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Adding code appendix

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PolishPigeon 2021-12-03 04:32:40 +01:00
parent 828a019165
commit e75b928c6e
12 changed files with 1307 additions and 171 deletions
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2
ENUME/projectB/Code/task1Bisection.m
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@ -12,7 +12,6 @@ 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()
@ -47,7 +46,6 @@ function [lastTwoA, i, a, lastTwoB, b, tolerance, zero, iterations] = insideLoop
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)

1
ENUME/projectB/Code/task2MM1.m
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@ -10,7 +10,6 @@ function y = polynomial(x)
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

72
ENUME/projectB/Code/task3.asv
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@ -1,72 +0,0 @@
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, 0], '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);
% update answer
zero = newzero;
iterations(:, size(iterations, 2) + 1) = [zero, polynomial(zero)];
end
function [derrivative0, derrivative1, derrivative2] = calculateDerrivatives(polynomial, zero)
% calculate derivatives
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 [] = chooseNewZero()
% choose value closer to current approximation
if abs(zPlus) < abs(zMinus)
newzero = zero - zPlus;
else
newzero = zero - zMinus;
end
% calculate the nth derivative of polynomial 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

122
ENUME/projectB/Report/report.aux
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@ -16,38 +16,108 @@
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23
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l.47 \chapter
{Find all zeros of function} [1
[2]
Chapter 1.
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Chapter 2.
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Chapter 3.
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Chapter 4.
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\BOOKMARK [1][-]{section.3.3}{Results}{chapter.3}% 21
\BOOKMARK [2][-]{subsection.3.3.1}{Comparison of results between MM1 and MM2}{section.3.3}% 22
\BOOKMARK [0][-]{chapter.4}{Code appendix}{}% 23
\BOOKMARK [1][-]{section.4.1}{Task 1}{chapter.4}% 24
\BOOKMARK [2][-]{subsection.4.1.1}{task1Bisection.m}{section.4.1}% 25
\BOOKMARK [2][-]{subsection.4.1.2}{task1Newton.m}{section.4.1}% 26
\BOOKMARK [1][-]{section.4.2}{Task 2}{chapter.4}% 27
\BOOKMARK [2][-]{subsection.4.2.1}{task2MM1.m}{section.4.2}% 28
\BOOKMARK [2][-]{subsection.4.2.2}{task2MM2.m}{section.4.2}% 29
\BOOKMARK [1][-]{section.4.3}{Task 3}{chapter.4}% 30
\BOOKMARK [2][-]{subsection.4.3.1}{rootBrackering.m}{section.4.3}% 31
\BOOKMARK [2][-]{subsection.4.3.2}{printGraph.m}{section.4.3}% 32
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@ -214,6 +214,816 @@ This formula is similar to the one from MM2 but also takes order of the polynomi
\chapter{Code appendix}
\section{Task 1}
\subsection{task1Bisection.m}
\subsubsection{Top of task1Bisection.m}
\begin{simplechar}
\begin{lstlisting}
interval = [-5, 10];
rootBrackets = rootBracketing(@taskFunction, interval(1), interval(2));
printGraph(@taskFunction, 'False Position', @falsePosition, interval, rootBrackets, 'Approximate zeros of function for method of ');
\end{lstlisting}
\end{simplechar}
\subsubsection{taskFunction}
\begin{simplechar}
\begin{lstlisting}
function y = taskFunction(x)
y = -2.1 + 0.3*x - x*exp(1)^(-x);
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{falsePosition}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{initialize}
\begin{simplechar}
\begin{lstlisting}
function [iterations, lastTwoA, lastTwoB, i] = initialize()
iterations = double.empty(2, 0);
lastTwoA = double.empty(2, 0);
lastTwoB = double.empty(2, 0);
i = 0;
end
\end{lstlisting}
\end{simplechar}
\subsubsection{taskFunction}
\begin{simplechar}
\begin{lstlisting}
function y = taskFunction(x)
y = -2.1 + 0.3*x - x*exp(1)^(-x);
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{firstTwoIterations}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{falsePositionLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{insideLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{changeLastTwoAB}
\begin{simplechar}
\begin{lstlisting}
function [lastTwoA, lastTwoB] = changeLastTwoAB(lastTwoA, lastTwoB, i, a, b)
lastTwoA(mod(i, 2) + 1) = a;
lastTwoB(mod(i, 2) + 1) = b;
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{calculateZero}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{newSubInterval}
\begin{simplechar}
\begin{lstlisting}
function [a, b] = newSubInterval(taskFunction, a, b, zero)
if sign(taskFunction(a)) ~= sign(taskFunction(zero))
b = zero;
else
a = zero;
end
end
\end{lstlisting}
\end{simplechar}
\subsection{task1Newton.m}
\subsubsection{Top of task1Newton.m}
\begin{simplechar}
\begin{lstlisting}
interval = [-5, 10];
rootBrackets = rootBracketing(@taskFunction, interval(1), interval(2));
printGraph(@taskFunction, 'Newton', @newtonMethod, interval, rootBrackets, 'Approximate zeros of function for method of ');
\end{lstlisting}
\end{simplechar}
\subsubsection{taskFunction}
\begin{simplechar}
\begin{lstlisting}
function y = taskFunction(x)
y = -2.1 + 0.3*x - x*exp(1)^(-x);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{newtonMethod}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{initialize}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{newtonLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{insideLoop}
\begin{simplechar}
\begin{lstlisting}
function [zero, iterations] = insideLoop(taskFunction, zero, iteration, iterations, a, b)
[zero, iterations] = calculateZeroIterations(taskFunction, zero, iteration, iterations);
checkForDivergence(zero, a, b);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{calculateZeroIterations}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{checkForDivergence}
\begin{simplechar}
\begin{lstlisting}
function checkForDivergence(zero, a, b)
if zero < a || zero > b
error('Divergent iteration');
end
end
\end{lstlisting}
\end{simplechar}
\newpage
\section{Task 2}
\subsection{task2MM1.m}
\subsubsection{Top of task2MM1}
\begin{simplechar}
\begin{lstlisting}
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, 0], 'Aproximate complex roots of polynomial');
\end{lstlisting}
\end{simplechar}
\subsubsection{polynomial}
\begin{simplechar}
\begin{lstlisting}
function y = polynomial(x)
y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
end
\end{lstlisting}
\end{simplechar}
\subsubsection{mm1}
\begin{simplechar}
\begin{lstlisting}
function [approximation, iterations] = mm1(polynomial, a, b, tolerance)
[approximation, approximationValue, iterations] = initialize(a, b, polynomial);
[approximation, iterations] = mm1Loop(approximation, tolerance, approximationValue, iterations, polynomial);
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{initialize}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{mm1Loop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{insideLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{createEquationSystem}
\begin{simplechar}
\begin{lstlisting}
function equationsSystem = createEquationSystem(approximation, approximationValue)
[z0, z1, difference0, difference1] = initializeEquationSystem(approximation, approximationValue);
equationsSystem = solveEquationSystem(z0, difference0, z1, difference1);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{rootsOfQuadraticFormula}
\begin{simplechar}
\begin{lstlisting}
function [zPlus, zMinus] = rootsOfQuadraticFormula(equationsSystem, approximationValue)
[a, b, c] = createApproximatedQuadraticFormula(equationsSystem, approximationValue);
[zPlus, zMinus] = findRootsOfQuadraticFormula(a, b, c);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{updateApproximations}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{initializeEquationSystem}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{solveEquationSystem}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{createApproximatedQuadraticFormula}
\begin{simplechar}
\begin{lstlisting}
function [a, b, c] = createApproximatedQuadraticFormula(equationsSystem, approximationValue)
a = equationsSystem(1, 3);
b = equationsSystem(2, 3);
c = approximationValue(3);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{findRootsOfQuadraticFormula}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{chooseNewRoot}
\begin{simplechar}
\begin{lstlisting}
function newApproximation = chooseNewRoot(zPlus, zMinus, approximation)
if abs(zPlus) < abs(zMinus)
newApproximation = approximation(3) + zPlus;
else
newApproximation = approximation(3) + zMinus;
end
end
\end{lstlisting}
\end{simplechar}
\subsubsection{addZeroToIterationVector}
\begin{simplechar}
\begin{lstlisting}
function iterations = addZeroToIterationVector(newApproximation, iterations, polynomial)
zero = newApproximation;
iterations(:, size(iterations, 2) + 1) = [zero, polynomial(zero)];
end
\end{lstlisting}
\end{simplechar}
\subsubsection{getWorstApproximationIndex}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{deleteWorstApproximation}
\begin{simplechar}
\begin{lstlisting}
function [approximation, approximationValue] = deleteWorstApproximation(worstApproximationIndex, approximation, polynomial, newApproximation)
approximation(worstApproximationIndex) = [];
approximation(3) = newApproximation;
approximationValue = arrayfun(polynomial, approximation);
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsection{task2MM2.m}
\subsubsection{top of task2MM2.m}
\begin{simplechar}
\begin{lstlisting}
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, 0], 'Aproximate complex roots of polynomial');
\end{lstlisting}
\end{simplechar}
\subsubsection{polynomial}
\begin{simplechar}
\begin{lstlisting}
function y = polynomial(x)
y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
end
\end{lstlisting}
\end{simplechar}
\subsubsection{mm2}
\begin{simplechar}
\begin{lstlisting}
function [approximation, iterations] = mm2(polynomial, a, b, tolerance)
[approximation, iterations] = initialize(a, b, polynomial);
[approximation, iterations] = mm2Loop(approximation, iterations, polynomial, tolerance);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{initialize}
\begin{simplechar}
\begin{lstlisting}
function [approximation, iterations] = initialize(a, b, polynomial)
approximation = (a + b) / 2;
iterations = [approximation; polynomial(approximation)];
end
\end{lstlisting}
\end{simplechar}
\subsubsection{mm2Loop}
\begin{simplechar}
\begin{lstlisting}
function [approximation, iterations] = mm2Loop(approximation, iterations, polynomial, tolerance)
while abs(polynomial(approximation)) > tolerance
[approximation, iterations] = insideLoop(approximation, polynomial, iterations);
end
end
\end{lstlisting}
\end{simplechar}
\subsubsection{insideLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{getABC}
\begin{simplechar}
\begin{lstlisting}
function [a, b, c] = getABC(approximation, polynomial)
c = polynomial(approximation);
b = derivative(polynomial, approximation, 1);
a = derivative(polynomial, approximation, 2) / 2;
end
\end{lstlisting}
\end{simplechar}
\subsubsection{findRoots}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{chooseNewApproximation}
\begin{simplechar}
\begin{lstlisting}
function newApproximation = chooseNewApproximation(zPlus, zMinus, approximation)
if abs(zPlus) < abs(zMinus)
newApproximation = approximation + zPlus;
else
newApproximation = approximation + zMinus;
end
end
\end{lstlisting}
\end{simplechar}
\subsubsection{updateApproximations}
\begin{simplechar}
\begin{lstlisting}
function [approximation, iterations] = updateApproximations(newApproximation, iterations, polynomial)
approximation = newApproximation;
iterations(:, size(iterations, 2) + 1) = [approximation, polynomial(approximation)];
end
\end{lstlisting}
\end{simplechar}
\subsubsection{derivative}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\section{Task 3}
\subsubsection{Top of Task 3}
\begin{simplechar}
\begin{lstlisting}
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, 0], 'Aproximate complex roots of polynomial');
\end{lstlisting}
\end{simplechar}
\subsubsection{polynomial}
\begin{simplechar}
\begin{lstlisting}
function y = polynomial(x)
y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
end
\end{lstlisting}
\end{simplechar}
\subsubsection{laguerre}
\begin{simplechar}
\begin{lstlisting}
function [zero, iterations] = laguerre(polynomial, a, b, tolerance)
[degree, zero, iterations] = initialize(a, b, polynomial);
[zero, iterations] = laguerreLoop(polynomial, zero, tolerance, iterations, degree);
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{initialize}
\begin{simplechar}
\begin{lstlisting}
function [degree, zero, iterations] = initialize(a, b, polynomial)
degree = 4;
zero = (a + b) / 2;
iterations = [zero; polynomial(zero)];
end
\end{lstlisting}
\end{simplechar}
\subsubsection{laguerreLoop}
\begin{simplechar}
\begin{lstlisting}
function [zero, iterations] = laguerreLoop(polynomial, zero, tolerance, iterations, degree)
while abs(polynomial(zero)) > tolerance
[iterations, zero] = insideLoop(polynomial, zero, degree, iterations);
end
end
\end{lstlisting}
\end{simplechar}
\subsubsection{insideLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{calculateDerrivatives}
\begin{simplechar}
\begin{lstlisting}
function [derrivative0, derrivative1, derrivative2] = calculateDerrivatives(polynomial, zero)
derrivative0 = polynomial(zero);
derrivative1 = derivative(polynomial, zero, 1);
derrivative2 = derivative(polynomial, zero, 2);
end
\end{lstlisting}
\end{simplechar}
\subsubsection{calculateZ}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\subsubsection{chooseNewZero}
\begin{simplechar}
\begin{lstlisting}
function newZero = chooseNewZero(zPlus, zMinus, zero)
if abs(zPlus) < abs(zMinus)
newZero = zero - zPlus;
else
newZero = zero - zMinus;
end
end
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{updateZeros}
\begin{simplechar}
\begin{lstlisting}
function [zero, iterations] = updateZeros(newZero, iterations, polynomial)
zero = newZero;
iterations(:, size(iterations, 2) + 1) = [zero, polynomial(zero)];
end
\end{lstlisting}
\end{simplechar}
\subsection{rootBrackering.m}
\subsubsection{updateZeros}
\begin{simplechar}
\begin{lstlisting}
% 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
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{initializeValues}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\newpage
\subsubsection{bracketingLoop}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}
\newpage
\subsection{printGraph.m}
\begin{simplechar}
\begin{lstlisting}
% 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
\end{lstlisting}
\end{simplechar}
\newpage
\subsection{printComplexGraph.m}
\begin{simplechar}
\begin{lstlisting}
% 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
\end{lstlisting}
\end{simplechar}
\begin{thebibliography}{9}
\bibitem{texbook}
Piotr Tatjewski (2014) \emph{Numerical Methods}, Oficyna Wydawnicza Politechniki Warszawskiej

120
ENUME/projectB/Report/report.toc
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@ -1,25 +1,95 @@
\contentsline {chapter}{\numberline {1}Find all zeros of function}{2}{chapter.1}%
\contentsline {section}{\numberline {1.1}a) False position method}{2}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Problem}{2}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Theoretical Introduction}{2}{subsection.1.1.2}%
\contentsline {subsubsection}{Properties of \emph {false position method}}{3}{section*.2}%
\contentsline {paragraph}{Improvement to the method}{3}{section*.3}%
\contentsline {subsection}{\numberline {1.1.3}Results}{4}{subsection.1.1.3}%
\contentsline {section}{\numberline {1.2}b) the Newton's method}{4}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Problem}{4}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Theoretical Introduction}{4}{subsection.1.2.2}%
\contentsline {subsection}{\numberline {1.2.3}Results}{4}{subsection.1.2.3}%
\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{5}{chapter.2}%
\contentsline {section}{\numberline {2.1}Problem}{5}{section.2.1}%
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{5}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}MM1}{6}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}MM2}{7}{subsection.2.2.2}%
\contentsline {section}{\numberline {2.3}Results}{8}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Comparison of results between MM1 and MM2}{8}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Comparison of results between Newton's method and MM2}{8}{subsection.2.3.2}%
\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{9}{chapter.3}%
\contentsline {section}{\numberline {3.1}Problem}{9}{section.3.1}%
\contentsline {section}{\numberline {3.2}Theoretical Introduction}{9}{section.3.2}%
\contentsline {section}{\numberline {3.3}Results}{10}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{10}{subsection.3.3.1}%
\contentsline {chapter}{\numberline {4}Code appendix}{11}{chapter.4}%
\contentsline {chapter}{\numberline {1}Find all zeros of function}{3}{chapter.1}%
\contentsline {section}{\numberline {1.1}a) False position method}{3}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Problem}{3}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Theoretical Introduction}{3}{subsection.1.1.2}%
\contentsline {subsubsection}{Properties of \emph {false position method}}{4}{section*.2}%
\contentsline {paragraph}{Improvement to the method}{4}{section*.3}%
\contentsline {subsection}{\numberline {1.1.3}Results}{5}{subsection.1.1.3}%
\contentsline {section}{\numberline {1.2}b) the Newton's method}{5}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Problem}{5}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Theoretical Introduction}{5}{subsection.1.2.2}%
\contentsline {subsection}{\numberline {1.2.3}Results}{5}{subsection.1.2.3}%
\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{6}{chapter.2}%
\contentsline {section}{\numberline {2.1}Problem}{6}{section.2.1}%
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{6}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}MM1}{7}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}MM2}{8}{subsection.2.2.2}%
\contentsline {section}{\numberline {2.3}Results}{9}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Comparison of results between MM1 and MM2}{9}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Comparison of results between Newton's method and MM2}{9}{subsection.2.3.2}%
\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{10}{chapter.3}%
\contentsline {section}{\numberline {3.1}Problem}{10}{section.3.1}%
\contentsline {section}{\numberline {3.2}Theoretical Introduction}{10}{section.3.2}%
\contentsline {section}{\numberline {3.3}Results}{11}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{11}{subsection.3.3.1}%
\contentsline {chapter}{\numberline {4}Code appendix}{12}{chapter.4}%
\contentsline {section}{\numberline {4.1}Task 1}{12}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}task1Bisection.m}{12}{subsection.4.1.1}%
\contentsline {subsubsection}{Top of task1Bisection.m}{12}{section*.4}%
\contentsline {subsubsection}{taskFunction}{12}{section*.5}%
\contentsline {subsubsection}{falsePosition}{13}{section*.6}%
\contentsline {subsubsection}{initialize}{13}{section*.7}%
\contentsline {subsubsection}{taskFunction}{13}{section*.8}%
\contentsline {subsubsection}{firstTwoIterations}{14}{section*.9}%
\contentsline {subsubsection}{falsePositionLoop}{14}{section*.10}%
\contentsline {subsubsection}{insideLoop}{15}{section*.11}%
\contentsline {subsubsection}{changeLastTwoAB}{15}{section*.12}%
\contentsline {subsubsection}{calculateZero}{16}{section*.13}%
\contentsline {subsubsection}{newSubInterval}{16}{section*.14}%
\contentsline {subsection}{\numberline {4.1.2}task1Newton.m}{16}{subsection.4.1.2}%
\contentsline {subsubsection}{Top of task1Newton.m}{16}{section*.15}%
\contentsline {subsubsection}{taskFunction}{17}{section*.16}%
\contentsline {subsubsection}{newtonMethod}{17}{section*.17}%
\contentsline {subsubsection}{initialize}{17}{section*.18}%
\contentsline {subsubsection}{newtonLoop}{17}{section*.19}%
\contentsline {subsubsection}{insideLoop}{18}{section*.20}%
\contentsline {subsubsection}{calculateZeroIterations}{18}{section*.21}%
\contentsline {subsubsection}{checkForDivergence}{18}{section*.22}%
\contentsline {section}{\numberline {4.2}Task 2}{19}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}task2MM1.m}{19}{subsection.4.2.1}%
\contentsline {subsubsection}{Top of task2MM1}{19}{section*.23}%
\contentsline {subsubsection}{polynomial}{19}{section*.24}%
\contentsline {subsubsection}{mm1}{19}{section*.25}%
\contentsline {subsubsection}{initialize}{20}{section*.26}%
\contentsline {subsubsection}{mm1Loop}{20}{section*.27}%
\contentsline {subsubsection}{insideLoop}{20}{section*.28}%
\contentsline {subsubsection}{createEquationSystem}{21}{section*.29}%
\contentsline {subsubsection}{rootsOfQuadraticFormula}{21}{section*.30}%
\contentsline {subsubsection}{updateApproximations}{21}{section*.31}%
\contentsline {subsubsection}{initializeEquationSystem}{22}{section*.32}%
\contentsline {subsubsection}{solveEquationSystem}{22}{section*.33}%
\contentsline {subsubsection}{createApproximatedQuadraticFormula}{23}{section*.34}%
\contentsline {subsubsection}{findRootsOfQuadraticFormula}{23}{section*.35}%
\contentsline {subsubsection}{chooseNewRoot}{23}{section*.36}%
\contentsline {subsubsection}{addZeroToIterationVector}{23}{section*.37}%
\contentsline {subsubsection}{getWorstApproximationIndex}{24}{section*.38}%
\contentsline {subsubsection}{deleteWorstApproximation}{24}{section*.39}%
\contentsline {subsection}{\numberline {4.2.2}task2MM2.m}{25}{subsection.4.2.2}%
\contentsline {subsubsection}{top of task2MM2.m}{25}{section*.40}%
\contentsline {subsubsection}{polynomial}{25}{section*.41}%
\contentsline {subsubsection}{mm2}{25}{section*.42}%
\contentsline {subsubsection}{initialize}{25}{section*.43}%
\contentsline {subsubsection}{mm2Loop}{26}{section*.44}%
\contentsline {subsubsection}{insideLoop}{26}{section*.45}%
\contentsline {subsubsection}{getABC}{26}{section*.46}%
\contentsline {subsubsection}{findRoots}{26}{section*.47}%
\contentsline {subsubsection}{chooseNewApproximation}{27}{section*.48}%
\contentsline {subsubsection}{updateApproximations}{27}{section*.49}%
\contentsline {subsubsection}{derivative}{27}{section*.50}%
\contentsline {section}{\numberline {4.3}Task 3}{28}{section.4.3}%
\contentsline {subsubsection}{Top of Task 3}{28}{section*.51}%
\contentsline {subsubsection}{polynomial}{28}{section*.52}%
\contentsline {subsubsection}{laguerre}{28}{section*.53}%
\contentsline {subsubsection}{initialize}{29}{section*.54}%
\contentsline {subsubsection}{laguerreLoop}{29}{section*.55}%
\contentsline {subsubsection}{insideLoop}{29}{section*.56}%
\contentsline {subsubsection}{calculateDerrivatives}{30}{section*.57}%
\contentsline {subsubsection}{calculateZ}{30}{section*.58}%
\contentsline {subsubsection}{chooseNewZero}{30}{section*.59}%
\contentsline {subsubsection}{updateZeros}{31}{section*.60}%
\contentsline {subsection}{\numberline {4.3.1}rootBrackering.m}{31}{subsection.4.3.1}%
\contentsline {subsubsection}{updateZeros}{31}{section*.61}%
\contentsline {subsubsection}{initializeValues}{32}{section*.62}%
\contentsline {subsubsection}{bracketingLoop}{33}{section*.63}%
\contentsline {subsection}{\numberline {4.3.2}printGraph.m}{34}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}printComplexGraph.m}{36}{subsection.4.3.3}%