\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, 1 + i], '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}

	\input{sections/11-task-3/01-rootbrackering-m}
	\input{sections/11-task-3/02-printgraph-m}
	\input{sections/11-task-3/03-printcomplexgraph-m}