\be
\item For Rosenbrock's banana function of Exercise 3, Section \ref{S:unconstrained_numeric}, how many iterations of the Levenberg-Marquardt Method are necessary before $\ds \left\|  \nabla f\left(\vecx_k\right)\right\| < .01$? How does
this result compare with the methods of Steepest Descent and Newton?

\item\footnote{Data taken from \cite{Reed_Holland} and \emph{The Quantitative Environmental Learning Project}
\texttt{http://www.seattlecentral.org/qelp}.}
A logistic function frequently models the growth of an organism, which is initially exponential in nature but is eventually slowed due to one or more environmental factors. Such growth might be mass, diameter, height, etc. The general form of such a function can be written as
\begin{equation}\label{E:LM_exercises_sunflower_logistic}
P(t)=\frac{MP_0e^{kt}}{M+P_0(e^{kt}-1)}.
\end{equation}
In this formula, $k$ is a positive growth parameter, whereas $P_0$ and $M$ represent the initial and maximum possible values
of $P$, respectively. A classic example illustrating how this nonlinear function models real world data is provided by the sunflower plant. Table \ref{Ta:LM_exercises_sunflower}, for example, displays data showing sunflower height measured weekly over
a period of 12 weeks.
\be
\item Find the logistic function $P$
of the form given in Equation \ref{E:LM_exercises_sunflower_logistic} that
best fits the data provided in Table \ref{Ta:LM_exercises_sunflower} in the sense of least squares.
That is, determine $P_0$, $M$, and $k$ using Table \ref{Ta:LM_exercises_sunflower}
and nonlinear regression.
Note: To determine sensible initial values of $P_0$, $M$, and $k$, you may wish to graph the data first and make some
crude estimates based upon the data.
\item What is the coefficient of determination associated with the best-fit function you just obtained?
\ee
\item \footnote{Data and background information provided by Professor George
McBane, Department of Chemistry, Grand Valley State University.}
In chemical kinetics, if an unstable chemical species spontaneously
decays with constant decay probability per unit time interval, its concentration
$C$ is governed by exponential decay and thus represented as
\begin{equation*}
  C(t) = C_0 e^{-kt},
\end{equation*}
where $C_0$ denotes the initial concentration and $k$ is a constant. The average lifetime of
the species is then $\tau = \frac{1}{k}$. One goal of some chemical kinetics experiments is to determine the value of
$\tau$ (or, equivalently, of $k$.) Frequently the concentration $C$ itself is not experimentally observable, but some
function $Q = aC + b$, where $a$ and $b$ are real but unknown constants, is observable instead. A typical data set
therefore consists of a set of observations of $Q$ at various times $t$: $\{Q_i, t_i\}, i = 1, N$.

\be
\item Show that
\begin{equation}\label{E:LM_exercises_chem_decay}
Q(t) = Ae^{-kt}+B,
\end{equation}
where $A$ and $B$ are constants related to $a$, $b$, and $C_0$.

\newpar
In one experiment, ions in an excited electronic state were suddenly produced by a pulsed laser. These ions then decayed by
emission of orange light. Since the intensity of the light is proportional to the concentration of the excited species, we may choose $Q$
as intensity in (\ref{E:LM_exercises_chem_decay}). A photomultiplier converted this intensity to a signal recorded by an oscilloscope as a voltage.
The result of fifty consecutive signal intensity samples is shown in
Table \ref{Ta:LM_exercises_chem_decay}.

\item Find the function (\ref{E:LM_exercises_chem_decay}) that best
fits the data in the sense of least-squares. That is, determine $A$, $B$, and $k$ using Table \ref{E:LM_exercises_chem_decay}
and nonlinear regression. Once you have determined $A$, $B$, and $k$, compute, $\tau$, the average lifetime of the ion, in microseconds.
\item What is the coefficient of determination associated with the best-fit function you just obtained?
\ee
\ee

\newpar
\begin{table}[h]
\centering
\caption{Time-height samples for sunflower plants}
\label{Ta:LM_exercises_sunflower}
\scalebox{.8}{
\begin{tabular}{|c|c|} \hline $t$ (days) & $H$ (centimeters) \\ \hline
0 &	0.00\\ \hline
7 &	17.93\\ \hline
14 &	36.36\\ \hline
21 &	67.76\\ \hline
28 &	98.10\\ \hline
35 &	131.00\\ \hline
42 &	169.50\\ \hline
49 &	205.50\\ \hline
56 &	228.30\\ \hline
63 &	247.10\\ \hline
70 &	250.50\\ \hline
77 &	253.80\\ \hline
84 &	254.50\\ \hline
\end{tabular}}
\end{table}
\newpage
\begin{table}[h]
\centering
\caption{Time-intensity samples for pulsed-laser experiment}
\label{Ta:LM_exercises_chem_decay}
\scalebox{.8}{
\begin{tabular}{|c|c|} \hline $t$ (microseconds) & $Q$ (millivolts) \\ \hline
0.075 &	1.90757\\ \hline
0.105 &	1.75498\\ \hline
0.135 &	1.40464\\ \hline
0.165 &	1.31492\\ \hline
0.195 &	1.31003\\ \hline
0.225 &	1.20139\\ \hline
0.255 &	1.14646\\ \hline
0.285 &	1.0543\\ \hline
0.315 &	0.967627\\ \hline
0.345 &	0.895605\\ \hline
0.375 &	0.785132\\ \hline
0.405 &	0.815039\\ \hline
0.435 &	0.702124\\ \hline
0.465 &	0.687476\\ \hline
0.495 &	0.605078\\ \hline
0.525 &	0.644141\\ \hline
0.555 &	0.516577\\ \hline
0.585 &	0.420142\\ \hline
0.615 &	0.387793\\ \hline
0.645 &	0.466528\\ \hline
0.675 &	0.443945\\ \hline
0.705 &	0.376196\\ \hline
0.735 &	0.337744\\ \hline
0.765 &	0.290137\\ \hline
0.795 &	0.259009\\ \hline
0.825 &	0.274268\\ \hline
0.855 &	0.188818\\ \hline
0.885 &	0.197363\\ \hline
0.915 &	0.204077\\ \hline
0.945 &	0.186987\\ \hline
0.975 &	0.150977\\ \hline
1.005 &	0.108252\\ \hline
1.035 &	0.0588135\\ \hline
1.065 &	0.130835\\ \hline
1.095 &	0.0472168\\ \hline
1.125 &	0.153418\\ \hline
1.155 &	0.0520996\\ \hline
1.185 &	0.0313477\\ \hline
1.215 &	0.0429443\\ \hline
1.245 &	0.0630859\\ \hline
1.275 &	0.0130371\\ \hline
1.305 &	0.0130371\\ \hline
1.335 &	0.00327148\\ \hline
1.365 &	-0.0125977\\ \hline
1.395 &	-0.0449463\\ \hline
1.425 &	-0.0199219\\ \hline
1.455 &	-0.0296875\\ \hline
1.485 &	-0.0516602\\ \hline
1.515 &	-0.0333496\\ \hline
1.545 &	-0.0528809\\ \hline
1.575 &	-0.0595947\\ \hline
1.605 &	-0.0602051\\ \hline
1.635 &	-0.0608154\\ \hline
 \end{tabular}}
 \end{table}




