\subsection{Gram-Schmid algorithm}

\begin{simplechar}
\begin{lstlisting}
% 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
\end{lstlisting}
\end{simplechar}

\subsubsection{initializeGramSchmid}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}

\subsubsection{factorizeColumnsOfQ}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}

\subsubsection{normalizeColumns}
\begin{simplechar}
\begin{lstlisting}
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
\end{lstlisting}
\end{simplechar}