
|
||||
for i = 1 : obj.n
|
||||
s = 0;
|
||||
for j = 1 : obj.n
|
||||
s = s+(-obj.A(i,j)*x1(j));
|
||||
end
|
||||
x1 = obj.x;
|
||||
s = (s+obj.B(i))/obj.A(i,i);
|
||||
obj.x(i) = obj.x(i) + s;
|
||||
if abs(s) > localMax
|
||||
localMax = abs(s);
|
||||
end
|
||||
end
|
||||
if err > localMax
|
||||
err = localMax ;
|
||||
end
|
||||
k = k+1;
|
||||
obj = GetErrors(obj);
|
||||
if(type == 'N')%norm
|
||||
array(end+1) = norm(obj.errors);
|
||||
elseif(type == 'E')%maximal value of s
|
||||
array(end+1) = localMax;
|
||||
end
|
||||
end
|
||||
disp(k - 1);
|
||||
end
|
||||
end
|
||||
methods
|
||||
function obj = TASK3()
|
||||
obj = obj@TASK2(4);
|
||||
end
|
||||
function obj = SetTask3(obj)
|
||||
obj = SetSize(obj, 4);
|
||||
obj.A = [13 2 -8 1; 1 10 5 -2; 6 2 -23 15; 1 22 -1 13];
|
||||
obj.B = [16 24 184 82];
|
||||
end
|
||||
function obj = Example(obj)
|
||||
obj = SetSize(obj, 4);
|
||||
obj.A = [5 -2 3 0; -3 9 1 -2; 2 -1 -7 1; 4 3 -5 7];
|
||||
obj.B = [-1 2 3 0.5];
|
||||
[obj, array] = gaussSeidel(obj, 'N');
|
||||
plot(array, '-o');
|
||||
end
|
||||
%type in 'G' for Gauss Seidel and 'J' for Jacobi method.
|
||||
function obj = Task3System(obj, alg)
|
||||
obj = SetSize(obj, 4);
|
||||
obj.A = [13 2 -8 1; 1 10 5 -2; 6 2 -23 15; 1 22 -1 13];
|
||||
obj.B = [16 24 184 82];
|
||||
if alg == 'G'
|
||||
[obj, array] = gaussSeidel(obj);
|
||||
plot(array, '-o');
|
||||
elseif alg == 'J'
|
||||
[obj, array] = jacobi(obj);
|
||||
plot(array, '-or');
|
||||
end
|
||||
end
|
||||
function obj = Task3a(obj, type)
|
||||
if type ~= 'N' && type ~= 'E'
|
||||
disp('Wrong type parameter!');
|
||||
return
|
||||
end
|
||||
obj = SetTask3(obj);
|
||||
[obj, array1] = gaussSeidel(obj, type);
|
||||
obj = SetTask3(obj);
|
||||
[obj, array2] = jacobi(obj, type);
|
||||
plot(array1, 'ob-');
|
||||
hold on
|
||||
plot(array2, 'or-');
|
||||
hold off
|
||||
end
|
||||
function obj = Task3With2a(obj, input, type)
|
||||
obj = SetSize(obj, 10);
|
||||
obj = TaskAArray(obj);
|
||||
if input == 'G'
|
||||
[obj, table] = gaussSeidel(obj, type);
|
||||
plot(table, '-o');
|
||||
elseif input == 'J'
|
||||
[obj, table] = jacobi(obj,type);
|
||||
plot(table, '-o');
|
||||
elseif input == 'B'
|
||||
[obj, table] = gaussSeidel(obj,type);
|
||||
obj = SetSize(obj, 10);
|
||||
obj = TaskAArray(obj);
|
||||
[obj, table2] = jacobi(obj,type);
|
||||
plot(table, '-o');
|
||||
hold on
|
||||
plot(table2, '-or');
|
||||
hold off
|
||||
end
|
||||
end
|
||||
function obj = Task3With2b(obj, input, type)
|
||||
obj = SetSize(obj, 10);
|
||||
obj = TaskBArray(obj);
|
||||
if input == 'G'
|
||||
[obj, table] = gaussSeidel(obj, type);
|
||||
plot(table, '-o');
|
||||
elseif input == 'J'
|
||||
[obj, table] = jacobi(obj,type);
|
||||
plot(table, '-o');
|
||||
elseif input == 'B'
|
||||
[obj, table] = gaussSeidel(obj,type);
|
||||
obj = SetSize(obj, 10);
|
||||
obj = TaskBArray(obj);
|
||||
[obj, table2] = jacobi(obj,type);
|
||||
plot(table, '-o');
|
||||
hold on
|
||||
plot(table2, '-o');
|
||||
hold off
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
@ -1,95 +0,0 @@
|
||||
classdef TASK4 < TASK2
|
||||
properties
|
||||
tol = 1e-6;
|
||||
imax = 10000;
|
||||
end
|
||||
methods
|
||||
function obj = TASK4()
|
||||
obj = obj@TASK2(5);
|
||||
end
|
||||
function [obj, Q, R]=QR(obj)
|
||||
Q=zeros(obj.n,obj.n);
|
||||
R=zeros(obj.n,obj.n);
|
||||
d=zeros(1,obj.n);
|
||||
for i=1:obj.n
|
||||
Q(:,i)=obj.A(:,i);
|
||||
R(i,i)=1;
|
||||
d(i)=Q(:,i)'*Q(:,i);
|
||||
for j=i+1:obj.n
|
||||
R(i,j)=(Q(:,i)'*obj.A(:,j))/d(i);
|
||||
obj.A(:,j)=obj.A(:,j)-R(i,j)*Q(:,i);
|
||||
end
|
||||
end
|
||||
for i=1:obj.n
|
||||
dd=norm(Q(:,i));
|
||||
Q(:,i)=Q(:,i)/dd;
|
||||
R(i,i:obj.n)=R(i,i:obj.n)*dd;
|
||||
end
|
||||
end
|
||||
function [obj, eig, i] = EigvalQRshifts(obj)
|
||||
obj.n=size(obj.A,1);
|
||||
eig=diag(ones(obj.n));
|
||||
INITIALsubmatrix=obj.A;
|
||||
for k=obj.n:-1:2
|
||||
DK=INITIALsubmatrix;
|
||||
i=0;
|
||||
while i<=obj.imax && max(abs(DK(k,1:k-1)))>obj.tol
|
||||
DD=DK(k-1:k,k-1:k);
|
||||
[ev1,ev2]=roots(1,-(DD(1,1)+DD(2,2)),DD(2,2)*DD(1,1)-DD(2,1)*DD(1,2));
|
||||
if abs(ev1-DD(2,2)) < abs(ev2-DD(2,2))
|
||||
shift=ev1;
|
||||
else
|
||||
shift=ev2;
|
||||
end
|
||||
DP=DK-eye(k)*shift;
|
||||
[Q1,R1]=qr(DP);
|
||||
DK=R1*Q1+eye(k)*shift;
|
||||
i=i+1;
|
||||
end
|
||||
if i > obj.imax
|
||||
error("Too many iterations!");
|
||||
end
|
||||
eig(k)=DK(k,k);
|
||||
if k > 2
|
||||
INITIALsubmatrix=DK(1:k-1,1:k-1);
|
||||
else
|
||||
eig(1)=DK(1,1);
|
||||
end
|
||||
DK
|
||||
end
|
||||
end
|
||||
function [obj, eigenvalues, i] = EigvalQRNoShift(obj)
|
||||
i=1;
|
||||
while i <= obj.imax && max(max(obj.A-diag(diag(obj.A)))) > obj.tol
|
||||
[obj, Q1,R1] = QR(obj);
|
||||
obj.A=R1*Q1;
|
||||
i=i+1;
|
||||
end
|
||||
if i > obj.imax
|
||||
error("Too many iterations!");
|
||||
end
|
||||
eigenvalues=diag(obj.A);
|
||||
end
|
||||
function obj = SetExampleShifts(obj)
|
||||
obj = SetSize(obj, 5);
|
||||
obj.A = [2 33 8 -3 4; 33 1 -6 5 -3; 8 -6 -5 -6 8; -3 5 -6 3 2; 4 -3 8 2 45];
|
||||
[obj, eig, k] = EigvalQRshifts(obj);
|
||||
eig
|
||||
k
|
||||
end
|
||||
function obj = SetExample(obj)
|
||||
obj = SetSize(obj, 5);
|
||||
obj.A = [2 33 8 -3 4; 33 1 -6 5 -3; 8 -6 -5 -6 8; -3 5 -6 3 2; 4 -3 8 2 45];
|
||||
[obj, eig, k] = EigvalQRNoShift(obj);
|
||||
eig
|
||||
k
|
||||
end
|
||||
end
|
||||
end
|
||||
function [x1, x2] = roots(a, b, c)
|
||||
first = -b + sqrt(b * b - 4 * a * c);
|
||||
second = -b - sqrt(b * b - 4 * a * c);
|
||||
li = max(abs(first), abs(second));
|
||||
x1 = li/(2*a);
|
||||
x2 = ((-b)/a);
|
||||
end
|
||||
Binary file not shown.
@ -1,8 +0,0 @@
|
||||
function [macheps] = macheps()
|
||||
%MACHEPS function calculates machine epsilon
|
||||
% finds smallest number E that 1+E>1
|
||||
macheps = 1;
|
||||
while( 1 + ( macheps / 2 ) > 1 )
|
||||
macheps = macheps/2;
|
||||
end
|
||||
end
|
||||
@ -1,21 +0,0 @@
|
||||
function [r] = euclideanNorm(rMatrix)
|
||||
%EUCLIDEANNORM calculates Euclidean Norm
|
||||
% yes
|
||||
sr = size(rMatrix);
|
||||
if( size(sr) > 2 )
|
||||
return;
|
||||
end
|
||||
if( sr(1) == 1 )
|
||||
n = sr(2);
|
||||
elseif( sr(2) == 1 );
|
||||
n = sr(1);
|
||||
else
|
||||
return;
|
||||
end
|
||||
sum = 0;
|
||||
for i = 1:n
|
||||
sum = sum + rMatrix(i)*rMatrix(i);
|
||||
end
|
||||
r = sqrt(sum);
|
||||
end
|
||||
|
||||
@ -1,21 +0,0 @@
|
||||
function [A, b] = matrixGen2a(n)
|
||||
%matrixGen2a Generates matrices for the task 2a
|
||||
% generates matrices for a system Ax = B of n linear equations
|
||||
|
||||
A = zeros(n, n);
|
||||
b = zeros(n, 1);
|
||||
|
||||
for i = 1:n
|
||||
b(i) = 0.9*i;
|
||||
for j = 1:n
|
||||
if(i == j)
|
||||
A(i, j) = 11;
|
||||
elseif (i == j-1)
|
||||
A(i, j) = 5;
|
||||
elseif (i == j+1)
|
||||
A(i, j) = 5;
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@ -1,18 +0,0 @@
|
||||
function [A,b] = matrixGen2b(n)
|
||||
%matrixGen2b Generates matrices for the task 2b
|
||||
% generates matrices for a system Ax = B of n linear equations
|
||||
|
||||
A = zeros(n, n);
|
||||
b = zeros(n, 1);
|
||||
|
||||
for i = 1:n
|
||||
if( mod(i,2) == 0 )
|
||||
b(i) = 2/(3*i);
|
||||
else
|
||||
b(i) = 0;
|
||||
end
|
||||
for j = 1:n
|
||||
A(i,j) = 7/(8*(i+j+1));
|
||||
end
|
||||
end
|
||||
end
|
||||
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|
Before Width: | Height: | Size: 25 KiB |
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|
Before Width: | Height: | Size: 26 KiB |
Binary file not shown.
Binary file not shown.
|
Before Width: | Height: | Size: 25 KiB |
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|
Before Width: | Height: | Size: 25 KiB |
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|
Before Width: | Height: | Size: 24 KiB |
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Binary file not shown.
|
Before Width: | Height: | Size: 22 KiB |
@ -1,17 +0,0 @@
|
||||
function [Rnew,Xnew] = residualCorrection(r,x,AT,b,A)
|
||||
%RESIDUALCORRECTION calculates an iteration of residual correction
|
||||
% currently not working properly
|
||||
n = length(x);
|
||||
dx = zeros(n, 1);
|
||||
for o = 1:n
|
||||
k = n-o+1;
|
||||
sum = 0;
|
||||
for j = k+1:n
|
||||
sum = sum + (AT(k,j) * dx(j));
|
||||
end
|
||||
dx(k) = (r(k) - sum)/AT(k,k);
|
||||
end
|
||||
Xnew = x - dx;
|
||||
Rnew = A*Xnew - b;
|
||||
end
|
||||
|
||||
@ -1,46 +0,0 @@
|
||||
function [x, A, b] = solveIndicated(A, b)
|
||||
%SOLVEINDICATED solves system of linear equations Ax=b
|
||||
% uses the indicated method (Gaussain elimination with partial pivoting)
|
||||
% returns x - the solution and A,b - the system after transformation
|
||||
|
||||
sa = size(A);
|
||||
sb = size(b);
|
||||
if( sa(1) ~= sa(2) )
|
||||
return;
|
||||
elseif ( sa(1) ~= sb(1) )
|
||||
return;
|
||||
end
|
||||
|
||||
n = sa(1);
|
||||
|
||||
for k = 1:n
|
||||
% Partial pivoting
|
||||
i = k;
|
||||
for j = k+1:n
|
||||
if ( abs(A(j, k)) > abs(A(i, k)) )
|
||||
i = j;
|
||||
end
|
||||
end
|
||||
if( k~=i )
|
||||
A([k i], :) = A([i k], :);
|
||||
b([k i]) = b([i k]);
|
||||
end
|
||||
% Gauss transform
|
||||
for j = k+1:n
|
||||
l = A(j,k) / A(k,k);
|
||||
A(j, :) = A(j, :) - A(k, :) * l;
|
||||
b(j) = b(j) - b(k) * l;
|
||||
end
|
||||
end
|
||||
%Backwards substitution
|
||||
x = zeros(n, 1);
|
||||
for o = 1:n
|
||||
k = n-o+1;
|
||||
sum = 0;
|
||||
for j = k+1:n
|
||||
sum = sum + (A(k,j) * x(j));
|
||||
end
|
||||
x(k) = (b(k) - sum)/A(k,k);
|
||||
end
|
||||
end
|
||||
|
||||
@ -1,130 +0,0 @@
|
||||
clear all;
|
||||
|
||||
repeatSmall = 4;
|
||||
repeatBig = 9;
|
||||
repeatEqua = 10;
|
||||
%-----TASK 2A
|
||||
%big graph
|
||||
repeats = zeros(1,repeatBig);
|
||||
errors = zeros(1,repeatBig);
|
||||
n = repeatEqua;
|
||||
for i = 1:1:repeatBig
|
||||
[A, b] = matrixGen2a( n );
|
||||
[x] = solveIndicated( A, b );
|
||||
repeats(i) = n;
|
||||
errors(i) = euclideanNorm( A*x - b );
|
||||
n = n*2;
|
||||
end
|
||||
figure(1)
|
||||
plot(repeats, errors, 'o');
|
||||
title(sprintf("Errors in task 2a, up to %d equations", n/2));
|
||||
xlabel('Number of equations');
|
||||
ylabel('Euclidean norm of errors');
|
||||
grid on;
|
||||
box off;
|
||||
saveas(1, "./plots/2aBIG.fig");
|
||||
saveas(1, "./plots/2aBIG.png");
|
||||
%small
|
||||
repeats = zeros(1,repeatSmall);
|
||||
errors = zeros(1,repeatSmall);
|
||||
n = repeatEqua;
|
||||
for i = 1:1:repeatSmall
|
||||
[A, b] = matrixGen2a( n );
|
||||
[x] = solveIndicated( A, b );
|
||||
repeats(i) = n;
|
||||
errors(i) = euclideanNorm( A*x - b );
|
||||
n = n*2;
|
||||
end
|
||||
figure(2)
|
||||
plot(repeats, errors, 'o');
|
||||
title(sprintf("Errors in task 2a, up to %d equations", n/2));
|
||||
xlabel('Number of equations');
|
||||
ylabel('Euclidean norm of errors');
|
||||
grid on;
|
||||
box off;
|
||||
saveas(2, "./plots/2aSMALL.fig");
|
||||
saveas(2, "./plots/2aSMALL.png");
|
||||
[Aa, ba] = matrixGen2a( 10 );
|
||||
[xa, AaT, baT] = solveIndicated( Aa, ba );
|
||||
Ra = Aa*xa - ba;
|
||||
%TODO: RESIDUAL
|
||||
repeatResidual = 10;
|
||||
residualA = zeros(1,repeatResidual+1);
|
||||
residualA(1) = euclideanNorm(Ra);
|
||||
xaR = xa;
|
||||
RaR = Ra;
|
||||
for i = 1:1:repeatResidual
|
||||
[RaR, xaR] = residualCorrection(RaR,xaR,AaT,ba,Aa);
|
||||
residualA(i+1) = euclideanNorm(RaR);
|
||||
end
|
||||
figure(5)
|
||||
plot(0:1:repeatResidual,residualA,'o');
|
||||
title("Results of residual correction");
|
||||
xlabel('Iteration');
|
||||
ylabel('Euclidean norm of errors');
|
||||
grid on;
|
||||
box off;
|
||||
saveas(5, "./plots/residualA.fig");
|
||||
saveas(5, "./plots/residualA.png");
|
||||
%-----TASK 2B
|
||||
%big graph
|
||||
repeats = zeros(1,repeatBig);
|
||||
errors = zeros(1,repeatBig);
|
||||
n = repeatEqua;
|
||||
for i = 1:1:repeatBig
|
||||
[A, b] = matrixGen2b( n );
|
||||
[x] = solveIndicated( A, b );
|
||||
repeats(i) = n;
|
||||
errors(i) = euclideanNorm( A*x - b );
|
||||
n = n*2;
|
||||
end
|
||||
figure(3)
|
||||
plot(repeats, errors, 'o');
|
||||
title(sprintf("Errors in task 2b, up to %d equations", n/2));
|
||||
xlabel('Number of equations');
|
||||
ylabel('Euclidean norm of errors');
|
||||
grid on;
|
||||
box off;
|
||||
saveas(3, "./plots/2bBIG.fig");
|
||||
saveas(3, "./plots/2bBIG.png");
|
||||
%small
|
||||
repeats = zeros(1,repeatSmall);
|
||||
errors = zeros(1,repeatSmall);
|
||||
n = repeatEqua;
|
||||
for i = 1:1:repeatSmall
|
||||
[A, b] = matrixGen2b( n );
|
||||
[x] = solveIndicated( A, b );
|
||||
repeats(i) = n;
|
||||
errors(i) = euclideanNorm( A*x - b );
|
||||
n = n*2;
|
||||
end
|
||||
figure(4)
|
||||
plot(repeats, errors, 'o');
|
||||
title(sprintf("Errors in task 2b, up to %d equations", n/2));
|
||||
xlabel('Number of equations');
|
||||
ylabel('Euclidean norm of errors');
|
||||
grid on;
|
||||
box off;
|
||||
saveas(4, "./plots/2bSMALL.fig");
|
||||
saveas(4, "./plots/2bSMALL.png");
|
||||
[Ab, bb] = matrixGen2b( 10 );
|
||||
[xb, AbT, bbT] = solveIndicated( Ab, bb );
|
||||
Rb = Ab*xb - bb;
|
||||
%TODO: RESIDUAL
|
||||
residualB = zeros(1,repeatResidual+1);
|
||||
residualB(1) = euclideanNorm(Rb);
|
||||
xbR = xb;
|
||||
RbR = Rb;
|
||||
for i = 1:1:repeatResidual
|
||||
[RbR, xbR] = residualCorrection(Rb,xb,AbT,bb,Ab);
|
||||
residualB(i+1) = euclideanNorm(RbR);
|
||||
end
|
||||
figure(6)
|
||||
plot(0:1:repeatResidual,residualB,'o');
|
||||
title("Results of residual correction");
|
||||
xlabel('Iteration');
|
||||
ylabel('Euclidean norm of errors');
|
||||
grid on;
|
||||
box off;
|
||||
saveas(6, "./plots/residualB.fig");
|
||||
saveas(6, "./plots/residualB.png");
|
||||
@ -1,40 +0,0 @@
|
||||
function [x, errors] = GaussSeidelMethod(A, b)
|
||||
%GAUSSSEIDELMETHOD solves a system Ax = b using the Gauss-Seidel iterative method
|
||||
% The accuracy target is 10e-10
|
||||
% returns x - the solution and errors - vector of errors for each
|
||||
% iteration
|
||||
|
||||
[L, D, U] = decomposeLDU(A);
|
||||
|
||||
%Checking covergence conditions
|
||||
if ~(rowDominant(A) || columnDominant(A))
|
||||
sr = max(abs(eig(-inv(D)*(L+U))));
|
||||
if sr >= 1
|
||||
x = sr;
|
||||
return
|
||||
end
|
||||
end
|
||||
|
||||
%Initial guess
|
||||
x = zeros(length(A), 1);
|
||||
|
||||
n = length(A);
|
||||
iteration = 1;
|
||||
errors(iteration) = vecnorm(A*x - b);
|
||||
|
||||
while errors(iteration) > 10^-10
|
||||
w = U*x - b;
|
||||
for i = 1:1:n
|
||||
sum = 0;
|
||||
for j = 1:1:i-1
|
||||
sum = sum - L(i,j) * x(j);
|
||||
end
|
||||
sum = sum - w(i);
|
||||
x(i) = sum / D(i,i);
|
||||
end
|
||||
|
||||
iteration = iteration + 1;
|
||||
|
||||
errors(iteration) = vecnorm(A*x - b);
|
||||
end
|
||||
|
||||
@ -1,32 +0,0 @@
|
||||
function [x, errors] = JacobiMethod(A, b)
|
||||
%JACOBIMETHOD solves a system Ax = b using the Jacobi iterative method
|
||||
% The accuracy target is 10e-10
|
||||
% returns x - the solution and errors - vector of errors for each
|
||||
% iteration
|
||||
|
||||
[L, D, U] = decomposeLDU(A);
|
||||
|
||||
%Checking covergence conditions
|
||||
if ~(rowDominant(A) || columnDominant(A))
|
||||
sr = max(abs(eig(-inv(D)*(L+U))));
|
||||
if sr >= 1
|
||||
x = sr;
|
||||
return
|
||||
end
|
||||
end
|
||||
|
||||
%Initial guess
|
||||
x = zeros(length(A), 1);
|
||||
|
||||
Di = inv(D);
|
||||
iteration = 1;
|
||||
errors(iteration) = vecnorm(A*x - b);
|
||||
|
||||
while errors(iteration) > 10^-10
|
||||
x = -Di * (L+U) * x + Di*b;
|
||||
|
||||
iteration = iteration + 1;
|
||||
|
||||
errors(iteration) = vecnorm(A*x - b);
|
||||
end
|
||||
|
||||
@ -1,20 +0,0 @@
|
||||
function [out] = columnDominant(A)
|
||||
%COLUMNDOMINANT checks column dominance of matrix A
|
||||
% returns true when dominant, false otherwise
|
||||
n = size(A);
|
||||
|
||||
for j = 1:1:n
|
||||
sum = 0;
|
||||
for i = 1:1:n
|
||||
if i == j
|
||||
continue
|
||||
end
|
||||
sum = sum + abs(A(i,j));
|
||||
end
|
||||
if abs(A(j,j)) <= sum
|
||||
out = false;
|
||||
return
|
||||
end
|
||||
end
|
||||
out = true;
|
||||
end
|
||||
@ -1,27 +0,0 @@
|
||||
function [L,D,U] = decomposeLDU(A)
|
||||
%DECOMPOSELDU decomposes the matrix A into L+D+U
|
||||
% returns L - lowerdiagonal, D - diagonal, U - upperdiagonal
|
||||
|
||||
n = size(A);
|
||||
L = zeros(n);
|
||||
D = zeros(n);
|
||||
U = zeros(n);
|
||||
|
||||
for i = 2:1:n
|
||||
for j = 1:1:i-1
|
||||
L(i,j) = A(i,j);
|
||||
end
|
||||
end
|
||||
|
||||
for i = 1:1:n
|
||||
D(i,i) = A(i,i);
|
||||
end
|
||||
|
||||
for i = 1:1:n
|
||||
for j = i+1:1:n
|
||||
U(i,j) = A(i,j);
|
||||
end
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
@ -1,21 +0,0 @@
|
||||
function [A, b] = matrixGen2a(n)
|
||||
%matrixGen2a Generates matrices for the task 2a
|
||||
% generates matrices for a system Ax = B of n linear equations
|
||||
|
||||
A = zeros(n, n);
|
||||
b = zeros(n, 1);
|
||||
|
||||
for i = 1:n
|
||||
b(i) = 0.9*i;
|
||||
for j = 1:n
|
||||
if(i == j)
|
||||
A(i, j) = 11;
|
||||
elseif (i == j-1)
|
||||
A(i, j) = 5;
|
||||
elseif (i == j+1)
|
||||
A(i, j) = 5;
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@ -1,18 +0,0 @@
|
||||
function [A,b] = matrixGen2b(n)
|
||||
%matrixGen2b Generates matrices for the task 2b
|
||||
% generates matrices for a system Ax = B of n linear equations
|
||||
|
||||
A = zeros(n, n);
|
||||
b = zeros(n, 1);
|
||||
|
||||
for i = 1:n
|
||||
if( mod(i,2) == 0 )
|
||||
b(i) = 2/(3*i);
|
||||
else
|
||||
b(i) = 0;
|
||||
end
|
||||
for j = 1:n
|
||||
A(i,j) = 7/(8*(i+j+1));
|
||||
end
|
||||
end
|
||||
end
|
||||
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|
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|
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|
Before Width: | Height: | Size: 23 KiB |
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|
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|
Before Width: | Height: | Size: 42 KiB |
@ -1,21 +0,0 @@
|
||||
function [out] = rowDominant(A)
|
||||
%ROWDOMINANT checks row dominance of matrix A
|
||||
% returns true when dominant, false otherwise
|
||||
n = size(A);
|
||||
|
||||
for i = 1:1:n
|
||||
sum = 0;
|
||||
for j = 1:1:n
|
||||
if i == j
|
||||
continue
|
||||
end
|
||||
sum = sum + abs(A(i,j));
|
||||
end
|
||||
if abs(A(i,i)) <= sum
|
||||
out = false;
|
||||
return
|
||||
end
|
||||
end
|
||||
out = true;
|
||||
end
|
||||
|
||||
@ -1,56 +0,0 @@
|
||||
clear all;
|
||||
%First case
|
||||
A = [20 10 8 1;
|
||||
1 10 6 2;
|
||||
2 1 8 4;
|
||||
1 1 1 4;];
|
||||
b = [100; 80; 60; 40];
|
||||
%From task 2
|
||||
[Aa, ba] = matrixGen2a( 10 );
|
||||
[Ab, bb] = matrixGen2b( 10 );
|
||||
|
||||
[xJ, errorsJ] = JacobiMethod(A, b);
|
||||
[xGS, errorsGS] = GaussSeidelMethod(A, b);
|
||||
[xJa, errorsJa] = JacobiMethod(Aa, ba);
|
||||
[xGSa, errorsGSa] = GaussSeidelMethod(Aa, ba);
|
||||
[sr] = JacobiMethod(Ab, bb);
|
||||
|
||||
%Plots
|
||||
figure(1);
|
||||
plot(1:1:length(errorsJ), errorsJ, '.')
|
||||
title("Jacobi method, matrices 3");
|
||||
xlabel("Iteration"); ylabel("Euclidean norm of errors");
|
||||
saveas(1, "./plots/Jacobi3.fig");
|
||||
saveas(1, "./plots/Jacobi3.png");
|
||||
|
||||
figure(2);
|
||||
plot(1:1:length(errorsJa), errorsJa, '.')
|
||||
title("Jacobi method, matrices 2a");
|
||||
xlabel("Iteration"); ylabel("Euclidean norm of errors");
|
||||
saveas(2, "./plots/Jacobi2a.fig");
|
||||
saveas(2, "./plots/Jacobi2a.png");
|
||||
|
||||
figure(3);
|
||||
plot(1:1:length(errorsGS), errorsGS, '.')
|
||||
title("Gauss-Seidel method, matrices 3");
|
||||
xlabel("Iteration"); ylabel("Euclidean norm of errors");
|
||||
saveas(3, "./plots/GaussSeidel3.fig");
|
||||
saveas(3, "./plots/GaussSeidel3.png");
|
||||
|
||||
figure(4)
|
||||
plot(1:1:length(errorsGSa), errorsGSa, '.')
|
||||
title("Gauss-Seidel method, matrices 2a");
|
||||
xlabel("Iteration"); ylabel("Euclidean norm of errors");
|
||||
saveas(4, "./plots/GaussSeidel2a.fig");
|
||||
saveas(4, "./plots/GaussSeidel2a.png");
|
||||
|
||||
figure(5);
|
||||
hold on;
|
||||
plot(1:1:length(errorsJ), errorsJ, 'o')
|
||||
plot(1:1:length(errorsGS), errorsGS, 'o')
|
||||
legend('Jacobi', 'Gauss-Seidel');
|
||||
title("Jacobi and Gauss-Seidel comparison");
|
||||
xlabel("Iteration"); ylabel("Euclidean norm of errors");
|
||||
xlim([0 40]);
|
||||
saveas(5, "./plots/compare.fig");
|
||||
saveas(5, "./plots/compare.png");
|
||||
@ -1,26 +0,0 @@
|
||||
function [Q,R] = QRfactorize(A)
|
||||
%QRFACTORIZE factorizes the matrix A using QR factorization
|
||||
% returns the matrices Q and R
|
||||
[m, n] = size(A);
|
||||
Q = zeros(m, n);
|
||||
R = zeros(m, n);
|
||||
d = zeros(m, n);
|
||||
%Factorization
|
||||
for i = 1:1:n
|
||||
Q(:, i) = A(:, i);
|
||||
R(i, i) = 1;
|
||||
d(i) = Q(:, i)' * Q(:, i);
|
||||
|
||||
for j = i+1:1:n
|
||||
R(i, j) = (Q(: ,i)' * A(:, j)) / d(i);
|
||||
A(:, j) = A(:, j) - R(i, j) * Q(:, i);
|
||||
end
|
||||
end
|
||||
%Normalization
|
||||
for i = 1:1:n
|
||||
dd = norm(Q(:, i));
|
||||
Q(:, i) = Q(:, i) / dd;
|
||||
R(i, i:n) = R(i, i:n) * dd;
|
||||
end
|
||||
end
|
||||
|
||||
@ -1,17 +0,0 @@
|
||||
function [eigenvalues,iteration,Afinal] = eigenvalueQRnoshift(A)
|
||||
%EIGENVALUEQRNOSHIFT calculates eigenvalues using the QR method with no
|
||||
%shifts
|
||||
% returns eigenvalues, the number of iterations and the transformed
|
||||
% matrix A
|
||||
iteration = 1;
|
||||
while max(max(A-diag(diag(A)))) > 10^-6
|
||||
[Q, R] = QRfactorize(A);
|
||||
A = R * Q;
|
||||
iteration = iteration+1;
|
||||
end
|
||||
|
||||
eigenvalues = diag(A);
|
||||
Afinal = A;
|
||||
|
||||
end
|
||||
|
||||
@ -1,37 +0,0 @@
|
||||
function [eigenvalues,iteration,Afinal] = eigenvalueQRshift(A)
|
||||
%EIGENVALUEQRSHIFT calculates eigenvalues using the QR method with shifts
|
||||
% returns eigenvalues, the number of iterations and the transformed
|
||||
% matrix A
|
||||
n = size(A, 1);
|
||||
eigenvalues = diag(ones(n));
|
||||
|
||||
initialSub = A;
|
||||
iteration = 0;
|
||||
for k = n:-1:2
|
||||
DK = initialSub;
|
||||
|
||||
while max(abs(DK(k, 1:k-1))) > 10^-6
|
||||
DD = DK(k-1:k, k-1:k);
|
||||
[ev1, ev2] = quadpolynroots(1, -(DD(1,1) + DD(2,2)), DD(2,2) * DD(1,1) - DD(2,1) * DD(1,2));
|
||||
|
||||
if abs(ev1 - DD(2, 2)) < abs(ev2 - DD(2, 2))
|
||||
shift = ev1;
|
||||
else
|
||||
shift = ev2;
|
||||
end
|
||||
DP = DK - eye(k) * shift;
|
||||
[Q, R] = QRfactorize(DP);
|
||||
DK = R * Q + eye(k) * shift;
|
||||
iteration = iteration + 1;
|
||||
end
|
||||
eigenvalues(k) = DK(k, k);
|
||||
A(1:k, 1:k) = DK(1:k, 1:k);
|
||||
if k > 2
|
||||
initialSub = DK(1:k-1, 1:k-1);
|
||||
else
|
||||
eigenvalues(1) = DK(1, 1);
|
||||
end
|
||||
end
|
||||
Afinal = A;
|
||||
end
|
||||
|
||||
@ -1,18 +0,0 @@
|
||||
function [x1, x2] = quadpolynroots(a,b,c)
|
||||
%UNTITLED4 Summary of this function goes here
|
||||
% Detailed explanation goes here
|
||||
|
||||
l1 = -b + sqrt(b*b - 4*a*c);
|
||||
l2 = -b - sqrt(b*b - 4*a*c);
|
||||
|
||||
if abs(l1) > abs(l2)
|
||||
ctr = l1;
|
||||
else
|
||||
ctr = l2;
|
||||
end
|
||||
|
||||
x1 = ctr/(2 * a);
|
||||
x2 = ((-b) / a) - x1;
|
||||
|
||||
end
|
||||
|
||||
@ -1,11 +0,0 @@
|
||||
clear all;
|
||||
|
||||
A = [23 12 3 5 10;
|
||||
12 14 8 5 22;
|
||||
3 8 9 13 11;
|
||||
5 5 13 10 17;
|
||||
10 22 11 17 25];
|
||||
|
||||
[eigNS, iteNS, finNS] = eigenvalueQRnoshift(A);
|
||||
[eigS, iteS, finS] = eigenvalueQRshift(A);
|
||||
eigE = eig(A);
|
||||
Binary file not shown.
|
Before Width: | Height: | Size: 471 KiB |
@ -1,10 +0,0 @@
|
||||
function [] = Zadanie1()
|
||||
x = 1.5; %poczštkowe zainicjalizowanie zmiennej
|
||||
g = 1.0;
|
||||
while( x > 1 ) %przechodzenie przez pętle i dzielenie epsilona
|
||||
g = g/2; %tak długo aż dodanie go nie wpłynie na wynik
|
||||
x = 1.0 + g;
|
||||
end
|
||||
g = g*2; %jeden przebieg pętli w tył
|
||||
fprintf('Wyznaczony epsilon maszynowy: %d \n',g);
|
||||
end
|
||||
@ -1,16 +0,0 @@
|
||||
function [] = Zadanie2a(n)
|
||||
A = a_genA(n); %generator macierzy wejściowej A
|
||||
b = a_genB(n); %generator macierzy wejściowej b
|
||||
b=b';
|
||||
|
||||
tic %rozpoczęcie pomiaru czasu
|
||||
L1 = cholesky(A,n); %rozkład metodą Cholesky'ego-Banachiewicza
|
||||
x = solveEq(L1,b); %rozwiązanie równania
|
||||
t = toc; %koniec pomiaru czasu
|
||||
|
||||
r = b - A * x; %obliczenie residuum
|
||||
br = norm(r); %obliczenie błędu rozwiązania jako normy z residuum
|
||||
|
||||
fprintf("Zmierzony czas rozwiązania: %d \n",t);
|
||||
fprintf("Liczba równań i błąd rozwiązania: \n %d %d \n",n,br);
|
||||
end
|
||||
@ -1,16 +0,0 @@
|
||||
function [] = Zadanie2b(n)
|
||||
A = b_genA(n);
|
||||
b = b_genB(n);
|
||||
b=b';
|
||||
|
||||
tic %rozpoczęcie pomiaru czasu
|
||||
L1 = cholesky(A,n); %rozkład metodš Cholesky'ego-Banachiewicza
|
||||
x = solveEq(L1,b); %rozwišzanie równania
|
||||
t = toc; %koniec pomiaru czasu
|
||||
|
||||
r = b - A * x; %obliczenie residuum
|
||||
br = norm(r); %obliczenie błędu rozwišzania jako normy z residuum
|
||||
|
||||
fprintf("Zmierzony czas rozwišzania: %d \n",t);
|
||||
fprintf("Liczba równań i błšd rozwišzania: \n %d %d \n",n,br);
|
||||
end
|
||||
@ -1,16 +0,0 @@
|
||||
function [] = Zadanie2c(n)
|
||||
A = c_genA(n);
|
||||
b = c_genB(n);
|
||||
b=b';
|
||||
|
||||
tic %rozpoczęcie pomiaru czasu
|
||||
L1 = cholesky(A,n); %rozkład metodš Cholesky'ego-Banachiewicza
|
||||
x = solveEq(L1,b); %rozwišzanie równania
|
||||
t = toc; %koniec pomiaru czasu
|
||||
|
||||
r = b - A * x; %obliczenie residuum
|
||||
br = norm(r); %obliczenie błędu rozwišzania jako normy z residuum
|
||||
|
||||
fprintf("Zmierzony czas rozwišzania: %d \n",t);
|
||||
fprintf("Liczba równań i błšd rozwišzania: \n %d %d \n",n,br);
|
||||
end
|
||||
@ -1,5 +0,0 @@
|
||||
function [A] = a_genA(n)
|
||||
v1 = ones(1,n)*10;
|
||||
v2 = ones(1,n-1)*4;
|
||||
A = diag(v1) + diag(v2,1) + diag(v2,-1);
|
||||
end
|
||||
@ -1,4 +0,0 @@
|
||||
function [b] = a_genB(n)
|
||||
bottom = 2.5 - 0.5*n;
|
||||
b = 2:-0.5:bottom;
|
||||
end
|
||||
@ -1,12 +0,0 @@
|
||||
function [A] = b_genA(n)
|
||||
v1 = 4*n^2-1:-1:4*n^2-n;
|
||||
A = diag(v1);
|
||||
for i = 1:n-1
|
||||
j=i+1;
|
||||
while(j <= n)
|
||||
A(i,j) = 2*(i+j)+1; %dodanie wartości do odpowiedniego indeksu macierzy
|
||||
j=j+1;
|
||||
end
|
||||
end
|
||||
A = triu(A)+triu(A,1)'; %odbicie symetrzyczne wobec przekątnej
|
||||
end
|
||||
@ -1,4 +0,0 @@
|
||||
function [b] = b_genB(n)
|
||||
top = 2.5 + 0.6*n;
|
||||
b = 3.1:0.6:top;
|
||||
end
|
||||
@ -1,7 +0,0 @@
|
||||
function [x] = backwardSubstitution(A,b)
|
||||
m = length(b); %wyłuskanie ostatniego indeksu
|
||||
x(m,1) = b(m)/A(m,m); %wyliczenie ostatniego elementu
|
||||
for i = m-1:-1:1 %iteracja po wszystkiech równaniach poczšwszy od przedostatniego wiersza
|
||||
x(i,1)=(b(i)-A(i,i+1:m)*x(i+1:m,1))./A(i,i); %wyznaczenie niewiadomych
|
||||
end
|
||||
end
|
||||
@ -1,12 +0,0 @@
|
||||
function [A] = c_genA(n)
|
||||
v1 = 0.2*n+0.3:0.3:0.2*n+0.3*n;
|
||||
A = diag(v1);
|
||||
for i = 1:n-1
|
||||
j=i+1;
|
||||
while(j <= n)
|
||||
A(i,j) = 1/(4*(i+j+1)); %dodanie wartości do odpowiedniego indeksu macierzy
|
||||
j=j+1;
|
||||
end
|
||||
end
|
||||
A = triu(A)+triu(A,1)'; %odbicie symetryczne wobec przekątnej
|
||||
end
|
||||
@ -1,4 +0,0 @@
|
||||
function [b] = c_genB(n)
|
||||
b = 1:1:n;
|
||||
b = (1./b).*(5/3); %odwrócenie i przemnożenie każdego elementu
|
||||
end
|
||||
@ -1,12 +0,0 @@
|
||||
function[L1] = cholesky(A,n)
|
||||
L1 = zeros(n,n);
|
||||
for i = 1:n %przechodzenie po kolumnach
|
||||
for j = i:n %przechodzenie wierszy wewnątrz kolumny
|
||||
if(i == j)
|
||||
L1(i,i) = sqrt(A(i,i)-sumDiag(L1,i)); %przypisanie wartości na przekątnej macierzy
|
||||
else
|
||||
L1(j,i) = (A(j,i)-sumRest(L1,i,j))/L1(i,i); %przypisanie wartości w reszcie przypadków
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
@ -1,7 +0,0 @@
|
||||
function [x] = forwardSubstitution(A,b)
|
||||
m = length(b); %wyłuskanie ostatniego indeksu
|
||||
x(1,1) = b(1)/A(1,1); %wyliczenie pierwszego elementu
|
||||
for i = 2:m %iteracja po wszystkich równaniach poczšwszy od drugiego wiersza
|
||||
x(i,1)=(b(i)-A(i,1:i-1)*x(1:i-1,1))./A(i,i); %wyznaczanie niewiadomych
|
||||
end
|
||||
end
|
||||
@ -1,5 +0,0 @@
|
||||
function [x] = solveEq(L1,b)
|
||||
b = b'; %transponowanie macierzy
|
||||
y = forwardSubstitution(L1,b);
|
||||
x = backwardSubstitution(L1',y);
|
||||
end
|
||||
@ -1,6 +0,0 @@
|
||||
function[sum] = sumDiag(L1,i)
|
||||
sum=0;
|
||||
for k = 1:1:i-1
|
||||
sum=sum+L1(i,k)^2;
|
||||
end
|
||||
end
|
||||
@ -1,6 +0,0 @@
|
||||
function[sum] = sumRest(L1,i,j)
|
||||
sum=0;
|
||||
for k = 1:1:i-1
|
||||
sum=sum+(L1(j,k)*L1(i,k));
|
||||
end
|
||||
end
|
||||
@ -1,21 +0,0 @@
|
||||
function[x,iter,od] = GaussSeidel(A,b,n,e)
|
||||
x = zeros(n,1);
|
||||
|
||||
%Stworzenie macierzy naddiagonalne, poddiagonalnej i diagonalnej
|
||||
[L,D,U] = rozkladGaussSeidel(A,n);
|
||||
b=b';
|
||||
r = 1;
|
||||
iter = 1;
|
||||
while(r>e || norm(A*x-b)>e) %kolejne iteracje
|
||||
y = x; %zapamietujemy wektor x z poprzedniej iteracji
|
||||
w = U*x - b;
|
||||
for i = 1:n
|
||||
x(i) = (-L(i,:)*x- w(i))/D(i,i);
|
||||
end
|
||||
r = norm(x-y); %liczmy blad z nomrmy euklidesowej, po kazdej iteracji
|
||||
iter = iter + 1; %zliczamy ilosc iteracji
|
||||
|
||||
end
|
||||
%fprintf('norm(A*x-b):%d\n',norm(A*x-b));
|
||||
od=norm(A*x-b);
|
||||
end
|
||||
@ -1,24 +0,0 @@
|
||||
function[x] = Zadanie3(n,e2)
|
||||
i=1;
|
||||
e=1;
|
||||
A = b_genA(n);
|
||||
b = b_genB(n);
|
||||
|
||||
%Sprawdzenie warunku dostatecznego zbieżności
|
||||
if(warDost(A,n) == 0)
|
||||
disp('Warunek silnej dominacji diagonalnej nie jest spelniony');
|
||||
return
|
||||
end
|
||||
|
||||
while e>e2 % 0.0000001
|
||||
[x,iter(i),od] = GaussSeidel(A,b,n,e);
|
||||
e = e/10;
|
||||
r(i) = od;
|
||||
i = i+1;
|
||||
end
|
||||
plot(iter, r);
|
||||
disp(norm(A*x-b));
|
||||
title('Zaleznosc bledu wyniku od ilosci iteracji')
|
||||
xlabel('Ilosc iteracji');
|
||||
ylabel('Blad wyniku');
|
||||
end
|
||||
Binary file not shown.
@ -1,15 +0,0 @@
|
||||
function[L,D,U] = rozkladGaussSeidel(A,n)
|
||||
L = zeros(n,n); %macierz poddiagonalna
|
||||
U = zeros(n,n); %macierz naddiagonalna
|
||||
D = diag(diag(A)); %macierz diagonalna
|
||||
|
||||
for i = 1:n
|
||||
for j = 1:n
|
||||
if(i<j)
|
||||
U(i,j) = A(i,j);
|
||||
elseif(i>j)
|
||||
L(i,j) = A(i,j);
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
@ -1,17 +0,0 @@
|
||||
function[bool] = warDost(A,n)
|
||||
for i = 1:n
|
||||
%Sumujemy wszystkie elementy w wierszu oprócz głównej przekštnej
|
||||
sum = 0;
|
||||
for j = 1:n
|
||||
if i~=j
|
||||
sum = sum + abs(A(i,j));
|
||||
end
|
||||
end
|
||||
%Sprawdzamy warunek silnej dominancji dla jednego wiersza
|
||||
if sum > abs(A(i,i))
|
||||
bool = 0;
|
||||
return;
|
||||
end
|
||||
end
|
||||
bool = 1;
|
||||
end
|
||||
Binary file not shown.
Binary file not shown.
|
Before Width: | Height: | Size: 854 KiB |
@ -1,73 +0,0 @@
|
||||
function [] = Zadanie1(czyPrzesun,czySym)
|
||||
% czyPrzesun - wartosc 1 jesli wersja z przesunieciami
|
||||
% czySym - wartosc 1 jestli macierz symetryczna
|
||||
|
||||
iterQR = 0; %liczba wartosci wlasnej QR bez przesuniec
|
||||
iterQRS = 0; %liczba wartosci wlasnej QR z przesunieciami
|
||||
|
||||
timeQR = 0; %czas wykonania alg QR bez przesuniec
|
||||
timeQRS = 0; %czas wykonania alg QR z przesunieciami
|
||||
timeE = 0; %czas wykonania alg eig() wbudowanym w MATLAB
|
||||
|
||||
SIZE = 20; %rozmiar maciery
|
||||
matrixNumberR = 30; %ilosc losowych macierzy
|
||||
|
||||
ILEMACQR = 0;
|
||||
ILEMACQRS = 0;
|
||||
|
||||
%Generowanie danych
|
||||
for i=1:matrixNumberR
|
||||
|
||||
A = genA(SIZE,czySym);
|
||||
tolerance = 0.0000001;
|
||||
imax = 200;
|
||||
|
||||
start = tic;
|
||||
[~,D] = eig(A);
|
||||
timeEig = toc(start);
|
||||
timeE = timeE + timeEig;
|
||||
|
||||
if czyPrzesun == 1 %QR z przesunieciami (eignes - wartWlasne)
|
||||
[eigens, iteracje, time, ok] = zPrzesun(A, tolerance, imax);
|
||||
if ok == 1
|
||||
ILEMACQRS = ILEMACQRS + 1;
|
||||
iterQRS = iterQRS + iteracje;
|
||||
timeQRS = timeQRS + time;
|
||||
end
|
||||
else %QR bez przesuniec (eigens - wartWlasne)
|
||||
[eigens, iteracje, time, ok] = bezPrzesun(A, tolerance, imax);
|
||||
if ok == 1
|
||||
ILEMACQR = ILEMACQR + 1;
|
||||
iterQR = iterQR + iteracje;
|
||||
timeQR = timeQR + time;
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
fprintf('Ilosc macierzy: %d\n',matrixNumberR);
|
||||
fprintf('Wielkosc macierzy: %d\n',SIZE);
|
||||
SREDNIAczasEig = timeE / matrixNumberR;
|
||||
|
||||
if czyPrzesun == 1
|
||||
%Wyniki z przesunieciami:
|
||||
SREDNIAQRSI = iterQRS / ILEMACQRS;
|
||||
SREDNIAczasQRS = timeQRS / ILEMACQRS;
|
||||
fprintf('Z przesunieciami:\n');
|
||||
fprintf('Ilosc zkonczonych sukcesem: %d\n', ILEMACQRS);
|
||||
fprintf('Srednia ilosci iteracji %d\n',SREDNIAQRSI);
|
||||
fprintf('Sredni czas obliczen %d\n',SREDNIAczasQRS);
|
||||
else
|
||||
%Wyniki bez przesuniec:
|
||||
SREDNIAQRI = iterQR / ILEMACQR;
|
||||
SREDNIAczasQR = timeQR / ILEMACQR;
|
||||
fprintf('Bez przesuniec:\n');
|
||||
fprintf('Ilosc zkonczonych sukcesem: %d\n', ILEMACQR);
|
||||
fprintf('Srednia ilosci iteracji %d\n',SREDNIAQRI);
|
||||
fprintf('Sredni czas obliczen %d\n',SREDNIAczasQR);
|
||||
end
|
||||
|
||||
fprintf('Sredni czas obliczen eig %d\n',SREDNIAczasEig);
|
||||
d = diag(D);
|
||||
disp(sort(d));
|
||||
disp(sort(eigens));
|
||||
end
|
||||
@ -1,20 +0,0 @@
|
||||
function [wartWlasne, iteracje, time, ok] = bezPrzesun(D, tolerance, imax)
|
||||
%imax - maksymalna liczba iteracji
|
||||
% ok - czy funkcja wykonala sie z przekroczeniem imax
|
||||
start = tic;
|
||||
ok = 1;
|
||||
i = 1;
|
||||
|
||||
while i <= imax && max(max(D-diag(diag(D)))) > tolerance
|
||||
[Q1, R1] = qrZmodGS(D);
|
||||
D = R1*Q1; %macierz po przekształceniu
|
||||
i = i + 1;
|
||||
end
|
||||
|
||||
if i > imax
|
||||
ok = 0;
|
||||
end
|
||||
iteracje = i;
|
||||
wartWlasne = diag(D); %wykstraktowanie wektora wartości własnych
|
||||
time = toc(start);
|
||||
end
|
||||
@ -1,10 +0,0 @@
|
||||
function [A] = genA(SIZE,czySym)
|
||||
A = rand(SIZE);
|
||||
while rank(A)~= SIZE %powstanie macierzy o pelnym rzedzie
|
||||
A = rand(SIZE);
|
||||
end
|
||||
|
||||
if czySym == 1
|
||||
A = A'+A; %dla symetrycznych
|
||||
end
|
||||
end
|
||||
@ -1,25 +0,0 @@
|
||||
%rozkład QR (waski) macierzy zmodyfikowanym alg. Grama-Schmidta dla
|
||||
%macierzy mxn (m>=n) o pelnym rzedzie, rzeczywistej lub zespolonej
|
||||
%na podstawie ksišzki prof. Tatjewskiego
|
||||
function [Q,R] = qrZmodGS(A)
|
||||
[m n] = size(A);
|
||||
Q = zeros(m,n);
|
||||
R = zeros(n,n);
|
||||
d = zeros(1,n);
|
||||
%rozkład A z kolumnami Q ortogonalnymi
|
||||
for i=1:n
|
||||
R(i,i)=1;
|
||||
Q(:,i)=A(:,i);
|
||||
d(i)=Q(:,i)'*Q(:,i);
|
||||
for j=i+1:n
|
||||
R(i,j)=(Q(:,i)'*A(:,j))/d(i);
|
||||
A(:,j)=A(:,j)-R(i,j)*Q(:,i);
|
||||
end
|
||||
end
|
||||
%normowanie rozkladu (kolumny Q ortogonalne)
|
||||
for i=1:n
|
||||
dd = norm(Q(:,i));
|
||||
Q(:,i) = Q(:,i)/dd;
|
||||
R(i,i:n) = R(i,i:n)*dd;
|
||||
end
|
||||
end
|
||||
@ -1,44 +0,0 @@
|
||||
function [wartWlasne, iteracje, time, ok] = zPrzesun(A,tolerance,imax)
|
||||
% ok - czy funkcja wykonala sie z przekroczeniem imax
|
||||
% tolerance - toleracnaj jako górna granica wartości elementów zerowanych
|
||||
start = tic;
|
||||
ok = 1;
|
||||
n = size(A,1);
|
||||
wartWlasne = diag(zeros(n));
|
||||
iteracje = 0;
|
||||
INITIALsubmatrix = A;
|
||||
for k=n:-1:2
|
||||
DK = INITIALsubmatrix(1:k, 1:k); %macierz potrzebna do wyznaczenia wartosci w1
|
||||
i = 0;
|
||||
|
||||
while i <= imax && max(abs(DK(k,1:k-1))) > tolerance
|
||||
|
||||
ev = roots([1, -(DK(k-1,k-1)+DK(k,k)), DK(k,k)*DK(k-1,k-1)-DK(k,k-1)*DK(k-1,k)]);
|
||||
if abs(ev(1)-DK(k,k)) < abs(ev(2)-DK(k,k))
|
||||
shift = ev(1); % nasze przesuniecie jako najbliższa DK(k,k) wartosc
|
||||
% wlasna analizowanej macierzy 2x2
|
||||
else
|
||||
shift = ev(2);
|
||||
end
|
||||
DK = DK - eye(k)*shift; %nasza macierz przesunięta
|
||||
[Q1, R1] = qrZmodGS(DK); %faktoryzacja QR
|
||||
DK = R1*Q1 + eye(k)*shift; %macierz przekształcona
|
||||
i = i+1;
|
||||
|
||||
iteracje = iteracje + 1;
|
||||
end
|
||||
|
||||
if i > imax
|
||||
ok = 0;
|
||||
break;
|
||||
end
|
||||
wartWlasne(k) = DK(k,k);
|
||||
|
||||
if k>2
|
||||
INITIALsubmatrix = DK(1:k-1,1:k-1); %definicja macierzy
|
||||
else
|
||||
wartWlasne(1) = DK(1,1); %ostatnia wartosc wlasna
|
||||
end
|
||||
end
|
||||
time = toc(start);
|
||||
end
|
||||
@ -1,25 +0,0 @@
|
||||
function [] = Zadanie2(STOPIEN)
|
||||
x = -5:1:5;
|
||||
x2 = -5:0.1:5;
|
||||
y = [-7.7743 -0.2235 1.9026 0.6572 0.1165 -1.8144 -1.0968 -0.8261 1.3327 6.1857 8.2891];
|
||||
x=x';
|
||||
y=y';
|
||||
a = najmnKwadratow(STOPIEN,x,y);
|
||||
|
||||
eukNorm = euklidesNorm(a,y,x);
|
||||
czebNorm = czebyszewNorm(a,y,x);
|
||||
|
||||
fprintf('Błšd aproksymacji w normie Czebyszewa dla równania rzędu %d wynosi %d\n',STOPIEN,czebNorm);
|
||||
fprintf('Błšd aproksymacji w normie Euklidesowej dla równania rzędu %d wynosi %d\n',STOPIEN,eukNorm);
|
||||
|
||||
%Rysowanie wykresu
|
||||
scatter(x,y,'filled')
|
||||
hold on
|
||||
p = polyval(a,x2);
|
||||
plot(x2,p);
|
||||
grid;
|
||||
title('Aproksymacja przy stopniu równym 10 - równania normalne');
|
||||
xlabel('x');
|
||||
ylabel('y');
|
||||
legend('Punkty oryginalne','funkcja aproksymujšca')
|
||||
end
|
||||
@ -1,13 +0,0 @@
|
||||
function [maxError] = czebyszewNorm(a,f,x)
|
||||
p = polyval(a,x);
|
||||
[s,~] = size(f);
|
||||
i = 1;
|
||||
maxError = 0;
|
||||
while i<=s
|
||||
err = abs(p(i)-f(i));
|
||||
if err > maxError
|
||||
maxError = err;
|
||||
end
|
||||
i = i + 1;
|
||||
end
|
||||
end
|
||||
Some files were not shown because too many files have changed in this diff Show More
Loading…
Reference in New Issue
Block a user