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\fancyhead[L]{\small Spring 2018 MATH-UA 252.001}
\fancyhead[C]{\small NYU Courant}
\fancyhead[R]{\small Prof. M. O'Neil}
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{\bf Homework 2}\\
\normalsize
Due: 12:30pm February 22, 2018
\end{center}
\normalsize
\vspace{.25in}
\horline
Notes on the assignment:
\textbf{Submission}: ({\color{red}New procedure.}) Homework assignments must be
submitted {\color{red} \textbf{via email}} to the following address:
\begin{center}
\texttt{hw2.rt0e5c61owt6ogqx@u.box.com}
\end{center}
by the start of class on the due date. Homework submitted after this time will
\textbf{not be accepted}. Your submission should take the form of a \emph{single
file} with the filename \texttt{yourNetID\_hw2.zip} attached to an email to
the above address. The subject line and body of the email will be ignored, so
please do not include any other comments aside from the attached file. For
example, if your NYU netID is \texttt{abc123}, then you should submit a single
file with filename \texttt{abc123\_hw2.zip}. The \texttt{.zip} archive should
include a PDF of the written portion of your homework, and any files required
for the programming aspect of your homework. Please prepare cleanly handwritten
or typed (preferably with LaTeX) homework, and make sure that your name is on
the homework. Feel free to use original homework LaTeX document to write-up your
homework. If you are required to hand in code, this will explicitly be stated on
that homework assignment.
\horline
\begin{enumerate}
\item Newton's method can be extended to
\emph{matrix-functions} as well. For example, given a square matrix
$\matrixsym{A}$
and real number $t$,
the \emph{matrix-exponential} $e^{t\matrixsym{A}}$ is defined via the Taylor
series for the exponential function:
\begin{equation}
e^{t\matrixsym{A}} = \mtx{I} + t\matrixsym{A} + \frac{(t\matrixsym{A})^2}{2!}
+ \frac{(t\matrixsym{A})^3}{3!} + + \frac{(t\matrixsym{A})^4}{4!} + \ldots
\end{equation}
Obviously, the matrix $\mtx{A}$ must be square.
\begin{enumerate}
\item \textbf{[4pts]}
Derive Newton's method for finding the root of an arbitrary
matrix-valued function $f = f(\matrixsym{X})$, where by \emph{root} we mean
that $\matrixsym{X}$ is a root of $f$ if $f(\matrixsym{X}) = \matrixsym{0}$,
where $\matrixsym{0}$ is the matrix of all zeros. Assume that the matrix
arguments of $f$ are square and invertible.
\item \textbf{[3pts]}
The square root of a matrix $\matrixsym A$ is a matrix $\matrixsym{X}$
such that $\matrixsym{X}^2 = \matrixsym{A}$. For a symmetric
positive-definite matrix $\matrixsym{A}$, derive the Newton iteration for
finding $\matrixsym{X} = \sqrt{\matrixsym{A}}$.
\item \textbf{[3pts]}
Write a program using the Newton iteration that you derived above
to find the square root of the matrix
\begin{equation}
\matrixsym{A} =
\begin{pmatrix}
8 & 4 & 2 & 1\\
4 & 8 & 4 & 2\\
2 & 4 & 8 & 4\\
1 & 2 & 4 & 8\\
\end{pmatrix}.
\end{equation}
The stopping criterion for your Newton iteration should be when the absolute
difference between elements of successive iterations is at most $10^{-10}$.
Submit your code for this part of the problem. Your code must be easily
executable, submit instructions if necessary.
\end{enumerate}
\vspace{\baselineskip}
\item \textbf{[10pts]}
For a yearly interest rate $0