Merge remote-tracking branch 'ENUME/main' into main

ENUME/README.md

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+# First project (Number 31) scored maximum (15/15) points
+# Second project (Number 32) scored maximum (15/15) points
+# Third project (Number 32) scored 8/20 points

ENUME/projectA/QRNoShifts.m

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+function [eigenValues, whichIterationAreWeOn, Matrix] = QRNoShifts(Matrix)
+
+    [whichIterationAreWeOn, threshold, startingMatrix, matlabEigen] = initializeValues(Matrix);
+    [Matrix, whichIterationAreWeOn] = QRNoShiftsLoop(threshold, Matrix, whichIterationAreWeOn);
+    eigenValues = diag(Matrix)';
+    %displayResults(eigenValues, whichIterationAreWeOn, Matrix, startingMatrix, matlabEigen);
+end
+
+function [Matrix, whichIterationAreWeOn] = QRNoShiftsLoop(threshold, Matrix, whichIterationAreWeOn)
+    while threshold > 1e-6
+        [Matrix, whichIterationAreWeOn, threshold] = QRNoShiftsInsideLoop(Matrix, whichIterationAreWeOn);
+    end
+end
+
+function [Matrix, whichIterationAreWeOn, threshold] = QRNoShiftsInsideLoop(Matrix, whichIterationAreWeOn)
+    [Q, R] = gramSchmidtAlgorithm(Matrix);
+    Matrix = R * Q;
+    whichIterationAreWeOn = whichIterationAreWeOn + 1;
+    
+    % iterate until all non-diagonal elements are below the threshold
+    matrixWithoutDiagonal = Matrix - diag(diag(Matrix)); % first diag converts Matrix 
+    % into vector consisting of values on the diagonal of matrix, 
+    % second diag converts this vector into matrix with zeros on
+    % everything except diagonal
+    % If we substract it from Matrix we get original Matrix with zeros
+    % on a diagonal
+    threshold = max(max(abs(matrixWithoutDiagonal)));
+    % first max returns vector of elements 
+    % second max returns max element from this vector
+end
+
+function displayResults(eigenValues, whichIterationsAreWeOn, Matrix, startingMatrix, matlabEigen)
+    disp("How many iterations it took:")
+    disp(whichIterationsAreWeOn)
+    disp("Starting Matrix:")
+    disp(startingMatrix)
+    disp("Final Matrix:")
+    disp(Matrix)
+    disp("eig(Matrix) eigen values:")
+    disp(matlabEigen)
+    disp("Our eigen values:")
+    disp(eigenValues);
+end
+
+function [whichIterationAreWeOn, threshold, startingMatrix, matlabEigen] = initializeValues(Matrix)
+    whichIterationAreWeOn = 0;
+    threshold = inf;
+    startingMatrix = Matrix;
+    matlabEigen = eig(Matrix);
+end
+
+% performs QR or QRdash decomposition of a matrix
+function [Q, R] = gramSchmidtAlgorithm(Matrix)
+    [columns, Q, R, d] = initializeGramSchmid(Matrix);
+    [Q, R] = factorizeColumnsOfQ(columns, Q, Matrix, R, d);
+    [Q, R] = normalizeColumns(columns, Q, R);
+end
+
+function [columns, Q, R, d] = initializeGramSchmid(Matrix)
+    % We start with empty matrices
+    [rows, columns] = size(Matrix);
+    Q = zeros(rows, columns);
+    R = zeros(columns, columns);
+    d = zeros(1, columns);
+end
+
+function [Q, R] = factorizeColumnsOfQ(columns, Q, Matrix, R, d)
+    for i = 1 : columns 
+        Q(:, i) = Matrix(:, i);
+        R(i, i) = 1;
+        d(i) = Q(:, i)' * Q(:, i);
+        for i2 = i + 1 : columns 
+            R(i, i2) = (Q(:, i)' * Matrix(:, i2)) / d(i);
+            Matrix(:, i2) = Matrix(:, i2) - R(i, i2) * Q(:, i);
+        end
+    end
+end
+
+function [Q, R] = normalizeColumns(columns, Q, R)
+    for i = 1 : columns
+        dd = norm(Q(:, i));
+        Q(:, i) = Q(:, i) / dd;
+        R(i, i:columns) = R(i, i : columns) * dd;
+    end
+end

ENUME/projectA/QRShifts.m

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+% finds the eigenValues of a matrix using QR with shifts
+function [eigenValues, whatIterationAreWeOn, Matrix] = QRShifts(Matrix)
+    eigenFromMatlab = eig(Matrix);
+    initialMatrix = Matrix;
+    [eigenValues, whatIterationAreWeOn, matrixSize, minThreshold] = initiateValues(Matrix);
+    [Matrix, whatIterationAreWeOn, eigenValues] = QRShiftLoop(matrixSize, Matrix, eigenValues, minThreshold, whatIterationAreWeOn);
+    dispResults(eigenValues, Matrix, whatIterationAreWeOn, eigenFromMatlab, initialMatrix);
+end 
+
+function [eigenValues, whatIterationAreWeOn, matrixSize, minThreshold] = initiateValues(Matrix)
+    eigenValues = double.empty(1, 0);
+    whatIterationAreWeOn = 0;
+    matrixSize = size(Matrix, 1);
+    minThreshold = 1e-6;
+end
+
+function [Matrix, whatIterationAreWeOn, eigenValues] = QRShiftLoop(matrixSize, Matrix, eigenValues, minThreshold, whatIterationAreWeOn)
+    while matrixSize >= 2
+        flag = 0;
+        [Matrix, matrixSize, whatIterationAreWeOn, eigenValues] = findEigenValue(Matrix, matrixSize, whatIterationAreWeOn, eigenValues, minThreshold, flag);
+        [matrixSize, Matrix] = deflateMatrix(Matrix, matrixSize);
+    end
+    eigenValues(size(eigenValues, 2) + 1) = Matrix(1, 1);
+end
+
+function [Matrix, matrixSize, whatIterationAreWeOn, eigenValues] = findEigenValue(Matrix, matrixSize, whatIterationAreWeOn, eigenValues, minThreshold, flag)
+    while flag == 0
+        eigenValueCorner = getEigenValueFromCorner(Matrix, matrixSize);
+        [Matrix, whatIterationAreWeOn] = shiftAndIterate(matrixSize, Matrix, eigenValueCorner, whatIterationAreWeOn);
+        [flag, eigenValues] = thresholdBreached(Matrix, matrixSize, minThreshold, eigenValues);
+    end
+end
+
+function eigenValueCorner = getEigenValueFromCorner(Matrix, matrixSize)
+    corner = Matrix((matrixSize - 1) : matrixSize, (matrixSize - 1) : matrixSize);
+    % get 2 x 2 corner of the matrix
+    eigenValueCorner = solveCharactersticEquation(corner);
+end
+
+function [Matrix, whatIterationAreWeOn] = shiftAndIterate(matrixSize, Matrix, eigenValueCorner, whatIterationAreWeOn)
+    % shift and iterate algorithm
+    identityMatrix = eye(matrixSize);
+    Matrix = Matrix - identityMatrix * eigenValueCorner;
+    [Q, R] = gramSchmidtAlgorithm(Matrix);
+    Matrix = R * Q + identityMatrix * eigenValueCorner;
+    whatIterationAreWeOn = whatIterationAreWeOn + 1;
+end
+
+function [matrixSize, Matrix] = deflateMatrix(Matrix, matrixSize)
+    matrixSize = matrixSize - 1;
+    Matrix = Matrix(1:matrixSize, 1:matrixSize);
+end
+
+function [flag, eigenValues] = thresholdBreached(Matrix, matrixSize, minThreshold, eigenValues)
+    flag = 0;
+    threshold = max(abs(Matrix(matrixSize, 1:(matrixSize - 1))));
+
+     % once we zero the row (or rather get close enough to zero) exit loop
+    if (threshold <= minThreshold)
+        eigenValues(size(eigenValues, 2) + 1) = Matrix(matrixSize, matrixSize);
+        flag = 1;
+    end
+end
+
+% finds the eigenvalue of a 2x2 matrix that is closer to the lower right corner
+function eigen = solveCharactersticEquation(Matrix)
+    [eigenOne, eigenTwo] = calculateZeros(Matrix);
+    eigen = valueCloserToLowerRightCorner(eigenOne, eigenTwo, Matrix);
+end
+
+function eigen = valueCloserToLowerRightCorner(eigenOne, eigenTwo, Matrix)
+    if abs(Matrix(4) - eigenOne) < abs(Matrix(4) - eigenTwo)
+        eigen = eigenOne;
+    else
+        eigen = eigenTwo;
+    end
+end
+
+function [zeroOne, zeroTwo] = calculateZeros(Matrix)
+    b = Matrix(1) + Matrix(4);
+    ac = Matrix(1) * Matrix(4) - Matrix(2) * Matrix(3);
+    delta = (b) ^ 2 - 4 * ac;
+    % get delta of quadratic equation
+    squareRootDelta = sqrt(delta);
+    % square delta
+    zeroOne = (b - squareRootDelta) / 2;
+    zeroTwo = (b + squareRootDelta) / 2;
+end
+
+% performs QR or QRdash decomposition of a matrix
+function [Q, R] = gramSchmidtAlgorithm(Matrix)
+    [columns, Q, R, d] = initializeGramSchmid(Matrix);
+    [Q, R] = factorizeColumnsOfQ(columns, Q, Matrix, R, d);
+    [Q, R] = normalizeColumns(columns, Q, R);
+end
+
+function [columns, Q, R, d] = initializeGramSchmid(Matrix)
+    % We start with empty matrices
+    [rows, columns] = size(Matrix);
+    Q = zeros(rows, columns);
+    R = zeros(columns, columns);
+    d = zeros(1, columns);
+end
+
+function [Q, R] = factorizeColumnsOfQ(columns, Q, Matrix, R, d)
+    for i = 1 : columns 
+        Q(:, i) = Matrix(:, i);
+        R(i, i) = 1;
+        d(i) = Q(:, i)' * Q(:, i);
+        for i2 = i + 1 : columns 
+            R(i, i2) = (Q(:, i)' * Matrix(:, i2)) / d(i);
+            Matrix(:, i2) = Matrix(:, i2) - R(i, i2) * Q(:, i);
+        end
+    end
+end
+
+function [Q, R] = normalizeColumns(columns, Q, R)
+    for i = 1 : columns
+        dd = norm(Q(:, i));
+        Q(:, i) = Q(:, i) / dd;
+        R(i, i:columns) = R(i, i : columns) * dd;
+    end
+end
+
+function dispResults(eigenValues, Matrix, whatIterationAreWeOn, eigenFromMatlab, initialMatrix)
+    disp("Initial matrix: ");
+    disp(initialMatrix);
+    disp("Final matrix: ");
+    disp(Matrix);
+    disp("Number of iterations: ");
+    disp(whatIterationAreWeOn);
+    disp("Eigen values from eig(Matrix):");
+    disp(eigenFromMatlab);
+    disp("Our eigen values: ");
+    disp(eigenValues);
+end

ENUME/projectA/checkIfDiagonallyDominant.m

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+function d = checkIfDiagonallyDominant(Matrix)
+    d = 1;
+    [Rows, ~] = size(Matrix);
+    for i = 1 : Rows
+        rowsSum = 0;
+        for j = 1 : Rows
+            rowsSum = rowsSum + abs(Matrix(i, j));
+        end
+        rowsSum = rowsSum - Matrix(i, i);
+        if Matrix(i, i) <= rowsSum
+            d = 0;
+            break
+        end
+    end
+end

ENUME/projectA/findmacheps.m

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+macheps = 1;
+while 1.0 + macheps / 2 > 1.0
+    macheps = macheps/2;
+end
+
+format long; 
+disp("Display calculated macheps:")
+disp(macheps)
+disp("Display actual eps:")
+disp(eps)
+disp("Display 2^(-52)")
+disp(2^(-52))
+disp("Display difference between calculated macheps and actual eps:")
+disp(macheps - eps)
+disp("Display difference between 2^(-52) and actual eps:")
+disp(2^(-52) - eps)
+disp("Display difference between calculated macheps and 2^(-52):")
+disp(macheps - 2^(-52))

ENUME/projectA/gaussSeidelMethod.m

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+function x = gaussSeidelMethod(Matrix, Vector)
+    [L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, flag, Rows] = initializeValues(Matrix);
+    [x, whichIterationAreWeOn, demandedTolerance] = jacobiLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, flag, Rows);
+    dispFinalResults(x, demandedTolerance, whichIterationAreWeOn, Matrix, Vector);
+end
+
+function [L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, flag, Rows] = initializeValues(Matrix)
+    [Rows, ~] = size(Matrix);
+    [L, D, U] = decomposeMatrix(Matrix);
+    initial_x = zeros(Rows, 1);
+    whichIterationAreWeOn = 0;
+    demandedTolerance = 10e-10; % as per task description
+    flag = 0;
+end
+
+function [L, D, U] = decomposeMatrix(Matrix)
+    D = diag(diag(Matrix));
+    U = triu(Matrix, 1); % Generates upper triangular part of matrix
+    % where the second variable denotes on which diagonal of matrix should we
+    % start
+    L = tril(Matrix, -1); % Generates lower triangular part of matrix
+    % where the second variable denotes on which diagonal of matrix should we
+    % start
+end
+
+function [x, whichIterationAreWeOn, demandedTolerance]  = jacobiLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, flag, Rows)
+    while flag ~= 1 % flag denotes whether norm(Matrix*x-Vector) <= demandedTolerance
+        [x, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = jacobiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, Rows);
+    end
+end
+
+function [x, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = jacobiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, Rows)
+    x = jacobiEquation(D, L, U, initial_x, Vector);
+    [flag, demandedTolerance] = checkError(x, initial_x, demandedTolerance, Matrix, Vector);
+    [initial_x, whichIterationAreWeOn] = endOfLoop(x, whichIterationAreWeOn);
+end
+
+function x = jacobiEquation(D, L, U, initial_x, Vector, Rows)
+    W = U*initial_x - Vector;
+    x(1, 1) = -W(1, 1) / D(1,1);
+    for i = 2 : Rows
+        x(i, 1) = calculateNominator(i, L, x, W) / D(i, i);
+    end
+end
+
+function nominator = calculateNominator(i, L, x, W)
+    nominator = 0;
+    for j = 1 : i - 1
+    nominator = nominator + L(i, j) * x(j);
+    end
+    nominator = - nominator - W(j + 1, 1);
+end
+
+function [flag, demandedTolerance] = checkError(x, initial_x, demandedTolerance, Matrix, Vector)
+    flag = 0;
+    currentError = norm(x - initial_x);
+    if currentError <= demandedTolerance
+        currentError = norm(Matrix*x-Vector);
+
+        if currentError <= demandedTolerance % if sequence as per textbook
+            flag = 1;
+        else 
+            demandedTolerance = demandedTolerance * 2; % arbitrary value
+        end
+    end
+end
+
+function [initial_x, whichIterationAreWeOn, flag] = endOfLoop(x, whichIterationAreWeOn)
+    initial_x = x;
+    whichIterationAreWeOn = whichIterationAreWeOn + 1;
+    flag = 0;
+end
+ 
+function dispFinalResults(x, demandedTolerance, whichIterationAreWeOn, Matrix, Vector)
+    disp("Final demandedTolerance");
+    disp(demandedTolerance);
+    disp("Final Iteration: ");
+    disp(whichIterationAreWeOn);
+    disp("Error:");
+    disp(norm(Matrix*x - Vector));
+    disp("A\b error:");
+    disp(norm(Matrix * (Matrix\Vector) - Vector));
+end

ENUME/projectA/indicatedMethod.m

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+function x = indicatedMethod(Matrix, Vector) % Name of the method as in the textbook
+% x stands for obtained result
+    disp(Matrix \ Vector);
+    checkIfMatrixIsSquareMatrix(Matrix);
+    [Matrix, Vector, x] = solveSystem(Matrix, Vector);
+    errorBeforeResidualCorrection = norm(Matrix * x - Vector);
+    disp("Solutions before residual correction:")
+    disp(x);
+    x = iterativeResidualCorrection(Matrix, x, Vector); % Improve on the solution
+    disp("errorBeforeResidualCorrection")
+    disp(errorBeforeResidualCorrection);
+    errorAfterResidualCorrection = norm(Matrix*x - Vector);
+    disp("errorAfterResidualCorrection")
+    disp(errorAfterResidualCorrection);
+    disp("A\b error:")
+    disp(norm(Matrix *  (Matrix \ Vector)  - Vector));
+    disp(Matrix\Vector);
+
+end % end function
+
+function [Matrix, Vector, x] = solveSystem(Matrix, Vector)
+    [~,Columns] = size(Matrix); % We need to know how big the matrix is in next steps
+    % notice the '~', since we assume we use square matrix, we do not need
+    % to have another variable for number of rows since it is the same as
+    % number of columns
+    [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector); 
+    % Change matrix to upper triangular matrix
+    [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector); 
+    % Get the solution
+end % end function
+
+function checkIfMatrixIsSquareMatrix(Matrix)
+    [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 j = 1 : Columns
+            centralElement = max(Matrix(j:Columns,j)); 
+            % we stay in the same row (j) but we change columns, as in the
+            % textbook
+            [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns);
+            % ensures that a_kk != 0 and reduces errors
+            [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector);
+            % change matrix into upper triangular matrix
+    end % end for
+end % end function
+
+function [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns)
+    for k = j : Columns
+        partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement);
+    end % end for
+end % end function
+
+function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement)
+    if Matrix(k,j) == centralElement
+    swapRowMatrix(Matrix, j, k); % swap jth row with kth row
+    swapValueVector(Vector, j, k); % swap jth value with kth value
+    end % end if
+end % end function
+
+function Matrix = swapRowMatrix(Matrix, j, k) 
+    temp =  Matrix(j , :); % ' : ' denote "all elements in jth row"
+    Matrix(j , :) = Matrix(k, :);
+    Matrix(k, :) = temp; % temp equal to previous value of jth row
+end
+
+function Vector = swapValueVector(Vector, j, k) 
+    temp = Vector(j);
+    Vector(j) = Vector(k);
+    Vector(k) = temp;  % temp equal to previous value of k element of vector
+end % end function
+
+function [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector)        
+    for i = j + 1 : Columns       
+        rowMultiplier = Matrix(i,j) / Matrix(j,j);
+        [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns);            
+    end % end for
+end % end function
+
+function [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns)
+    Vector(i) = Vector(i) - rowMultiplier * Vector(j);
+    for curentColumn = 1 : Columns
+        Matrix(i,curentColumn) = Matrix(i,curentColumn) - rowMultiplier * Matrix(j, curentColumn);
+    end % end for
+end % end function
+
+function [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector)
+    for k = Columns : -1 : 1 
+    % Start at final column and move by -1 each iteration until we reach 1
+        equation = 0;
+        for j = k+1 : Columns
+            equation = equation + Matrix(k,j) * x(j, 1); 
+            % even though x is a vector we still need to put '1' to ensure
+            % that number of columns in the first matrix matches number of
+            % rows in second matrix
+        end % end for
+      
+        x(k, 1) = (Vector(k,1) - equation) / Matrix(k,k);
+        % even though x is a vector we still need to put '1' to ensure
+        % that we do not exceed array bounds
+    end % end for
+end % end function
+
+
+function x = iterativeResidualCorrection(A, x, b)
+    r = A*x - b;
+    for i = 1 : 100
+        [~, ~, deltaX] = solveSystem(A, r);
+        newX = x - deltaX;
+        r = A*newX - b;
+        x = newX;
+    end 
+
+end % end function

ENUME/projectA/iterative.m

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+function [x_j, x_g, whichIterationAreWeOnJ, whichIterationAreWeOnG] = iterative(Matrix, Vector)
+    [L, D, U, initial_x, whichIterationAreWeOnJ, whichIterationAreWeOnG, demandedToleranceJ, demandedToleranceG, flag, Rows] = initializeValues(Matrix);
+    [x_j, whichIterationAreWeOnJ, demandedToleranceJ] = jacobiLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOnJ, demandedToleranceJ, Vector, flag);
+    [x_g, whichIterationAreWeOnG, demandedToleranceG] = gaussSeidelLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOnG, demandedToleranceG, Vector, flag, Rows);
+    dispFinalResults(x_j, x_g, demandedToleranceJ, demandedToleranceG, whichIterationAreWeOnJ, whichIterationAreWeOnG, Matrix, Vector);
+end
+
+function [L, D, U, initial_x, whichIterationAreWeOnJ, whichIterationAreWeOnG, demandedToleranceJ, demandedToleranceG, flag, Rows] = initializeValues(Matrix)
+    [Rows, ~] = size(Matrix);
+    [L, D, U] = decomposeMatrix(Matrix);
+    initial_x = zeros(Rows, 1);
+    whichIterationAreWeOnJ = 0;
+    whichIterationAreWeOnG = 0;
+    demandedToleranceJ = 10e-10; % as per task description
+    demandedToleranceG = 10e-10; % as per task description
+    flag = 0;
+end
+
+function [L, D, U] = decomposeMatrix(Matrix)
+    D = diag(diag(Matrix));
+    U = triu(Matrix, 1); % Generates upper triangular part of matrix
+    % where the second variable denotes on which diagonal of matrix should we
+    % start
+    L = tril(Matrix, -1); % Generates lower triangular part of matrix
+    % where the second variable denotes on which diagonal of matrix should we
+    % start
+end
+
+function [x_j, whichIterationAreWeOn, demandedTolerance]  = jacobiLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, flag)
+    while flag ~= 1 % flag denotes whether norm(Matrix*x_g-Vector) <= demandedTolerance
+        [x_j, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = jacobiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector);
+    end
+end
+
+function [x_g, whichIterationAreWeOn, demandedTolerance]  = gaussSeidelLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, flag, Rows)
+    while flag ~= 1 % flag denotes whether norm(Matrix*x_g-Vector) <= demandedTolerance
+        [x_g, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = gaussiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, Rows);
+    end
+end
+
+function [x_j, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = jacobiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector)
+    x_j = jacobiEquation(D, L, U, initial_x, Vector);
+    [flag, demandedTolerance] = checkError(x_j, initial_x, demandedTolerance, Matrix, Vector);
+    [initial_x, whichIterationAreWeOn] = endOfLoop(x_j, whichIterationAreWeOn);
+end
+
+function [x_j, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = gaussiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, Rows)
+    x_j = gaussSeidelEquation(D, L, U, initial_x, Vector, Rows);
+    [flag, demandedTolerance] = checkError(x_j, initial_x, demandedTolerance, Matrix, Vector);
+    [initial_x, whichIterationAreWeOn] = endOfLoop(x_j, whichIterationAreWeOn);
+end
+
+function x = jacobiEquation(D, L, U, initial_x, Vector)
+    x = - D \ ( L + U ) * initial_x + D \ Vector; % As per formula
+    % We will be using D \ Vector and D \ ( ) instead of inverseD since
+    % this is faster according to matlab
+end
+
+function x_g = gaussSeidelEquation(D, L, U, initial_x, Vector, Rows)
+    W = U*initial_x - Vector;
+    x_g(1, 1) = -W(1, 1) / D(1,1);
+    for i = 2 : Rows
+        x_g(i, 1) = calculateNominator(i, L, x_g, W) / D(i, i);
+    end
+end
+
+function nominator = calculateNominator(i, L, x_g, W)
+    nominator = 0;
+    for j = 1 : i - 1
+    nominator = nominator + L(i, j) * x_g(j);
+    end
+    nominator = - nominator - W(j + 1, 1);
+end
+
+function [flag, demandedTolerance] = checkError(x_g, initial_x, demandedTolerance, Matrix, Vector)
+    flag = 0;
+    currentError = norm(x_g - initial_x);
+    if currentError <= demandedTolerance
+        currentError = norm(Matrix*x_g-Vector);
+
+        if currentError <= demandedTolerance % if sequence as per textbook
+            flag = 1;
+        else 
+            demandedTolerance = demandedTolerance * 1; % arbitrary value
+        end
+    end
+end
+
+function [initial_x, whichIterationAreWeOn, flag] = endOfLoop(x_g, whichIterationAreWeOn)
+    initial_x = x_g;
+    whichIterationAreWeOn = whichIterationAreWeOn + 1;
+    flag = 0;
+end
+ 
+function dispFinalResults(x_j, x_g, demandedToleranceJ, demandedToleranceG, whichIterationAreWeOnJ, whichIterationAreWeOnG, Matrix, Vector)
+    disp("Final demandedTolerance for Jacobi method");
+    disp(demandedToleranceJ);
+    disp("Final demandedTolerance for Gaussian-Seidel method:");
+    disp(demandedToleranceG);
+    disp("Final Iteration for Jacobi method: ");
+    disp(whichIterationAreWeOnJ);
+    disp("Final Iteration for Gaussian-Seidel method: ");
+    disp(whichIterationAreWeOnG);
+    disp("Error for Jacobi method:");
+    disp(norm(Matrix*x_j - Vector));
+    disp("Error for Gaussian-Seidel method:");
+    disp(norm(Matrix*x_g - Vector));
+    disp("A\b error:");
+    disp(norm(Matrix * (Matrix\Vector) - Vector));
+    disp("Answer for Jacobi method: ");
+    disp(x_j);
+    disp("Answer for Gaussian-Seidel method: ");
+    disp(x_g);
+end

ENUME/projectA/jacobiMethod.m

@@ -0,0 +1,77 @@
+function x = jacobiMethod(Matrix, Vector)
+    [L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, flag] = initializeValues(Matrix);
+    [x, whichIterationAreWeOn, demandedTolerance] = jacobiLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, flag);
+    dispFinalResults(x, demandedTolerance, whichIterationAreWeOn, Matrix, Vector);
+end
+
+function [L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, flag] = initializeValues(Matrix)
+    [Rows, ~] = size(Matrix);
+    [L, D, U] = decomposeMatrix(Matrix);
+    initial_x = ones(Rows, 1);
+    whichIterationAreWeOn = 0;
+    demandedTolerance = 10e-10; % as per task description
+    % Minimal values I got: 3.202372833989376e-15 for both system of
+    % equations - original and task 2a)
+    flag = 0;
+end
+
+function [L, D, U] = decomposeMatrix(Matrix)
+    D = diag(diag(Matrix));
+    U = triu(Matrix, 1); % Generates upper triangular part of matrix
+    % where the second variable denotes on which diagonal of matrix should we
+    % start
+    L = tril(Matrix, -1); % Generates lower triangular part of matrix
+    % where the second variable denotes on which diagonal of matrix should we
+    % start
+end
+
+function [x, whichIterationAreWeOn, demandedTolerance]  = jacobiLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector, flag)
+    while flag ~= 1 % flag denotes whether norm(Matrix*x-Vector) <= demandedTolerance
+        [x, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = jacobiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector);
+    end
+end
+
+function [x, whichIterationAreWeOn, demandedTolerance, flag, initial_x] = jacobiInsideLoop(Matrix, L, D, U, initial_x, whichIterationAreWeOn, demandedTolerance, Vector)
+    x = jacobiEquation(D, L, U, initial_x, Vector);
+    [flag, demandedTolerance] = checkError(x, initial_x, demandedTolerance, Matrix, Vector);
+    [initial_x, whichIterationAreWeOn] = endOfLoop(x, whichIterationAreWeOn);
+end
+
+function x = jacobiEquation(D, L, U, initial_x, Vector)
+    x = - D \ ( L + U ) * initial_x + D \ Vector; % As per formula
+    % We will be using D \ Vector and D \ ( ) instead of inverseD since
+    % this is faster according to matlab
+end
+
+function [flag, demandedTolerance] = checkError(x, initial_x, demandedTolerance, Matrix, Vector)
+    flag = 0;
+    currentError = norm(x - initial_x);
+    if currentError <= demandedTolerance
+        currentError = norm(Matrix*x-Vector);
+
+        if currentError <= demandedTolerance % if sequence as per textbook
+            flag = 1;
+        else 
+            demandedTolerance = demandedTolerance * 1; % arbitrary value
+        end
+    end
+end
+
+function [initial_x, whichIterationAreWeOn, flag] = endOfLoop(x, whichIterationAreWeOn)
+    initial_x = x;
+    whichIterationAreWeOn = whichIterationAreWeOn + 1;
+    flag = 0;
+end
+ 
+function dispFinalResults(x, demandedTolerance, whichIterationAreWeOn, Matrix, Vector)
+    disp("Final demandedTolerance");
+    disp(demandedTolerance);
+    disp("Final Iteration: ");
+    disp(whichIterationAreWeOn);
+    disp("A\b matlab:");
+    disp(Matrix \ Vector);
+    disp("Error:");
+    disp(norm(Matrix*x - Vector));
+    disp("A\b error:");
+    disp(norm(Matrix * (Matrix\Vector) - Vector));
+end

ENUME/projectA/matrix4.m

@@ -0,0 +1,6 @@
+function A = matrix4(n)
+    A = 10 * rand(n); % rand generates 5x5 matrix filled with random numbers
+    % we multiply by 10 to get at lest one digit in front of the dot
+    A = floor(A); % we floor the matrix we got to get nice natural numbers matrix
+    A = A * A'; % we get symmetric matrix
+end

ENUME/projectA/matrixA.m

@@ -0,0 +1,14 @@
+function a = matrixA(n) % We want n rows and columns in the matrix
+a = zeros(n, n); % in order to save speed we preallocate zeros to the vector
+    for i = 1 : n % we iterate through rows
+        for j = 1 : n % we iterate through columns
+            if i == j % as per problem description on how to fill the matrix
+                a(i, j) = 9;
+            elseif i == j - 1 || i == j + 1
+                a(i, j) = -3;
+            else
+                a(i, j) = 0;
+            end % end if
+        end % end column for 
+    end % end row for
+end  % end function

ENUME/projectA/matrixB.m

@@ -0,0 +1,8 @@
+function a = matrixB(n) % We want n rows and columns in the matrix
+a = zeros(n, n); % in order to save speed we preallocate zeros to the vector
+    for i = 1 : n % we iterate through rows
+        for j = 1 : n % we iterate through columns
+            a(i, j) = 5 / (8*(i + j + 1));
+        end % end column for 
+    end % end row for
+end  % end function

ENUME/projectA/plotErrorsGaussian.m

@@ -0,0 +1,36 @@
+function plotErrorsGaussian(maxMatrixSize)
+
+    errorsA = zeros(maxMatrixSize);
+    errorsB = zeros(maxMatrixSize);
+    errorsAR = zeros(maxMatrixSize);
+    errorsBR = zeros(maxMatrixSize);
+    for i = 1 : maxMatrixSize
+        [~, errorBeforeResidualCorrection, errorAfterResidualCorrection] = indicatedMethod(matrixA(i), vectorA(i));
+        errorsA(i) = errorBeforeResidualCorrection;
+        errorsAR(i) = errorAfterResidualCorrection;
+        [~, errorBeforeResidualCorrection, errorAfterResidualCorrection] = indicatedMethod(matrixB(i), vectorB(i));
+        errorsB(i) = errorBeforeResidualCorrection;
+        errorsBR(i) = errorAfterResidualCorrection;
+    end
+    nexttile
+    plot(errorsA, '.');
+    title('Errors before residual correction for task 2a:');
+    xlabel('Size of matrix A');
+    ylabel('Errors');
+    nexttile
+    plot(errorsAR, '.');
+    title('Errors after residual correction for task 2a:');
+    xlabel('Size of matrix A');
+    ylabel('Errors');
+    nexttile
+    plot(errorsB, '.');
+    title('Errors before residual correction for task 2b:');
+    xlabel('Size of matrix A');
+    ylabel('Errors');
+    nexttile
+    plot(errorsBR, '.');
+    title('Errors after residual correction for task 2b:');
+    xlabel('Size of matrix A');
+    ylabel('Errors');
+   
+end

ENUME/projectA/plotIterations.m

@@ -0,0 +1,21 @@
+function plotIterations()
+    maxMatrixSize = 500;
+    iterationsJ = zeros(maxMatrixSize);
+    iterationsG = zeros(maxMatrixSize);
+    for i = 1 : maxMatrixSize
+        [~, ~, whichIterationAreWeOnJ, whichIterationAreWeOnG] = iterative(matrixA(i), vectorA(i));
+        iterationsJ(i) = whichIterationAreWeOnJ;
+        iterationsG(i) = whichIterationAreWeOnG;
+    end
+    nexttile
+    plot(iterationsJ, '.');
+    title('Number of iterations for different sizes of matrices for Jacobi method');
+    xlabel('Size of matrix A');
+    ylabel('Number of iterations');
+    nexttile
+    plot(iterationsG, '.');
+    title('Number of iterations for different sizes of matrices for Gauss-Seidel method');
+    xlabel('Size of matrix A');
+    ylabel('Number of iterations');
+   
+end

ENUME/projectA/task2Matrix.m

@@ -0,0 +1,2 @@
+function Matrix = task2Matrix
+    Matrix = [10 -4 1 2; 2 -6 3 -1; 1 4 -12 1; 2 3 -3 -10];

ENUME/projectA/task2MatrixAndVector.m

@@ -0,0 +1,4 @@
+function Matrix = task2Matrix
+    Matrix = [10 -4 1 2; 2 -6 3 -1; 1 4 -12 1; 2 3 -3 -10];
+function Vector = task2Vector
+    Vector = [-8 -12 4 1];

ENUME/projectA/task2Vector.m

@@ -0,0 +1,6 @@
+function Vector = task2Vector
+    Vector = zeros(4, 1);
+    Vector(1, 1) = -8;
+    Vector(2, 1) = -12;
+    Vector(3, 1) = 4;
+    Vector(4, 1) = 1;

ENUME/projectA/task4.m

@@ -0,0 +1,4 @@
+function [eigenValuesNoShifts, iterationsNoShifts, finalMatrixNoShifts, eigenValuesShifts, iterationsShifts, finalMatrixShifts] = task4(Matrix)
+    [eigenValuesNoShifts, iterationsNoShifts, finalMatrixNoShifts] = QRNoShifts(Matrix);
+    [eigenValuesShifts, iterationsShifts, finalMatrixShifts] = QRShifts(Matrix);
+end

ENUME/projectA/task4Plot.m

@@ -0,0 +1,19 @@
+function task4Plot(maxMatrixSize)
+    iterationsNoShiftsVector = zeros(maxMatrixSize);
+    iterationsShiftsVector = zeros(maxMatrixSize);
+    for i = 1 : maxMatrixSize
+        [~, iterationsNoShifts, ~, ~, iterationsShifts, ~] = task4(matrix4(i));
+        iterationsNoShiftsVector(i) = iterationsNoShifts;
+        iterationsShiftsVector(i) = iterationsShifts;
+    end
+    nexttile
+    plot(iterationsNoShiftsVector, '.');
+    title('Number of iterations for different sizes of matrices for no shift method');
+    xlabel('Size of matrix A');
+    ylabel('Number of iterations');
+    nexttile
+    plot(iterationsShiftsVector, '.');
+    title('Number of iterations for different sizes of matrices for shift method');
+    xlabel('Size of matrix A');
+    ylabel('Number of iterations');
+end

ENUME/projectA/usefullinks.txt

@@ -0,0 +1,64 @@
+Useful links:
+
+Problem 1:
+https://stackoverflow.com/questions/27511349/how-to-calculate-machine-epsilon-in-matlab - How to calculate machine epsilion
+
+Problem 3:
+https://www.mathworks.com/help/matlab/learn_matlab/matrices-and-arrays.html - Matrices and arrays MATLAB
+
+https://www.mathworks.com/help/matlab/ref/break.html - Get out of while loop
+
+mathworks.com/help/matlab/ref/triu.html - upper triangular matrix in matlab
+
+LaTeX related:
+https://latex-tutorial.com/symbols/text-formatting/ - Italics
+
+https://www.overleaf.com/learn/latex/Table_of_contents - Table of contents
+
+https://www.overleaf.com/learn/latex/Sections_and_chapters - Sections and chapters
+
+https://www.overleaf.com/learn/latex/Bibliography_management_with_bibtex - Bibliography management
+
+https://en.wikibooks.org/wiki/LaTeX/Title_Creation - Title Creation
+
+https://tex.stackexchange.com/questions/73862/how-can-i-make-a-clickable-table-of-contents - Clickable table of contents
+
+https://tex.stackexchange.com/questions/75116/what-can-i-use-to-typeset-matlab-code-in-my-document - Matlab in LaTeX
+
+https://tex.stackexchange.com/questions/269841/1-5e-10-style-scientific-notation-looks-ugly-in-latex-math-mode-how-can-i-forma - Pretty scientific notation LaTeX
+
+https://tex.stackexchange.com/questions/422197/latex-environment-to-write-in-plain-text-mode - How to treat special characters as simple characters (like '^')
+
+https://www.overleaf.com/learn/latex/Subscripts_and_superscripts - Subscripts and superscripts latex
+
+https://tex.stackexchange.com/questions/32217/3-dots-in-matrix/32221 - 3 dots latex
+
+https://stackoverflow.com/questions/48240982/how-do-you-align-a-system-of-equations - System of equations latex
+
+https://tex.stackexchange.com/questions/526025/problem-with-vdotswithin - vdots alligned to equal sign
+
+https://www.overleaf.com/learn/latex/Brackets_and_Parentheses - Curly brackets, double 'pipes'
+
+https://tex.stackexchange.com/questions/130516/how-to-write-something-vertically-below-another-math-symbol - Symbol vertically below another
+
+https://tex.stackexchange.com/questions/103998/how-to-force-a-table-of-contents-to-start-new-page - new page at the start of chapter in table of contents
+
+https://www.codegrepper.com/code-examples/whatever/column+vector+latex - Vector in latex
+
+https://tex.stackexchange.com/questions/9901/how-to-have-matrices-side-by-side-in-latex - Matrices side by side in LaTeX
+
+https://latex.wikia.org/wiki/Percent_sign_(LaTeX_symbol) - percentage sign latex
+
+https://tex.stackexchange.com/questions/508341/writing-a-system-of-linear-equations - Writing system of equations LaTeX
+
+https://www.overleaf.com/learn/latex/Matrices - matrixes latex
+
+https://tex.stackexchange.com/questions/294561/using-textbf-vs-mathbf-in-math-mode - mathbf vs textbf
+
+https://www.math-linux.com/latex-26/faq/latex-faq/article/how-to-get-dots-in-latex-ldots-cdots-vdots-and-ddots - Dots LaTeX
+
+https://www.overleaf.com/learn/latex/Tables - Tables LaTeX
+
+https://en.wikibooks.org/wiki/LaTeX/Special_Characters - an arrow (circumflex) above letter in latex
+
+https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols - For every symbol in LaTeX 

ENUME/projectA/vectorA.m

@@ -0,0 +1,8 @@
+function b = vectorA(n) % We want n elements in the vector 
+b = zeros(n, 1); % in order to save speed we preallocate zeros to the vector
+    for i = 1 : n % We loop from first element to the last element of the vector
+        b(i, 1) = -5 + 0.3 * i; % Formula as in the task description
+        % We need to put '1' in b(i, 1) so that matlab acknowledges that
+        % this is a column and not a row
+    end % end for
+end % end function

ENUME/projectA/vectorB.m

@@ -0,0 +1,10 @@
+function b = vectorB(n) % We want n elements in the vector 
+b = zeros(n, 1); % in order to save speed we preallocate zeros to the vector
+    for i = 1 : n % We loop from first element to the last element of the vector
+        if mod(i, 2) == 0 % as per problem description
+            b(i, 1) = 9 / (2*i);
+        else
+            b(i, 1) = 0;
+        end % end if
+    end % end for
+end % end function

ENUME/projectB/Code/printComplexGraph.m

@@ -0,0 +1,28 @@
+% graph the complex roots of a function
+function printComplexGraph(printComplexGraph, algorithmName, algorithm, rootBrackets, plottitle)
+    figure();
+    grid on; % Get y values lines
+    hold on; % Retain current plot when adding new plots
+    title([plottitle, algorithmName]);
+    xlabel("Real part");
+    ylabel("Imaginary part");
+    set(gca, 'XAxisLocation', 'origin'); % Set properties of current axis 
+    
+    % find all zeros within the bracket using the given algorithm
+    [~, steps] = algorithm(printComplexGraph, rootBrackets(1), rootBrackets(2), 1e-15);
+    
+    % plot first step
+    text(real(steps(1, 1)), imag(steps(1, 1)), 'start', 'HorizontalAlignment', 'center', 'VerticalAlignment', 'top');
+
+    % plot steps on graph
+    plot(real(steps(1, :)), imag(steps(1, :)), '-x');
+    
+    % plot last step
+    text(real(steps(1, end)), imag(steps(1, end)), 'end', 'HorizontalAlignment', 'center', 'VerticalAlignment', 'top');
+
+    
+    % print root table
+    disp([plottitle, ' (', algorithmName, ')']);
+    columns = {'step', 'root', 'abs value at root'};
+    disp(table([1:size(steps, 2)]', steps(1, :)', abs(steps(2, :))', 'VariableNames', columns));
+end

ENUME/projectB/Code/printGraph.m

@@ -0,0 +1,34 @@
+% graph the real roots of a function
+function printGraph(taskFunction, algorithmName, algorithm, interval, rootBrackets, plotTitle)
+    figure()
+    grid on; % Get y values lines
+    hold on; % Retain current plot when adding new plots
+    title([plotTitle, algorithmName]);
+    set(gca, 'XAxisLocation', 'origin'); % Set properties of current axis 
+    x = interval(1):0.01:interval(2); 
+    % x is a vector of values between left and right interval with every value being higher by 0.01
+    y = arrayfun(taskFunction, x); % We sketch the function from task for each x
+    plot(x, y);
+    
+    % iterate over rootBrackets and add them to the plot
+    for rootBracket = rootBrackets   
+        % find all zeros within the bracket using the given algorithm
+        % Get all steps from the algorithm we use
+        [~, steps] = algorithm(taskFunction, rootBracket(1), rootBracket(2), 1e-10);
+        
+        firstStepColor = [1 0 0]; % Red
+        otherStepsColor = 	[0 1 0]; % Green
+        % plot first steps
+        scatter(steps(1, 1), steps(2, 1), [], firstStepColor); 
+        text(steps(1, 1), steps(2, 1), 'firstStep', 'HorizontalAlignment', 'center', 'VerticalAlignment', 'top'); % It makes text appear neatly
+        % plot other steps
+        scatter(steps(1, 2:end), steps(2, 2:end), [], otherStepsColor);
+
+
+            
+    % print root table
+    disp([plotTitle, ' (', algorithmName, ')']);
+    columns = {'step', 'root', 'value at root'};
+    disp(table((1:size(steps, 2))', steps(1, :)', steps(2, :)', 'VariableNames', columns));
+    end
+end

ENUME/projectB/Code/rootBracketing.m

@@ -0,0 +1,34 @@
+% find the root brackets of a function within the given range
+function rootBrackets = rootBracketing(givenFunction, intervalLeft, intervalRight)
+    [a, b, rootBrackets, resolution] = initializeValues(intervalLeft, intervalRight);
+    rootBrackets = bracketingLoop(a, b, rootBrackets, intervalRight, resolution, givenFunction);
+end
+
+function [a, b, rootBrackets, resolution]  = initializeValues(intervalLeft, intervalRight)
+    % define search resolution
+    resolution = (intervalRight - intervalLeft) / 6;
+    % The higher the value of denominator the less iterations will it take
+    % to reach the roots, however in order to have nice graph showing those
+    % brackets I will choose relatively small denominator - I have choosen
+    % the smallest natural number that still generates brackets on a graph
+    
+    % start search at the start of the range
+    a = intervalLeft;
+    b = intervalLeft + resolution;
+    rootBrackets = double.empty(2, 0); % initialize empty vector of size 2
+end
+
+function rootBrackets = bracketingLoop(a, b, rootBrackets, intervalRight, resolution, givenFunction)
+    while b ~= intervalRight % if the bracket can't be expanded end loop
+        % if the function changes sign inside the interval that means that we passed through a root that means that a bracket has been found
+        if sign(givenFunction(a)) ~= sign(givenFunction(b))
+            % save bracket
+            rootBrackets(:, size(rootBrackets, 2) + 1) = [a, b]; % Add the new bracket to existing ones
+        end
+        % check next bracket
+        a = b;
+        b = min(a + resolution, intervalRight); 
+        % Once a + resolution > intervalRight, then we will know that we
+        % reached beyond the interval and we must stop
+    end
+end

ENUME/projectB/Code/task1Bisection.m

@@ -0,0 +1,73 @@
+interval = [-5, 10];
+rootBrackets = rootBracketing(@taskFunction, interval(1), interval(2));
+
+printGraph(@taskFunction, 'False Position', @falsePosition, interval, rootBrackets, 'Approximate zeros of function for method of ');
+    
+function y = taskFunction(x)
+    y = -2.1 + 0.3*x - x*exp(1)^(-x);
+end
+
+function [zero, iterations] = falsePosition(taskFunction, a, b, tolerance)
+
+    [iterations, lastTwoA, lastTwoB, i] = initialize();
+    [zero, iterations, a, b, lastTwoA, lastTwoB] = firstTwoIterations(a, b, taskFunction, iterations, lastTwoA, lastTwoB);
+    [zero, iterations] = falsePositionLoop(taskFunction, zero, tolerance, lastTwoA, lastTwoB, i, a, b, iterations);
+end
+
+function [iterations, lastTwoA, lastTwoB, i] = initialize()
+    iterations = double.empty(2, 0);
+    lastTwoA = double.empty(2, 0);
+    lastTwoB = double.empty(2, 0);
+    i = 0;
+end
+
+function [zero, iterations, a, b, lastTwoA, lastTwoB] = firstTwoIterations(a, b, taskFunction, iterations, lastTwoA, lastTwoB)
+    for j = 1 : 2
+        zero = (a*taskFunction(b) - b * taskFunction(a)) / (taskFunction(b) - taskFunction(a));
+        iterations(:, size(iterations, 2) + 1) = [zero, taskFunction(zero)];
+        if sign(taskFunction(a)) ~= sign(taskFunction(zero))
+            b = zero;
+        else
+            a = zero;
+        end
+        lastTwoA(j) = a;
+        lastTwoB(j) = b;
+    end
+end
+
+function [zero, iterations] = falsePositionLoop(taskFunction, zero, tolerance, lastTwoA, lastTwoB, i, a, b, iterations)
+    while abs(taskFunction(zero)) > tolerance
+        [lastTwoA, i, a, lastTwoB, b, tolerance, zero, iterations] = insideLoop(lastTwoA, i, a, lastTwoB, b, tolerance, taskFunction, iterations);
+    end
+end
+
+function [lastTwoA, i, a, lastTwoB, b, tolerance, zero, iterations] = insideLoop(lastTwoA, i, a, lastTwoB, b, tolerance, taskFunction, iterations)
+    [lastTwoA, lastTwoB] = changeLastTwoAB(lastTwoA, lastTwoB, i, a, b);
+    zero = calculateZero(lastTwoB, tolerance, a, b, lastTwoA, taskFunction);
+    iterations(:, size(iterations, 2) + 1) = [zero, taskFunction(zero)];
+    [a, b] = newSubInterval(taskFunction, a, b, zero);
+end
+
+function [lastTwoA, lastTwoB] = changeLastTwoAB(lastTwoA, lastTwoB, i, a, b)
+    lastTwoA(mod(i, 2) + 1) = a;
+    lastTwoB(mod(i, 2) + 1) = b;
+end
+
+function [zero] = calculateZero(lastTwoB, tolerance, a, b, lastTwoA, taskFunction)
+    if(abs(lastTwoB(1) - lastTwoB(2)) < tolerance) 
+        zero = (a*(taskFunction(b) / 2) - b * taskFunction(a)) / (taskFunction(b) / 2 - taskFunction(a));
+    elseif (abs(lastTwoA(1) - lastTwoA(2)) < tolerance) 
+        zero = (a*taskFunction(b) - b * (taskFunction(a) / 2)) / (taskFunction(b) - (taskFunction(a) / 2));
+    else
+        zero = (a*taskFunction(b) - b * taskFunction(a)) / (taskFunction(b) - taskFunction(a));
+    end
+end
+
+function [a, b] = newSubInterval(taskFunction, a, b, zero)
+    if sign(taskFunction(a)) ~= sign(taskFunction(zero))
+        b = zero;
+    else        
+        a = zero;
+    end
+end
+

ENUME/projectB/Code/task1Newton.m

@@ -0,0 +1,48 @@
+interval = [-5, 10];
+rootBrackets = rootBracketing(@taskFunction, interval(1), interval(2));
+
+% printGraph(@taskFunction, 'Newton', @newtonMethod, interval, rootBrackets, 'Approximate zeros of function for method of ');
+printGraph(@polynomial, 'Newton', @newtonMethod, interval, rootBrackets, 'Approximate zeros of function for method of ');    
+
+function y = taskFunction(x)
+    y = -2.1 + 0.3*x - x*exp(1)^(-x);
+end
+
+function y = polynomial(x)
+    y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
+end
+
+function [zero, iterations] = newtonMethod(taskFunction, a, b, tolerance)
+    [iterations, iteration, zero] = initialize(a, b);
+    [zero, iterations] = newtonLoop(iterations, iteration, zero, a, b, tolerance, taskFunction);
+end
+
+function [iterations, step, zero] = initialize(a, b)
+    iterations = double.empty(2, 0);
+    step = sqrt(eps);
+    zero = (a + b) / 2;
+    iterations(:, size(iterations, 2) + 1) = [zero, taskFunction(zero)];
+end
+
+function [zero, iterations] = newtonLoop(iterations, iteration, zero, a, b, tolerance, taskFunction)
+    while abs(taskFunction(zero)) > tolerance
+        [zero, iterations] = insideLoop(taskFunction, zero, iteration, iterations, a, b);
+    end
+end
+
+function [zero, iterations] = insideLoop(taskFunction, zero, iteration, iterations, a, b)
+    [zero, iterations] = calculateZeroIterations(taskFunction, zero, iteration, iterations);
+    checkForDivergence(zero, a, b);
+end
+
+function [zero, iterations] = calculateZeroIterations(taskFunction, zero, iteration, iterations)
+    derivative = (taskFunction(zero + iteration) - taskFunction(zero - iteration)) / (2 * iteration);
+    zero = zero - taskFunction(zero) / derivative;
+    iterations(:, size(iterations, 2) + 1) = [zero, taskFunction(zero)];
+end
+
+function checkForDivergence(zero, a, b)
+    if zero < a || zero > b
+        error('Divergent iteration');
+    end
+end

ENUME/projectB/Code/task2MM1.m

@@ -0,0 +1,109 @@
+interval = [-5, 10];
+rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
+
+printGraph(@polynomial, 'MM1', @mm1, interval, rootBrackets, 'Approximate zeros of function for method of ');
+     
+printComplexGraph(@polynomial, 'MM1', @mm1, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
+
+function y = polynomial(x)
+    y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
+end
+
+function [approximation, iterations] = mm1(polynomial, a, b, tolerance)
+    [approximation, approximationValue, iterations] = initialize(a, b, polynomial);
+    [approximation, iterations] = mm1Loop(approximation, tolerance, approximationValue, iterations, polynomial);
+end
+
+function [approximation, approximationValue, iterations] = initialize(a, b, polynomial)
+    approximation = [a, b, (a + b) / 2];
+    approximationValue = arrayfun(polynomial, approximation);
+    iterations = [approximation(3); polynomial(approximation(3))];
+end
+
+function [approximation, iterations] = mm1Loop(approximation, tolerance, approximationValue, iterations, polynomial)
+    while abs(polynomial(approximation(3))) > tolerance
+        [approximation, approximationValue, iterations] = insideLoop(approximation, approximationValue, polynomial, iterations);
+    end
+end
+
+function [approximation, approximationValue, iterations] = insideLoop(approximation, approximationValue, polynomial, iterations)
+    equationsSystem = createEquationSystem(approximation, approximationValue);
+    [zPlus, zMinus] = rootsOfQuadraticFormula(equationsSystem, approximationValue);
+    [approximation, approximationValue, iterations] = updateApproximations(zPlus, zMinus, approximation, iterations, polynomial);
+end
+
+function equationsSystem = createEquationSystem(approximation, approximationValue)
+    [z0, z1, difference0, difference1] = initializeEquationSystem(approximation, approximationValue);
+    equationsSystem = solveEquationSystem(z0, difference0, z1, difference1);
+end
+
+function [zPlus, zMinus] = rootsOfQuadraticFormula(equationsSystem, approximationValue)
+    [a, b, c] = createApproximatedQuadraticFormula(equationsSystem, approximationValue);
+    [zPlus, zMinus] = findRootsOfQuadraticFormula(a, b, c);
+end
+
+function [approximation, approximationValue, iterations] = updateApproximations(zPlus, zMinus, approximation, iterations, polynomial)
+    newApproximation = chooseNewRoot(zPlus, zMinus, approximation);
+    iterations = addZeroToIterationVector(newApproximation, iterations, polynomial);
+    worstApproximationIndex = getWorstApproximationIndex(approximation, newApproximation);
+    [approximation, approximationValue] = deleteWorstApproximation(worstApproximationIndex, approximation, polynomial, newApproximation);
+end
+
+function [z0, z1, difference0, difference1] = initializeEquationSystem(approximation, approximationValue)
+    z0 = approximation(1) - approximation(3);
+    z1 = approximation(2) - approximation(3);
+    difference0 = approximationValue(1) - approximationValue(3);
+    difference1 = approximationValue(2) - approximationValue(3);
+end
+
+function equationsSystem = solveEquationSystem(z0, difference0, z1, difference1)
+    equationsSystem = [z0 ^ 2, z0, difference0; z1 ^ 2, z1, difference1];
+    reductor = equationsSystem(2, 1) / equationsSystem(1, 1);
+    equationsSystem(2, :) = equationsSystem(2, :) - reductor * equationsSystem(1, :);
+    equationsSystem(2, 1) = 0;
+    equationsSystem(2, :) = equationsSystem(2, :) ./ equationsSystem(2, 2);
+    equationsSystem(1, :) = equationsSystem(1, :) - equationsSystem(1, 2) * equationsSystem(2, :);
+    equationsSystem(1, :) = equationsSystem(1, :) ./ equationsSystem(1, 1);
+end
+
+function [a, b, c] = createApproximatedQuadraticFormula(equationsSystem, approximationValue)
+    a = equationsSystem(1, 3);
+    b = equationsSystem(2, 3);
+    c = approximationValue(3);
+end
+
+function [zPlus, zMinus] = findRootsOfQuadraticFormula(a, b, c)
+    zPlus = -2 * c / (b + sqrt(b ^ 2 - 4 * a * c));
+    zMinus = -2 * c / (b - sqrt(b ^ 2 - 4 * a * c));
+end
+
+function newApproximation = chooseNewRoot(zPlus, zMinus, approximation)
+    if abs(zPlus) < abs(zMinus)
+        newApproximation = approximation(3) + zPlus;
+    else
+        newApproximation = approximation(3) + zMinus;
+    end
+end
+
+function iterations = addZeroToIterationVector(newApproximation, iterations, polynomial)
+    zero = newApproximation;
+    iterations(:, size(iterations, 2) + 1) = [zero, polynomial(zero)];
+end
+
+function worstApproximationIndex = getWorstApproximationIndex(approximation, newApproximation)
+    worstApproximationIndex = -1;
+    worstApproximationDifference = 0;
+    for i = 1:size(approximation, 2)
+        diff = abs(approximation(i) - newApproximation);
+        if diff > worstApproximationDifference
+            worstApproximationIndex = i;
+            worstApproximationDifference = diff;
+        end
+    end
+end
+
+function [approximation, approximationValue] = deleteWorstApproximation(worstApproximationIndex, approximation, polynomial, newApproximation)
+    approximation(worstApproximationIndex) = [];
+    approximation(3) = newApproximation;
+    approximationValue = arrayfun(polynomial, approximation);
+end

ENUME/projectB/Code/task2MM2.m

@@ -0,0 +1,70 @@
+interval = [-5, 10];
+rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
+
+printGraph(@polynomial, 'MM2', @mm2, interval, rootBrackets, 'Approximate zeros of function for method of ');
+    
+printComplexGraph(@polynomial, 'MM2', @mm2, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
+
+% the polynomial function for task 2
+function y = polynomial(x)
+    y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
+end
+
+% find roots of polynomial using MM2
+function [approximation, iterations] = mm2(polynomial, a, b, tolerance)
+    [approximation, iterations] = initialize(a, b, polynomial);
+    [approximation, iterations] = mm2Loop(approximation, iterations, polynomial, tolerance);
+end
+
+function [approximation, iterations] = initialize(a, b, polynomial)
+    approximation = (a + b) / 2;
+    iterations = [approximation; polynomial(approximation)];
+end
+
+function [approximation, iterations] = mm2Loop(approximation, iterations, polynomial, tolerance)
+    while abs(polynomial(approximation)) > tolerance
+       [approximation, iterations] = insideLoop(approximation, polynomial, iterations);
+    end
+end
+
+function [approximation, iterations] = insideLoop(approximation, polynomial, iterations)
+        [a, b, c] = getABC(approximation, polynomial);
+        [zPlus, zMinus] = findRoots(a, b, c);
+        newApproximation = chooseNewApproximation(zPlus, zMinus, approximation);
+        [approximation, iterations] = updateApproximations(newApproximation, iterations, polynomial);
+end
+
+function [a, b, c] = getABC(approximation, polynomial)
+    c = polynomial(approximation);
+    b = derivative(polynomial, approximation, 1);
+    a = derivative(polynomial, approximation, 2) / 2;
+end
+
+function [zPlus, zMinus] = findRoots(a, b, c)
+    zPlus = -2 * c / (b + sqrt(b ^ 2 - 4 * a * c));
+    zMinus = -2 * c / (b - sqrt(b ^ 2 - 4 * a * c));
+end
+
+function newApproximation = chooseNewApproximation(zPlus, zMinus, approximation)
+    if abs(zPlus) < abs(zMinus)
+        newApproximation = approximation + zPlus;
+    else
+        newApproximation = approximation + zMinus;
+    end
+end
+
+function [approximation, iterations] = updateApproximations(newApproximation, iterations, polynomial)
+    approximation = newApproximation;
+    iterations(:, size(iterations, 2) + 1) = [approximation, polynomial(approximation)];
+end
+
+% calculate the nth derivative of func at x
+function y = derivative(function_, x, degree)
+    if degree == 0
+        y = function_(x);
+        return
+    end
+
+    step = sqrt(eps);
+    y = (derivative(function_, x + step, degree - 1) - derivative(function_, x - step, degree - 1)) / (2 * step);
+end

ENUME/projectB/Code/task3.m

@@ -0,0 +1,72 @@
+interval = [-5, 10];
+rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
+
+printGraph(@polynomial, 'Laguerre', @laguerre, interval, rootBrackets, 'Approximate zeros of function for method of ');
+     
+printComplexGraph(@polynomial, 'Laguerre', @laguerre, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
+
+function y = polynomial(x)
+    y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
+end
+
+function [zero, iterations] = laguerre(polynomial, a, b, tolerance)
+    [degree, zero, iterations] = initialize(a, b, polynomial);
+    [zero, iterations] = laguerreLoop(polynomial, zero, tolerance, iterations, degree);
+
+end
+
+function [degree, zero, iterations] = initialize(a, b, polynomial)
+    degree = 4;
+    zero = (a + b) / 2;
+    iterations = [zero; polynomial(zero)];
+end
+
+function [zero, iterations] = laguerreLoop(polynomial, zero, tolerance, iterations, degree)
+    while abs(polynomial(zero)) > tolerance
+        [iterations, zero] = insideLoop(polynomial, zero, degree, iterations);
+    end
+end
+
+function [iterations, zero] = insideLoop(polynomial, zero, degree, iterations)
+    [derrivative0, derrivative1, derrivative2] = calculateDerrivatives(polynomial, zero);
+    [zPlus, zMinus] = calculateZ(degree, derrivative0, derrivative1, derrivative2);
+    newZero = chooseNewZero(zPlus, zMinus, zero);
+    [zero, iterations] = updateZeros(newZero, iterations, polynomial);
+end
+
+function [derrivative0, derrivative1, derrivative2] = calculateDerrivatives(polynomial, zero)
+    derrivative0 = polynomial(zero);
+    derrivative1 = derivative(polynomial, zero, 1);
+    derrivative2 = derivative(polynomial, zero, 2);
+end
+
+function [zPlus, zMinus] = calculateZ(degree, derrivative0, derrivative1, derrivative2)
+    expressionUnderSquareRoot = (degree - 1) * ((degree - 1) * derrivative1 ^ 2 - degree * derrivative0 * derrivative2);
+    lagsqrt = sqrt(expressionUnderSquareRoot);
+
+    zPlus = degree * derrivative0 / (derrivative1 + lagsqrt);
+    zMinus = degree * derrivative0 / (derrivative1 - lagsqrt);
+end
+
+function newZero = chooseNewZero(zPlus, zMinus, zero)    
+    if abs(zPlus) < abs(zMinus)
+        newZero = zero - zPlus;
+    else
+        newZero = zero - zMinus;
+    end
+end
+
+function [zero, iterations] = updateZeros(newZero, iterations, polynomial)
+    zero = newZero;
+    iterations(:, size(iterations, 2) + 1) = [zero, polynomial(zero)];
+end
+
+function y = derivative(function_, x, degree)
+    if degree == 0
+        y = function_(x);
+        return
+    end
+
+    step = sqrt(eps);
+    y = (derivative(function_, x + step, degree - 1) - derivative(function_, x - step, degree - 1)) / (2 * step);
+end

ENUME/projectB/Report/task1Newtonzommedleft.eps

@@ -0,0 +1,998 @@
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